IliNE-'S NDUCTIVEIIiLGEBRA •r-^^a ■■ '^•^ ■'■;? Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031218997 Comeri University Library arV19272 The inductive aigebra 3 1924 031 218 997 oiin.anx Inductiu4i $ems. THE INDUCTIVE ALGEBE A EMBEACING A COMPLETE COURSE FOB SCHOOLS AND ACADEMIES. BY WILLIAM J. MILNE, Ph.D., LL.D., Principal of State Normal &:hool, Qeneseo, N. Y.; Author of Milne's Zndiictive Arithmetics, etc. VAN ANTWERP, BRAGG & CO. CINCINNATI. NEW YORK. (S) Eclectic Educational Series. McQuffey's Eemse8^Eeadera, SpeUar and Uhm-is. Milne's Inductive Arithmetics and Algebra. EckcHc School Qeometry. Marvey's Revised Grammars. SolbrooWs Normal Grammar^ JRidpath's Inductive Gramma^ New Eclectic Geographies. Neva Eclectic Penmanship. Eclectic System of Drawing. Forbriger's Drawing Tablets. Eclectic Histories of the United States. Thalheimer's Historical Series. Sidpath's Histories of the United States. Eclectic Complete Book-Keeping. Murdoch's Analytic Eheution. Kidd's New Elocution. Etc., Etc., Etc. Descriptive Csitalogue and Priee List on application. Copyright, 1881, by John T. Jones. ECLECTIC PRESS, Van Antwerp, Bragg & Co., Cincinnati, O. PEEFAOE. In this work on tte Elements of Algebra, the author has followed, the same plan that was adopted in his works on arithmetic. He has endeavored to present the subject in such a manner as to make it simple and attractive by ren- dering the transition from arithmetic to algebra easy and natural, while he has preserved all the valuable features which give discipline and skill in algebraic processes. The method of teaching the subject, as given in this text-book, has been thoroughly tested in the class-room, and the results attained through its use have been more gratifying than could have been expected or hoped. The student is led, step by step, to a thorough and accurate comprehension of the principles of the science, and then they are fixed in the mind by abundant practice upon appropriate examples. The order and treatment of the subjects will be found to be difierent from that given by most authors, yet it is confidently believed that the candid instructor will find the changes introduced to be of great assistance in inter- esting his students, and in inspiring them with a desire to investigate the beauties of this most attractive science. iv PREFACE. The number of problems and examples given is unusu- ally large, and the variety great. The deiiuitions, princi- ples, explanations, and demonstrations are brief, accurate, clear, and comprehensive, while they are free from the ver- bosity which commonly accompanies technical accuracy of statement. A cursory perusal of the work will disclose many new features, which the author feels sure will commend themselves to progressive and intelligent instructors. With the hope that this book may prove a valuable aid alike to student and teacher in the investigation of the Science of Algebra, the author presents his work to the public. W. J. M. State Norhal Schooi,, Gknesbo, N. Y., Januai-y, 1881. OOS"TENTS. Faob AixjEBKAic Processes . . 7 Definitions 8 PkELIMINABY DEriNITIONS 13 Algebraic Expressions . 16 Definitions 17 Addition 19 Equations and Problems 24 Subtraction 26 The Parenthesis ... 32 Transposition in Equa- tions 34 Equations and Problems 36 Multiplication .... 40 Equations and Problems 48 Special Cases .... 50 Division ....... 54 Zero and Negative Ex- ponents 62 Equations and Problems 64 Eeview Exercises ... 66 Factoring ■ 68 Common Divisors ... 80 Common Multiples ... 87 Fractions 90 Keduction of Fractions . 92 Pmom Clearing Equations of Fractions .... 102 Addition of Fractions . 108 Subtraction of Fractions 111 Multiplication of Frac- tions 114 Division of Fractions . 120 Complex Fractional Forms 125 Keview 126 Simple Equations . . 128 General Problems . . 141 Simultaneous Simple Equations 142 Two Unknown Quanti- ties 145 Three or More Unknown Quantities .... 157 Zero and Infinity . . . 163 General Problems . . 165 Involution 167 Involution of Binomial Theorem 172 Evolution 176 Square Eoot of Numbers 183 (V) Vl CONTENTS. Fagb Cube Koot of Numbers . 191 Radical Quantities . . 195 Eeduction of Radicals . 197 Addition and Subtrac- tion of Radicals . . 202 Multiplication of Radi- cals 203 Division of Radicals. . 206 Involution of Radicals . 209 Evolution of Radicals . 210 Rationalization . . . 211 Imaginary Quantities . 215 Review Exercises . . . 217 Radical Equations . . 219 Quadratic Equations . . 223 Pure Quadratics . . . 224 Affected Quadratics . . 227 Equations in the Qua- dratic Form . . . 235 Formation of Quadratics 241 Simultaneous Quadrat- ics 242 Ratio 250 Propobtion 253 Principles of Proportion 255 Equations Solved by Pro- portion 264 Peogbessions 267 Arithmetical Progression 268 Geometrical Progression 276 Loqabithms 285 Miscellaneous Examples 294 Test Questions .... 307 Answers 315 ELEMENTS OF ALGEBRA. ALGEBRAIC PROCESSES. Article 1. Example 1. Two boys had together 121. If the elder had twice as much as the younger, how much had each? ARITHMETICAL SOLUTION. A certain sum = what money the younger had. 2 times that sum = what money the elder had. 3 times that sura = what money both had. Therefore, 3 times that sum = 821. The sum = $7, what the younger had. 2 times $7 = $14, what the elder had. The above solution may be abridged by using the letter « for the expressions, a certain mm and timt sum. In Algebra it is common to use the letter x, or some other one of the last letters of the alphabet, for a number whose value is un- known, but is to be determined. Therefore, the following is the ALGEBRAIC SOLUTION. Let X = money of the younger. Then 2x = money of the elder. And 3x = money of both. Therefore, 3a; = $21. X = $7, the money of the younger. 2x = $14, the money of the elder. (7) ELEMENTS OF ALGEBRA. DEFINITIONS. 2. An Equation is an expression of equality between two numbers or quantities. Thus, 4 + 7 = 11, and 2j; = 16, are equations. 3. A Problem is a question requiring solution. 4. A Solution of a problem is a process of finding the result sought. 5. A Statement of a problem is an equation which ex- presses the conditions of the problem. Solve algebraically the following: 2. A man paid $30 for a coat and a vest. If the coat cost 4 times as much as the vest, what was the cost of each? 3. Two boys earned together $36. If James earned 3 times as much as Henry, how much did each earn? 4. A farmer picked 24 bushels of apples from two trees. If one tree bore twice as many bushels as the other, how many bushels did each bear? 5. A and B together furnish $800 capital, of which A furnishes 3 times as much as B. How much does each furnish? 6. A man had 450 sheep in three fields. In the second he had twice as many as in the first, and in the third 3 times as many as in the second. How many were there in each field? 7. Two boys together solved 350 problems, of which Will- iam solved 4 times as many as Charles. How many did each solve? 8. A certain number added to itself is equal to 260. What is the number? ALGEBRAIC PROCESSES. 9 9. A farmer sold a horse and a cow for 8250, receiving 4 times as much for the horse as for the cow. How much did he receive for each? 10. A has 3 times as many sheep as B, and both have 420. How many has each? 11. A farm of 480 acres was divided between a brother and a sister, the brother having 3 times as many acres as the sister. How many acres had eacli? 12. The greater of two numbers is 5 times the less, and their sum is 540. What are the numbers? 13. A and B had a joint capital of $1750. A furnished 4 times as much as B. How much did each furnish ? 14. A farmer raised 1320 bushels of grain. If he raised 5 times as much corn as wheat, how many bushels of each did he raise? 15. A farmer raised 1350 bushels of wheat, corn, and rye. If he raised twice as much corn as rye, and 3 times as much wheat as corn, how many bushels of each did he raise ? 16. A, B, and C contributed 8560 for the relief of the sick. A gave a certain sum, B gave twice as much as A, and C gave twice as much as B. How much did each give ? 17. The number 169 can be divided into three integral parts such that the second part is 3 times the first, and the third 9 times the first. What are the parts? 18. The profits of a business for 3 years were $10890. The second year the gain was twice the gain of the first year, and the gain the third year was twice as much as that of both previous years. What was the gain the third year? 19. The expenses of a manufactory doubled each year for three years. The third year they were $13800. What were the expenses for each of the other years? 10 ELEMENTS OF ALGEBRA. 20. A lecturer received $300 for 2 lectures. For the second lecture he received 3- times as much as he did for the first. How much did he receive for each? 21. A, B, and C own 10000 head of cattle, B owns 3 times as many as A, and C owns i as many as are owned by A and B. How many does each own? 22. A number plus twice itself, plus 3 times itself, plus 4 times itself, equals 30. "What is the number? 23. John has 5 times as many hens as ducks. He has in all 12 fowls. How many ducks has he? 24. A man has two daughters and one son. He wishes to divide $6000 among them so as to give the elder daugh- ter twice as much as the younger, and the son as much as both the daughters. How much must he give each? 25. Walter has 3 times as many slate-pencils as Albert has lead-pencils. The lead-pencils cost 3 cents apiece, and the slate-pencils 1 cent apiece, and together they cost 30 cents. How many slate-pencils has Walter? 26. Divide 36 into 4 parts so that the second shall be 8 times the first, the third shall be ^ of the first and second, and the fourth shall be ^ of the other three. 27. What number added to 5 times itself equals 90 ? 28. What number added to twice itself, and that sum added to 4 times the number, equals 28 ? 29. What number added to 7 times itself equals 104 ? 30. A and B enter into partnership to do busi^i^. A furnishes 4 times as much of the capital as B, and "-both together furnish $15500. How much does each furnish? 31. A gentleman dying, bequeathed his property of $14400 as follows: To his son 3 times as much as to his daughter, and to his widow twice as much as to both son and daughter. What was the share of each? 32. A farmer bought some grain for seed — in all, 32 ALGEBRAIC PROCESSES. 11 Ijushels. He purchased 3 times as many bushels of oats as of barley, and as many bushels of wheat as of oats and barley. How many bushels of each kind did he purchase? 33. A merchant bought three pieces of cloth which to- gether measured 144 yards. The second was 3 times as long as the first, and the third was 8 times as long as the first. What was the length of each piece ? 34. A farmer had an orchard containing 560 trees. The number of peach trees was 3 times the number of cherry trees, and the number of apple trees 8 times the number of peach trees. How many were there of each? 35. James has 6 times as much money as John. He finds also that he has 30 cents more than John. How much has each? 36. A library contains 10000 volumes. The books of fiction are 9 times as many as the scientific works, the books of travel and biography each one-third as many as the 'books of fiction, and all the other works 4 times as many as the scientific works. How many books of fiction are there in the library? 37. Mary has 40 cents more than Sarah, and Mary's money is 5 times as much as Sarah's. How much money has each? 38. A farmer had 217 cattle in three fields. The first field contained twice as many as the third, and the second twice as many as the first. How many were there in each field? 39. The earnings of a manufactory doubled each year. If, at the end of four years, they amounted to $15000, what were the earnings the first year and the fourth year? 40. Three men engaged in business with a joint capital of $6000. A furnished three times as much as C, and B fur- nished \ as much as A and C. How much did each furnish ? 12 ELEMENTS OF ALQEBRA. INDUCTIVE EXERCISES. 6. 1. How many cubic feet are there in a block of mar- ble containing 1 cubic yard ? 2. What do the expressions 1 cubic yard and £7 cubic feet tell about the block of marble? 3. A cask was found to contain 35 gallons of water. What does the expression S5 gallons tell about the water? 4. When it is said that a room contains 2000 cubic feet of space, what does the expression 8000 cubic feet tell about the space? 5. When it is said that two places are 5 miles apart, what does the expression 6 miles tell about the distance apart? 6. What may the amount or extent of any thing be called ? 7. Name something that can be measured. Express some quantity of that thing. 8. In the expression 5 acres of land, what expresses that which is measured ? What expresses the quantity of land ? What is that called which expresses the quantity or acres of land? 9. When any thing is measured, by what is the quantity expressed ? 10. How will the price of any number of acres of land, at $10 per acre, compare with the number of acres? What expresses the quantity of land ? What is that called which expresses the quantity or acres of land ? 11. How do the expressions 5 acres and any number of acres compare in definiteness ? 12. In the problem, "How many dollars will 3 yards of cloth cost at $4 per yard," how many numbers are referred to? What numbers are given or known? What is the number sought or unknowtif DEFINITIONS. 13 DEFINITIONS AND SIGNS. 7. Quantity is the amount or extent- of any thing. Numbers are used to express quantity. In Algebra, however, the word quantity is frequently used for the word number. 8. Known Numbers, or Quantities, are such as have definite values, or those whose values are given, or to which any value can be assigned. They are represented by figures and the first letters of the alphabet. Thus, 6, 8, 215, representing given numbers, and a, b, c, etc., representing any numbers, are known numbers, or quantities. 9. Unknown Numbers, or Quantities, are those whose values are to be found. They are represented by the last letters of the alphabet. Thus, X, y, ii, V, w, etc., are u.sed to represent unknown numbers, or quantities. 10. Algebra is that branch of mathematics which treats of general numbers, or quantities, and the nature and use of equations. The Signs in Algebra are, for the most part, the same as those used in Arithmetic. 11. The Sign of Addition is an upright cross: +. It is called Phis. Placed between quantities, it shows that they are to be added. Thus, a + 6 is read a plus 6, and means that a and 6 are to be added. 12. The Sign of Subtraction is a short horizontal line: — . It is called Minus. Placed between two quanti- 14 ELEMENTS OF ALGEBRA. ties it shows that the second is to be subtracted from the first. Thus, a — 6 is read u. minus 6, and means that 5 is to be sub- tracted from a. 13. The Sign of Multiplication is an oblique cross: X- It is read muUiplied by or times. Placed between two quan- tities, it shows that they are to be multiplied together. Multiplication may also be indicated by a dot (.), or by writing the literal factors side by side. Thus, oX6, a.b, and ab, each shows that o is to be multiplied by 6. 14. The Sign of Division is a short horizontal line be- tween two dots: -=-. It is read divided by. Placed between two quantities, it shows that the one at the left is to be divided by the one at the right. Division may also be indicated by writing the dividend above the divisor, with a line between them. Thus, a -^5 and — each shows that a is to be divided by b. 16. The Sign of Equality is two short horizontal lines : = . It is read equals, or is equal to. When it is placed between two equal expressions an Equation is formed. Thus, a + 6 = 4 is an equation. 16. The Signs of Aggregation are: The Parenthesis, (); the Vineulum, ; the Bracket, []; and the jBroce, { }. They show that the quantities included by them are to be subjected to the same process. Thus, {a+b)c, a + bxc, [a + 6]c, and {a + b}c, each shows that the sum of a and b is to be multiplied by c. 17. The Sign of Involution is a small figure or letter. DEFINITIONS. 15 called an Exponent, written a little above and at the right of a quantity to indicate how many times the quantity is used as a factor. Thus, a^ shows that a is to be used as a factor 5 times, and is equal to aXaXaXaXa. When no exponent is written, the exponent is 1. Thus, o is regarded as a^, 6 as 6^. 18. A Power of a quantity is the product arising from using the quantity a certain number of times as a factor. Thus, 4 is the second power of 2; a' the third power of a. 19. Powers are tumied from the number of times the quantity is used as a factor. Thus, a^ is called the fifth power of a, or o fifth. The second power of a quantity is also called the square, and the third power the cube of the quantity. 20. A Root of a quantity is one of the equal factors of the quantity. Thus, 2 is a root of 4; u. is a root of a'. 21. Roots are rwimed from the number of equal factors into which the quantity is separated. Thus, one of two equal factors is the secmd root, one of three equal factors the third root, etc. The second root of a quantity is also called the square root, and the third root the cube root of the quantity. 22. The Sign of Evolution is V~, called the Radical iSign. When it is placed before a quantity it shows that a root of the quantity is required. When no quantity or Index is written at the opening of the radical sign, the square root is indicated; if 3, as ^^ the third root; if 4, as ]/ , the fourth root, etc. ThuSj l^ a is read the fourth root of a; ^'h, the seventh root of b. 16 ELEMENTS OF ALGEBRA. 23. The Ambiguous Sign is ±, a combination of the sign of Addition and the sign of Subtraction. ThuB, a d= 6 shows that 6 may be added to or subtracted from o. 24. A Coefficient is a figure or letter placed before a quantity to show how many times the quantity is taken. Thus, in the expression 76, 7 is the coefficient of 6, and it shows that 76 is equal to 6 + 6 + 6 + 6 + 6 + 6 + 6. In the expression 3aa;, 3 may be regarded as the coefficient of ax, or 3a may be regarded as the coefficient of x. 25. Coefficients expressed by numbers are called Nu- meral Coefficients; those expressed by letters, Literal CoSfficients; those expressed by figures and letters, Mixed Coefficients. When no coefficient is expressed, the coefficient is 1. ALGEBRAIC EXPRESSIONS. 26. An Algebraic Expression is the expression of a quantity in algebraic language. EXERCISES. 1. Interpret in ordinary language a* -f" ^l/a^ — x^. Ikteepketation. — The algebraic expression interpreted or read is, the sum of a square and 3 times the square root of the remainder when X square is subtracted from a square. Or, the sum of a square and 3 times the square root of the quantity a square minus x square. Copy and read the following expressions: 2. a + 6. 3. 36— a. 4. a2 + 6. 5. a — -j/ST 6. a;2 + 6— c2. 7. 4(a + 6) — c. ALGEBRAIC EXPRESSIONS. 17 8. x» + \/x^—y. 9. ^ja + VW+7. 10 x + ijx — By) 2 — 1/43; — 8 11. i/r+l + a;2 — 4 12. t/« + a: + y'' l/a — (a; + ?/) 13_ Bx + y^ — V^ When a = 1, 5 = 2, c = 3, d = 4, e = 5, find the nu- merical value of each of the following expressions by using the number for the letter which represents it: Thus, a + 6 + 3d — 6 = 1 + 2 + 12 — 5 = 10. 1. 3a + 6. 2. 2c — 6. 3. M + a — h. 4. 2c2 — a — 6. 5. d + e — 2a. 6. d— (a + 6). 7. a2 + 62 _ d 8. (a-\-h)d—c 9. (a+-6)(d — c). 10. (a2 + 62) -- (a + 6). 11. 4(3a — 5). 12. 7a(3d — 2a). 13. a6cd(a + 6 + c + d). 14. (a + 6 + c)(a + 6 + c). 15. (d + e — 6) — (c — 6). 16. v'5e + a+-^- jg (a2+ 62)35 26 + a 19. a2 + 62 + e2 + d:2— e2. 20. v''d+(a + 6)2— e. 21. (a + 6)(5 — a)4a. 22. a+3V2e+l/4e+3d+26. / oe ^\2, \ a-l-c c / 23 24. a-\-e 3eCd« — qS) — g 2c?+6 DEFINITIONS. 27. The Terms of an algebraic expression are the parts connected by -|- or — . Thus, in the expression 2a + 3a; — 2cd, there are three terms. 28. A Positive Term is one that has the sign -f before it. 2 18 ELEMENTS OF ALGEBRA. When the first term of an expression is positive, the sign + is usually omitted. Thus, in the expression a + 3c — 2d+5e, the first, second, and fourth terms are positive. 29. A Negative Term is one that has the sign — be- fore it. Thus, in the expression 3a — 2d — 3c + 26 — e, the second, third, and fifth terms are neyaiive. 39. Similar Terms are such as are formed of the same letters with the same exponents. Thus, 3x'' and 12i-^ are similar terms, as are also 2{x-{-y)'^ and 4{x-\-y)^. la' «nd bx'^ are similar terms when a and 6 are regarded as coeflicients. 31. Dissimilar Terms are such as contain different let- ters, or the same letters with different exponents. Thus, Zxy and 2yz are dissimilar terms, as are also Znj and 'iry^. 33. A Monomial is an algebraic expression consisting of one term. Thus, xy, 3a5, and 2y are monomials. 33. A Polynomial is an algebraic expression consisting of more than one term. Thus, x-\-y-\-z and 3a + 26 are polynomials. 34. A Binomial is a name applied to a polynomial of two terms. Thus, 2a + 36 and x — y are binomials. 35. A Trinomial is a name applied to a polynomial of three terms. Thus, x-]-y-\-z and 2« + 35 — 2c are trinomials. ADDITION. INDUCTIVE EXERCISES. 36. 1. How many apples are 5 apples, 3 apples, and 7 apples ? 2. How many oranges are 5 oranges, 3 oranges, and 7 oranges? 3. How many things are 5 things, 3 things, and 7 things ? 4. How many a's are 5a, 3a, and 7a? 5. How many 6's are 46, 36, 56, and 26 ? 6. How many xs are Bx, 5x, Qx, 13x, and 10a;? 7. How many a6's are 2a6, 3a6, 4a6, 6a6, and 9a6? 8. How many a^x's are 3a^a;, 7a^a;, 4a^a;, and 2a^a; ? 9. How many a^m^'s are 3a^m^, 2a^m'^, 4a^ni'^, and 9a*m'* ? 10. James has no money, and owes one person 5 cents, another 3 cents, and another 2 cents. What is his finan- cial condition ? 11. If the sign — is placed before each sum which he owes, what sign should be placed before the entire amount? 12. What financial condition is represented by — 5 dol- lars, — 7 dollars, — 9 dollars, — 3 dollars ? 13. What sign will the sum of negative quantities have? 14. How many — a's are — 9a, — 3a, — 7a, — 8a? 15. How many —a^x*'s are —da^x*, —laH*, — Sa^a;*? 16. Asa owes one person 10 cents, another 12 cents, and another 15 cents. If James owes him 5 cents and Henry (19) 20 ELEMENTS OF ALGEBRA. owes him 9 cents, what is Asa's financial condition? What is the value of — 10, — 12, — 15, 5, and 9 ? 17. How much is the debt in excess in the following: — 8 dollars, — 7 doDars, — 9 dollars, 5 dollars, and 12 dollars ? 18. Which is in excess, and how much, in the following: 3a, —5a, — 2a, 7a, —6a, 9a, —2a? 19. How many (a + 6)'s are 2 (a + 6), and 3 (a + 6)? 20. When no sign is prefixed to a number, or quantity, what sign is it assumed to have? DEFINITIONS. 37. Addition is the process of uniting several quantities so as to express their value in the simplest form. 38. The Sum is the result obtained by adding. 39. Peinciples. — 1. Ordy similar quantities can be united in one term, 2. Dissimilar quantities are added by loriiing them one after tJ^e other with their proper signs. CASE I. 40. To add similar monomials. 1. What is the sum of 3a, a, 4a, and 5a? PROCESS. Explanation. — Since the quantities are drmhir— 3a that is, have the same letter and same exponents — a they are written in a column. The sum of 5a, 4a, 4a a, and 3a is determined by adding the coeffxiimts, or gi^ numbers, which tell how many o!s there are. Hence, 1^ the sum is 13a. ADDITION. 21 2. What is the value of 2a -f 4a — 2a + 3a — a — 3a? ElxPLAifATioii. — Since the quantities are similar, they are written in columns. The sum of the positive quantities is 9a, and the sum of the negative quantities, or quantities to be subtracted, is 6a. 0^ 6a=:3a ^" — 6a ^3o. Hence, the value is 3a. Find the sum of each of the following : 2a — 2a 4a — a 3a — 3a 9a — 6a (3.) 46 (4.) 3aa; (5.) 4a;22/ (6.) — 43^2/2 (7.) — 2ca;* h 2aa; Wy -3aV — cx^ Ih ax Bx^y — ^V — 8ex» 96 Aax 2x^y — 822y2 — cx^ 56 9aa; 9x^y — 7z^y^ — eai» 8. Find the sum of ax, Sax, lax, 9aa;, Soa;, and 2aa;. 9. Find the sum of Imn, mn, 2mn, 8mn, 3mn, and 5mn. 10. Find the sum of — 3x^y^, —x^y^, —5x^y'', —Ix^y^, — 9x^y^, and — x^y^. 11. Find the sum of 3x^y^, 4x^y^, Sx^y^, x^y^, Ix'y^, and x^y^- 12. Express 3a + 4a — 2a-\-7a — 3o — 6a + a in the simplest form. 13. Express 9a^a; — 3a*a; + a^x -J- 2a^x — 7a^x — a^x in the simplest form. 14. Express A\/ xy -\- 2 [/ xy — ZVxy -\-V xy -\- 4:\/xy — 2'\/xy in the simplest form. 15. Express 3 (xyY + 4 (xyy—Z{xyy — (xyy —1 (as/)« in the simplest form. 16. Express 2(a; + 2/)*+6(«+2/)*— 7(a;-|-2/)* — 3(a;+y)* —4(x+yy+9{x + yy-9{x-\-yY + 8{x+yy-{x + yy in the simplest form. 22 ELEMENTS OF ALGEBRA. CASE II. 41. To add when some terms are dissimilar. 1. Find the sum of a; + 21/ + 3, x — y, and x+Sy. Explanation.— For convenience in adding, PROCESS. similar terms are written in the same column. Z-\-1y -\-S Since there are three different seta of similar y quantities, their sum, or the simplest expres- X-\-Zv ^^°"' ^® ^'^ ^""^ °^ *® different sets of quan- titles connected by their proper signs, for only- similar quantities can be united in one term. X 3a + 4?/ -|- z 2. Express in its simplest form the following: Zx-\-2xy + 3 — 3a;i/ + 2x — 3z + 4a; — 3x2/ — 2*2/ + Gz — 7a; + 2w. ■nD^r.Ti.0.. Explanation. — The quantities are arranged ,so that similar terms are writ- "■" ™ "I" ten in the same column. Beginning at ■^*' "*y "^^ either hand, each column is added sep- 4x — 3x!/ -|- 6z arately, and tlie dissimilar terms of the ■ 7x — 2x!/ -|- 2m) result connected by their proper signs, 2j; g™ _|_ 43 _i_ 2m; ^°'' *^® dissimilar terms can not be united in one term. KtiLE. — Write similar terms in the same column. Add each column separately by finding the difference of the sums of the positive and negative terms. Connect the results with ihdr proper signs. EXAMPLES. (3.) (4.) (5.) 3a 4- 26 5x + 3xy Sx + 4z — xz — 2a + 36 — e 2x — 7xy 2x — 4z 2a -\-2e — 3x — Qxy Zz — 4x2 36 — 7c 4x2/ — 32 3x + 6a — 4xa 3a — 46 3x +42 7a» ADDITION. 23 Express in their simplest form the following : 6. 3a; + 2!/— 3? — 22/+33 — 6a; + 42/ + 32 + 3a; + 3z— 6?/. 7. 4xy + e—y + Sz—y—3xy + xy—y + z-Jr4x—3y + z. 8. Sac + 4ay -\- 2ac — Say -f- 2ay -f- 2ac — Sac -j- ay. 9. 9&+ 2cd — 3e — 3cd+ 96 + 3cd— 6e — 26 - 4e + 3cd 10. 3a;22/ + 3a^ — Ss + 6aa/ — Gai^t/ + 23 — Srw/ + 62 — 4z. 11. a + 66 + 3c — 4a + 3c+3a — 66-fdH-2c— 3a+7rf. 12. a;2y + 2/ + w — 32/ + 2M) + 2a;2y_(-2— 3a;22/— 32/ + 2u). 13. 9a262 — 3c3?/3 + 2d^ — 4c^y^ + 4a262 — Sd^ 4- 2d2 — 3a262. 14. Add 3a6 + 3i/^+4, 4i/^— 2a6 + 7, 7a6 + 3 + 2 1/^, 2 1/«^ + 4 — 4a6, and 3a6 — 2 v'xy -j- 7. 15. Add 32;3— 4a;2— a;+7, 2a;3 — a;^ + 3a;— 10, 2^2 — 7a;3— 2a;+4, Sx^ — 2a;2 + 12 — 3a;, lla;^ 4- 5a;2 _[_ 6a; — 7. 16. Add |aa;2 -(- |a2 4- 2a;3?/ + 6^ 3aa;2 + ^a;^?/ + Sa^ — 263, 2ax2 + 3a;3!/— a2— f63, and Jjaa;^ +^a;32/H-3a2 — f6». 17. Add ae2+a62 + ia3— a26 + fa6c + ^a2c, a26-|-63 + a62 + 6c2 -I- 2a6c + ^b'^c, and a^c _ ac^ + 6^0 — 60^ + c* + a6c. 18. Add 3(x + y), 4Cx + y), 9(x + y), -10(a; + y), S(x + y), -5(x + y), 7 (x + y), &nd-3(x + y). 19. Add 5(a— 6)2 + 3(a; — 2/)2, 4 (a; — y) 2 — 2 (a — 6)2, 7(a — by—3(x — yy, and 5 (a;-.y) 2 — 3 (a — 6)2. 20. What is the sum of 4x^ -\- ax^ — bx^ -\- 2x^ ? Explanation. — Since tlie dissimilar terms ~r ^ have a common factor, x', 4, «, — 6, and 2 may -|- ax^ be regarded as the coefficients of x^, and their — bx^ sum, which is 6 -f a — 6, will be the coefficient _j_ 2x^ °-f ^' i" '■'^^ sura. Cfi 4- a My3 Therefore, the sum is (6 + a — 5)j;'. 24 ELEMENTS OF ALGEBRA 21. What is the sum of 2aa; — 36a; + 4ca; -f Scfo ? 22. What is the sum of 2ax'^—4:hx'^ + Zex^ + 4a:2 ? 23. Add 2 (a + 6), 3a (a + 6), 4 (a + 6), and 2a (a + 6). 24. Add 5 (a + 3), 2 (a + 3), 3a (a + 3), and 26 (a + 3). 25. Add 3a-\/x-{-y, 2 i/a; + 3/, 2a\/x-\-y, and 3i/a;-|-yT EQUATIONS AND PROBLEMS. 42. Simplify the following and find the value of x: 1. 3a; + 4a;+2x — 3a; — 2a; + 4a;=16. SOLUTION. 3a; + 42 + 2a; — 3a; — 2ai + 4a; = 16 Uniting terms, 8a; = 16 Whence, a; = 2 2. 5a;+2a; — 3a; + 4a;— 6a; + 7a; = 18. 3. 5a; + 6a; — 9a; — 3a; + 2a; + 4a; = 20. 4. 3a; — 2a; + 5a; + 7a; + 4a; — 3a; = 26 -(- 2. 5. 3a; — 4a; + 2ie + 6.t; — 4a;+ a;=15 + 3 — 2. 6. a; 4- 4a; + 6a; — 3a; + 7a; — 9a; = 21 + 7 — 4. 7. 9a; — 2a; — 3a; + 7a; — 5a; + 4a; = 35 + 9 — 4. 8. 8a; — 4a;-f 7a; + 3a; — 6a; — 4a; = 37 — 3 + 2. 9. 11a;— 3a;+7a; — 4a; + 6a; — 3a; = 23 + 7— 2. 10. 10a; — 4x + 2a; + 7a; — 6a; + 2a; = 35 + 6 + 3. Solve the following problems: 11. James solved twice as many problems as Henry, and Henry solved 3 times as many as Harvey. If they all solved 70 problems, how many did each solve? ADDITION. 25 12. A had twice as much money as B, and B had twice as much as C. K they all had $140, how much had each? 13. William had twice as many marbles as Henry, and Henry had 3 times as many as Samuel. How many had each, if they all had 50 marbles? 14. A merchant owes B a certain sum of money, and C twice as much. Various persons owe him in all 10 times as much as he owes B. After paying all his debts he will have $1400 left. How much does he owe B and C? 15. After taking 5 times a number from 13 times a number and adding to the remainder 8 times the number, the result was 5 more than 155. What was the number? 16. A circulating library contained 10 times as many books of reference and 3 times as many historical books as works of fiction. The works of reference exceeded the works of fiction and history by 12000 volumes. How many volumes were there of each? 17. A merchant failed in business, owing A 10 times as much as B, C three times as much as B, and D twice the difierence of his indebtedness to B and C. The entire debt to these persons was $36000. How much did he owe each? 18. At a local election there were three candidates for an ofiice who polled the following vote respectively: A received twice as many as B, and B IJ times as many as C. The vote for all lacked 3 votes of being 1125. How large a vote did each receive? 19. A man earned daily for 5 days 3 times as much as he paid for his board, after which he was obliged to be idle 4 days. Upon counting his money after paying for his board he found that he had 2 ten-dollar bills and 4 dollars. How much did he pay for his board, and what were his wages ? SUBTRACTIOK INDUCTIVE EXERCISES. 43. 1. What is the difference between 5 dollars and 3 dollars ? 2. What is the difference between 7 miles and 9 miles? 3. What is the difference between 9m and 3m? 4. What is the remainder when 8a is taken from 12a? 5. What is the remainder when 3a^a^ is taken from Sa^a^? 6. What is the remainder when 8x^y^2* is taken from 15x31/22*? What is the sum of —Sx^y^s* and 15x^y^z*l 7. What is the remainder when 9y*x^ is taken from 18y*x^7 What is the sum of —9y*x^ and 18y*x^? 8. What is left when Sa'^b is taken from 12a26? What is the sum of 12a^b and — Sa^i? 9. What is left when 5pq^ is taken from ISpq'^ ? What is the sum of 13p5^ and — 5pq^ ? 10. What is left when Z(x + y) is taken from 8(x + y)? What is the sum of 8(x-\- y) and — 3(x-\-y)^ 11. How much less than zero is — 2? — 4? — 9? 12. When 8 is subtracted from 0, how is the result expressed? If 2 should be subtracted from that result, what would be the result? How many are — 8 less 2? •— 8a less 2a ? What is the sum of — 8a and — 2a ? 18. What is the result when Ix^y is taken from — 3x^y^ What is the sum of — Sx'^y and — 7x^1/? (26) SUBTRACTION. 27 14. Instead of subtracting a positive quantity, what may be done to secure the same result? 15. What is the result when 7 — 3 is subtracted from 13? 7 from 13? 8a — 5a from 11a? 8a from 11a? 9x^—2x^ from 12a;2? 9x^ from 12a;2? 16. How does the result, when 7 — 8 is subtracted from 13, compare with the result when 7 is subtracted from 13? How does the result, when 8a — 5a is subtracted from 11a, compare with the result, when 8a is subtracted from 11a? 17. What is the result when 7 — 3 is subtracted from 8? What is the sum of 8— 7 + 3? 18. What is the result when Qx^ — 3a;^ is subtracted from 8x2 ? "vVhat is the sum of 8a;2 — 6a;2 + 3a;2 ? 19. What is the remainder when &xy — ixy is subtracted from %xy ? What is the sum of %xy — Qxy -{- 4xy ? 20. Instead of subtracting a negative quantity, what may be done to secure the same result? DEFINITIONS. 44. Subtraction is the process of finding the diflference between two quantities; or. The process of finding a quantity which, added to one given quantity, will produce another. 45. The Minuend is the quantity from which another is to be subtracted. 46. The Subtrahend is the quantity to be subtracted. 47. The DifiFerenee, or Bemainder, is the result ob- tained by subtracting. 28 ELEMENTS OF ALGEBRA. 48. Principles. — 1. The difference between similar quanti- ties only can be expressed in one term. 2. Subtracting a positive quantity is tlie same as adding a numerically equal negative quantity. 3. Subtracting a negative quantiiy is the same as adding a numerically equal positive quantity. CASE I. 49. To subtract when the terms are positive. 1. From 9a subtract 3a. Explanation. — When 3 times any number is PROCESS. ■' subtracted from 9 times tliat number, tlie remainder Q„ is 6 times tlie number; therefore, when 3a is sub- ""^ tracted from 9a, tlie remainder is 6a. Or, since sub- e^ trading a positive number or quantity is the same as adding an equal negative quantity {Prin. 2), 3a may be subtracted from 9a by changing the sign of 3a and adding the quantities. Therefore, to subtract 3a from 9a, we find the sum of 9a and — 3a, wliicli is 6a. 2. rrom 13a take 15a. PROCESS. Explanation.— After subtracting from 13a as 13(j much as we can of 15«, there will be 2a yet to be jgjj subtracted, or tlie result will be —2a. Or, since - subtracting a positive quantity is the same as add- — 2a ing an equal negative quantity (Prin. 2), 15o may be subtracted from 13a by finding the sum of 13a and —15a, which is —2a. Therefore, when 15a is taken from 13o, the result is — 2a. (3.) (4.) (5.) (6.) (7.) (8.) From 15a ISm/ lox^y^ 19xyz Sx'^y^z lOa^JSc Take 6a 8xy llx^y^ 22xyz 15x^y^e 13a26«e SUBTRACTION. 29 Cop)' and subtract the following: 9. 8a; + 2y from 12a; + &y. 10. 9a +36 from 10a +26. 11. Ixy + 22 from bxy + 4?. 12. 3*22/2 + 63 from bx^y^ + 82. 13. &xy^z + 3a^ from Qxy^z + 2a;y. 14. 4p2js + 3pgi2g from 5p*gs + Qpq'^s. 15. dm^nx + 3mna; from Vm^na; + 2mna;. 16. 5x'^y + 2y^ from 9a;2y + 7?/«. 17. Sa;^/^ + ^2 from a;^^ +2. 18. 5p^q^ + Spj from p^q^ + 'ipg. 19. a;22/2s2 + 4y^ from ISas^j/^a* + 2y^. 20. 82/2* + 2/*2 from 3yz* + Sy^a. 21. 3p^q^ + 4gs from 9p2gr2 + 2qs. 22. IQxyz^ + 4a^2 from a^* + a:^2. 23. Sa^a^ + 5aa;2 from 4a^xy + Toa;*. 24. Qr^s^z + 8r822 from lOr^s^g _(. 4^32. CASE' II. 50. To subtract when some terms are negative. 1. From 6a subtract — 3a. PKOCESS. Explanation. — If were subtracted from 6a, the 6a remainder would be 6a; therefore, when — 3a, which D is 3a less than 0, is subtracted from 6a, the differ- + ence, or remainder, is 9a. Or, since subtracting a 9a negative quantity is the same as adding an equal positive quantity (Prin. 3), we may subtract by changing the sign of — 3a and adding the quantities, obtaining for a result 9a. 30 ELEMENTS OF ALGEBRA. 2. From 6a — 26 subtract 3a — 46. PROCESS. Explanation. — The subtrahend is written under g^ 26 t'le minuend, so that similar terms stand in the q ^7 same column. Since the subtrahend is compo.«ed of - + two terms, each term must be subtracted separately. 3a + 26 Subtracting 3a from 6a.— 26 leaves 3tt — 26, or the result may be obtained by adding — 3a to 6a — 26. But since the subtrahend was 46 less than So, to obtain the true remainder, 46 must be added to 3a — 26, which gives 3a + 26. There- fore, the subtraction may be performed by changing the sign of each term of the subtrahend and adding the quantities. Rule. — Write simUar terms in the same column. Change the sign of each term of the subtrahend from + to — , or from — to +, or conceive it to be changed, and proceed as in Addition. Peoof. — Add together the remainder and the subtrahend. If the result is equal to the minuend, the work is correct. (3.) (4.) (5.) (6.) (7.) Prom 4a3a; 3x^y^ 2a; + y Qy — 2z lax — Aby Take —2a^x — 5a;V — 2x— 2i/ 3y + 4g 3aa:— 9% (8-) (9.) (10.) Prom 3a + 26 — 3c Ax-\-^y — 'iz Axy + Bz + x"^ Take 2a — 46 + 5o 2x—4y — 5z 2xy — ^z-\-Ax^—y 11. Prom a + 6 + c subtract a + 26 — c. 12. From ^x-\-2y—^z subtract 2x — Sy-{-'4s. 13. Prom 6a2+262 + 3c2 subtract 3a^ — 3b'^ — 2c^. 14. Prom 3a»— 2c3— 4(£» subtract 4c» — Ba^+2d^. 15. From 8x* — By^+2z^ subtract 4y^ — Bx* + 2z^. 16. From 9p2+4g2 + rS subtract 3r^—4p^—2q^. 17. From ax-\-2ay-\-z subtract 2aa; — 2ay-\-s. SUBTRACTION. 31 18. From 2xy -^ 5yz -\- Bxz subtract 2x>f — 3yz — 4xz. 19. From 8x^y^ -{- 16xy^ -\- lOxy subtract lise^y^ — 8xy^ — 4xy. 20. From 5x^y^-^10x*y — 6yz^ subtract 10x*y — 4x^y^ + 5yz^. 21. From Sx^ + 2xy + s'^ + w subtract 2x''—Sxy — 4z^. 22. From 15x^+10y^+8z^^r^ subtract 5y^+4z^+6r^. 23. From 43iy^ +3x^y + 4x—3 subtract 4xy'' — 3x — 7. 24. . From 46a; * -f 3ay^ -\-4- — by subtract by — 5 — bx^. 25. ¥i6m:.3x^y* + 3xy — 5x subtract 2x''y* — 2xy-\-4x — 5. .26.. From 4x^y^ — 3xy^ — 7z* subtract 2x^y^ + Qxy^ + 24* + 9." ' 27. From Tar^ — 46s» + 3rs subtract 3ar^ +p + 2bs» + 7. 28. From 15x^ — 24x«y^ — 16y^ subtract 15x^y»+4z — by* + z«. 29. From 3af" — 4a;"2/'" + 42/" subtract 4a;"' + 2a;"?/'" — 4x'^'". 30. From 3a;2''— 2a;*"2/'" — 2/""-* subtract 32/'"-i + 2a;3«2/"' 31. From 3\/'^-\-2z—f'f subtract 2Vl^—3z-^2fy^. 32. From 4(a-\-hy—3a+4e subtract a—2(a-\-by—2c. 33i From- 5 V^a +**■ — 3 if a; + 2/ subtract G^'ai + j/ — 7l/a; + ^. 34. From 5 -[/ a + b^ — 3 fc + ~d subtract 4i/a + 62 + 2if^Trf^ 35. From aa; -|- % subtract ex — dy. ExPLAiTATiON. — Since the terms, though dissimilar, have a common factor, u, and PKOCESS. ' ; f may be regarded as the coefficients of ' " X, and 6 and d as the coefficients of i/, ca; dy ^^^ tjjg difference indicated by placing (a — G)x + (6 + d)y the difference between the coefficients in parentheses. 32 ELEMENTS OF ALGEBRA. 36. From ay + 2a! subtract cy — xyhj +8. 14. How does the product of 4 X 5 compare with the product of 5 X 4? 6x7 with 7X6? +4aX+5 with -f 5X +4a? — 4X +3 with +3X —4? What, then, is the product of — 4X +3a? Of -f 3aX —4? Of — 5:eX +7? Of +7X —5a;? 15. "When a positive quantity is multiplied by a negative quantity, what is the sign of the product? 16. What is the product of —3 X 6? 17. Since —3X 6 is —18, if —3 is multiplied by 6 — 2, how many times — 3 must be subtracted from — 18 to obtain the true result? 18. If the subtraction is indicated, what are the signs of the remainder when — 6 is subtracted from — 18? 19. What is the product of — 5 X 4? 20. Since —5X4 is —20, if —5 is multiplied by 4 — 3, how many times — 5 must be subtracted from — 20? If the subtraction is indicated, what are the signs of the remainder when — 15 is subtracted from — 20 ? 21. Since, in the results just obtained, — 3 multiplied by — 2 gives + 6 and — 5 X — 3 gives + 15, what may be inferred as to the sign of the product when a negative quantity is multiplied by a negative quantity? 22. What is an exponent? "What does it show? In the expression 5*, what does the 3 show? In the expression o^, what does the 5 show? In a*, what does the 6 show? 42 ELEMENTS OF ALGEBRA. 23. When a^ is multiplied by a^, how many times is a used as a factor? How many times is a used as a factor when a 2 is multiplied by a' ? 24. How, then, may the number of times a quantity is used as a factor in multiplication, be determined from the exponents of the quantity in the expressions which are multiplied? 25. How is the exponent of a quantity in the product determined? DEFINITIOKS. 63. Multiplication is the process of repeating one quan- tity as many times as there are units in another. 64. The MultipUoand is the quantity to be repeated or multiplied. 65. The Multiplier is the quantity showing how many times the multiplicand is to be repeated. 66. The Product is the result obtained by multiplying. 67. The multiplicand and multiplier are called the/ociors of the product. 68. The Signs of Multiplication. (See Art. 13.) 69. Peinciples.— 1. Either factor may he used as multi- plier or multiplicand when hoih are abstract. 2. The sign of any term of the product is + when its factors have LIKE signs, and — when they have unlike signs. 3. The Goeffident of a quantity in the product is equal to the product of the coefficients of its factors. 4. The exponent of a quantity in the product, is equal to the Slim of its exponents in the factors. MULTIPLICATION. 43 70. The principle relating to the signs of the terms of the product is illustrated as follows : -|- a multiplied by -|- 6 = -\- ah — a multiplied by + ^ = — ^ -{- a multiplied by — 6 = — ab — a multiplied by — 6 = -\- ah CASE I. 71. To multiply when the multiplier is a monomial. 1. What is the product of Sa^x multiplied by 2a*a;2y? PROCESS. ExPLAifATioN. — Since the multiplier is com- Ba^X posed of the factors 2, a', x^, and y, the multi- 2„3„2y plicand may be multiplied by each successively. „ , , ■ 2 times 3a^x^6a'x: a' times 6a^x = 6a^x (Prin. 4); x^ times 6a*i = 6a*a:' (Prin. 4); y times 6a'a;' ^Ga^x^y, since literal quantities when multiplied may be written one after another without the sign of multiplication. The coefficient of the product is obtained by multiplying 3 by 2 (Prin. 3). The literal quantities are multiplied by adding their exponents (Prin. 4). Hence, the product is Ba'i'y. 2. What is the product of 2a — 6 2 multiplied by —36? PROCESS. Explanation. — Since 2a multiplied by 2a — 6 2 — 3^ is the same as 2a times — 36 (Prin. 1), 3 J the product of 2a multiplied by — 36 is „ , , oTi" — ^"^^ But, since the entire multiplicand is 2a — b', the product of 6^ multiplied by — 36 must be mhiraeled from — 6a6. 6^ multiplied by — 36 gives as a product — 36', which subtracted from — 6a6 gives the entire product 7-6a6 + 36«; or. Since 2o and — 36 have unlike signs, the sign of their product is — (Prin. 2) ; and, since — 5" and —36 have like signs, the sign of their product is -j- (Prin. 2). Hence, the product is — 6'a6+36'. 44 ELEMENTS OF ALGEBRA. EuLB. — Multiply each term of the multiplicand by the multi- plier, as follows : To the product of the numerical coefficients, annex each literal factor uMi an exponent equal to the sum of the exponents of that letter in both factors. Write the sign -{- before each term of the produ,ct when its factors have like signs, and — when they have unlike signs. EXAMPLES. Multiply By (3.) (4.) — 8 4 3 —3 (5.) (6.) (7.) (8.) 7a —3x 4x 3x^ 3 4—5 2a;* Multiply By (9.) (10.) 3x^ , 4x»y 2x* 2x^y (11.) (12.) 3x^y^ — 4xmy 2x^y* 3x^my (13.) — l^xyz-'' 2 4a;22/*a Multiply By (14.) — 2x^y'^z'^ — 8xyz (15.) Baix^ — 4a262a. (16.) — 5a862^ — 3a»bHy Multiply By (17.) — 3cHy 4eH (20.) 4a^x^y^ — 32/2z2 - (24.) — B{y + z)^ (18.) 5a^x^y^ — 3aHH (21.) (22.) 5aa;22/ {x^y) '3hxH 2 (25.) {a-bY 4{a—bY (19.) — Gx^y'^z^ Multiply By (23.) 4(a + b) — 3 Multiply By (26.) 2{c + d)^ 3(c + (^3 MVLTIPLIOA TION. 46 (27.) (28.) (29.) (30.) Multiply 2 (» + y + a) * 3x» 4a" — 5aa; By _5(a;-|-2, + a)3 4ii» — 5a2" Sfti'a;'" (31.) (32.) (33.) Multiply 2aa^ SaV — ix^y" By 4aa;"' — 5a''"a;8» — bxfy" 34. Multiply x^ — 2y by 3^/. 35. Multiply x^y — 23 by 23. 36. Multiply 4x^ —2mi by 3aa/. 37. Multiply —dx^—2y^ by 2a;22/. 38. Multiply 4x^y^ + 23^ by — 4xH^. 39. Multiply Sx^y'' — 2^3 by 3otz. 40. Multiply 4x^ + 2y + Zs by a;^/. 41. Multiply 3x^y + y — 8xz by 2x3. 42. Multiply &x^y^ -\-4y^ —^z"^ by 3a;j/2, 43. Multiply 4a6 — 3ac — Zed by Socd. 44. Multiply bcui — 6ax -\- 4a6 by — bacx. 45. Multiply 5a6e — 3acd — 36cd by — 4ahcd. 46. Multiply Za^xy — 2a^ho-{- ^axy by — 2aa;2. CASE II. Ti. To multiply when the multiplier is a polynomial. 1. Multiply X — 2y by 2x-\- y. PROCESS. X —2y 2x +y 2x times (x — 2y) ^=2x^ — 4aiy y times (cc — 2y) = xy — 2y^ (2x + y) times {x — 2y) = 2x^ — 3xy — 2y^ 46 ELEMENTS OF ALGEBRA. Explanation.— Since the multiplier is 2i + y, the multiplicand is to be multiplied by 2j; and by y. 2x times (x — 2y) = 2x^—4jy; y times {x — 2y) = xy — 2y'. Therefore, the sum of these two partial products is the entire product. Hence, the product is 2x' — 3xy — 2y'. Rule. — Multiply each term of the multiplicand by each term of the multiplier, and add the partial products. (2.) (3.) Multiply o6 + 2c Bx^ — axy By 2ab — 3e 1x^ + Bojcy 2ffi262 ^ 4a6c 6a;* — lax^y — 3a&c — 6eg ' + 9aa:«y — ZaH'^y'^ Product, 2a^62 -f ahc — 60^ 6x'* -|- lax^y — 'Sa'^x^y'^ 4. Multiply X -\- y by x — y. 5. Multiply 3a + c by a + 3c. 6. Multiply 4a— 26 by 3a— 36. 7. Multiply 2y + 3z by 3;/ — 4z. 8. Multiply 2a; + 2/ by 2a; + 2y. 9. Multiply 3x — 4y by 3a; — 4y. 10. Multiply 5a + 2c by 3a — 7c. 11. Multiply ax -\- by by aa; -f- by. 12. Multiply 2ac + 36c by 2ac — 36c. 13. Multiply 36d — 46c by 2bd + 36c. 14. Multiply 3x2^/2 _ 4^2 i,y ^x^y"^ -\- 3z^ 15. Multiply 3x^22 + 2y by 2xy^ -\- z. 16. Multiply 4a62 -f 36c2 by 2a6 + 26e2. 17. Multiply 5x^y — 3ax by 5xy^^2ax. 18. Multiply a2 + 2a6 + 62 by a + 6. 19. Multiply a;2 + 4a; + 4 by a; + 2. 20. Multiply a2 + ay — 2/2 by a — y. 21. Multiply 2a2 -(- a6 — 262 by 3a — 36 22. Multiply aS -|- a* + a2 ^y a2 — 1. MUL TIPLICA TION. 47 23. Multiply a;* + x^y'^ + y* by x"^ — y"^. 24. Multiply 2x — By + is by 3x + 2y — 53. 25. Multiply 2a^ + 5a6 — Sc^ by Sa^ — 2ab + 5e^. 26. Multiply Sx^ — 4xy + 5y^ by 7a;2 — 2aa/ — Si/^. 27. Multiply 1 — 3a; + Sa;^ by 1 — 2a; + 2a;2. 28. Multiply a^ + aa; + a;^ by a^—ax + x^. 29. Multiply a; + 21/ — « by a; + 2/ — 22. 30. Multiply a" -|- fi" by oT — fe". 31. Multiply a;" -|- y" by a;" + j/". 32. Multiply a;" + 2/" by a;" + 1/". 33. Multiply a;'"^-" + 2/™ + " by a;'"^-" + 2/"+". 34. Multiply a*"-" + 6""-" by a™-" — 6™-". The multiplication of polynomials is sometimes indicated by placing them in parentheses. When the multiplication is per- formed, they are said to be expanded. 35. Expand (x -j-y) (x-\- y). 36. Expand {2x — y)(2x — y). 37. Expand (3a; — 4y) (3a; + 4!/). 38. Expand {4x-{- &y) {ix — Qy). 39. Expand (3aa; + 2y) (3aa; + 2z). 40. Expand (2a; — Axy) (2x — 2z). 41. Expand (Sa^ _ 26c) (3a + 26c). 42. Expand (a^ + 6) (a + 62). 43. Expand (a + 6 + c) (a — 6 — c). 44. Expand (a + 6) (a + 6) (a + 6). 45. Expand (a — 6) (a + 6) (a — 6) (a + 6). 46. Expand (a;^ + 2a; + 1) (a;^ — 2a; + 1). 47. Expand (a^ — 2a6 + 6^) (a^ + 2a6 + 6^). 48. Expand (1 + a) (1 — a) (1 + a) (1 + a). 49. Expand (a;^ — 2/^*) (x^ — 2/^) (a''^ — 2/*) (a;^ — y"). 50. Expand (a^ -|- 6^) («« — 6^) (a* — 6*) (a* — &»). 51. Expand (a^h^—b''){a^b^-\-h'^){a*h*^—h^){a»h»—h^) 48 ELEMENTS OF ALGEBRA. EQUATIONS AND PROBLEMS. 73. 1. 5 (a: — 3) = 2 (a; + 3) + 3. Find the value of x. PROCESS. 5 (« — 3) = 2 (a; + 3) + 3 Multiplying, 5a;— 15 =2a; + 6 + 3 Transposing, 5a; — 2a; = 15 + 6-1-3 Uniting, 3a; == 24 x= 8 Explanation. — Since the multiplication is indicated in one term on each side of the equation, in finding the value of x, the multi- plication must be performed. The known quantities are then transposed to the second member and the unknown quantities to the first member, similar terms united, and the value of x found. Verification. — Since the value of x is 8, the substitution of 8 for X in the original equation will give an equation in which the members are identical, if the result is correct. Substituting, the original equation becomes 5(8 — 3)=2(8-|-3)-|-3, which, simplified, becomes 25 = 25. Hence, 8 is the value of x. Find the value of x and verify the result in the following : 2. 3(2a; — 5)=21. 3. 4 -I- 3 (3a; — 7) = 19. 4. 3 (4a; + 7) -F 5 = 50. 5. 5a;-|-3(2— a;)=40. 6. 6a; -t- 3 (4a; -I- 3) = 41. 7. 5 (a; -I- 6) z=: 2 (a; + 3) H- 30. 8. 3(2a;— 4)=4(a;— 5)-^32. 9. 3(a;-l-2) = 4(a; — 2)-f 15. 10. 3a; — 2 (a; -1-1) = 13 — 7. 11. 5x— 3 (a; — 4) = 4a; 4- 7, multiplication: 49 12. 4(a; — 5) — 3(a; + 6)=0, 13. (2 + a;) (a; + 3) =a!2 + 2a! + 18. 14. 5 (2»— 2) =27 + 3(2a! + 1). 15. 10 (a; — 5) = (a; + 1) + 5 (a; + 1). 16. 5(a; + 3)— 2(2a; — 7) = 3(a; — 7). 17. 3 + 7(a! — 2) — 4(2a! — 7) = 164-(a; — 2). 18. 6a;=15 + 3(a; — 3)— 3(a; — 10). 19. 19 = 2(4 — a;) + 5 (7 + 2a;)— 48. 20. 2a; + 3(6a; — 5) — 5 = a;— 1. 21. 3(a; — 7) = 14 + 2(a;— 10) + 2. Solve the following problems, and verify the result: 22. There are two numbers whose sum is 40. One is twice the other increased by 5. AVhat are they? PROCESS. Let X represent the first number. Then, 2(a; + 5) will represent the second number. And, a; + 2(a; + 5) = 40 a; + 2a; +10 = 40 3a; = 40 — 10 3a; = 80 a; = 10, the first number 2 (10 -|- 5) ^ 30, the second number 23. What number is that to which, if 3 times the sum of the number and 2 is added, the result will be 22? 24. If B were 5 years younger, A's age would be twice B's. The sum of their ages is 20. How old is each? 25. Two boys find that they have together 21 cents. They discover that if Henry had 5 cents less, John's money would be just 3 times Henry's. How much has each? 26. Two pedestrians travel toward each other at the rate 50 ELEMENTS OF ALGEBRA. of 5 miles per hour until they meet. When they meet they discover that one has traveled 3 hours longer than the other, and that the entire distance traveled by both is 55 miles. How far does each travel? 27. Three men, A, B, and C, each had a sum of money. A had twice as much as B, and B twice as much as C. A and B each lost 10 dollars and C gained 5 dollars, when the difference between what A and B had was equal to what C then had. How much had each? 2&. A farmer plowed two fields containing together 50 acres. If the smaller field had contained 10 acres more, it would have been half the size of the larger. How many acres were there in each field? 29. A commenced business with twice as much capital as B. During the first year A gained $500 and B lost $300, when A had 3 times as much money as B. What was the original capital of each? 30. A man wishing to buy a quantity of butter found two firkuis, one of which lacked 6 pounds of containing enough, and the other weighed 14 pounds more than he wanted. If three times the quantity in the first firkin was equal to twice the quantity in the second, how many pounds did he wish to purchase? How many pounds were there in each firkin? SPECIAL CASES IN MULTIPLICATION. 74. By performing the operations indicated in the follow- ing examples, it is found that, (a + 6) (a + 6) = a2 + 2a6 + 62 (« + y) ( y>+ y)=x^ + 2xy + 2/2 (a" + g2) (^2 ^ j,2) =^4 _,. 2a;222 ^ ^4 MULTIPLICATION. 51 1. When a quantity is multiplied by itself, what power is obtained? 2. In the square of (a + 6) or (x-\-y), how is the first term of the power obtained from the first term of the quantity to be squared? 3. How is the second term of the power obtained? 4. How is the third term of the power obtained? 5. What signs connect the terms of the power? 75. Peinciple. — The square of the sum of two quantities is eqiud to ffie square of the first quantify, ■plvs twice the product of the first and second, plus the square of the second. IIXAMFI.ES. Write out the products or powers of the following : 1. (c + eOCc + rf)- 2. (m + «) (wi + n). 3. (r + s)(r + s). 4. Oc + 2)(x + 2). 5. (a +3) (a + 3). 6. (Sa + x)(Sa + x). 7. Square 2x-{-4y. 8. Square 3a + 26. 9. Square x^ + y^. 10. Square 4x -\- 3y. 11. Square Bp -\- 2q. 12. Square 2a;2 -(- 5j,2. 76. By performing the operations indicated in the follow- ing examples, it is found that, (a — 6)(a — 6)=a2 — 2a6+62 Ct>! — y)(x — y)=x^—2xy + y^ (a;2 — s2) (a;2 — g2) == a;* — 2*222 + g* 1. How are the terms of the power obtained from the terms of the quantity squared? 2. What signs connect the terms of the power? 3. How does the square of (a — b) difier from the square of (a + 6)? 52 ELEMENTS OF ALGEBRA. 77. Peinciple. — Ths square of the difference of two quan- tities is equal to the square of the frit quantity, minus twice the product of the first and second, plus the square of the second. EXAMPtES. Write out the products or powers of the following : 13. (a — c) (a — c). 14. (y — z)(y—3). 15. (r — s) (r — s). 16. (6 — c)(6 — c). 17. (« — l)(a;— 1). 18. (x — 2y)ix — 2y). 19. Square o — d. 20. Square 2r — 3s. 21. Square 2s — q. 22. Square 3m — 4n. 23. Square 2v — w. 24. Square 2x^—2y^. 78. By performing the operations indicated in the follow- ing examples, it is found that, ((i + 6)(a — 6)=a2— 62 (x — y)(x + y)=x^—y^ (X' + 32) (a;2 —82) :^ a;4 _g4 1. How are the terms of the product obtained from the quantities ? 2. What sign connects the terms? 79. Peinciple. — The product of tJw sum and the difference of two quantities is equal to the difference of their squares. EXAMPLES. 25. (c + d!)(c — (f). 26. (r + s)(r — s). 27. (m -j- n) (m — ri). 28. (c + a)(c — a). 29. (x—l)(x+l). 30. (2 — a;) (2 + a;). 31. (2a; + 4)(2x — 4). 32. (2x^+y){2x^—y). 33. (a;2+t/2)(2;2— 2,2). 34. (x*-y^)(x*+y^). 35. {3v + 2w)(3v — 2w). 36. (5a;y — 3)(5a^ + 3). MULTIPLICATION. 53 80. By performing the operations indicated in the follow- ing examples, it is found that, fa; 4- 2) (a; + 3) =a;2 + 5a; + 6 (a; + 2) (a; — 3) =a;2 — a; — 6 (a; — 2) (a; — 3) =a;2 — 5a; + 6 1. How many terms are alike in each factor? 2. How is the first term of each product obtained from the factors? 3. How is the second term of the product in the first example obtained from the factors? In the second exam- ple? In the third example? 4. How is the third term of the product in each exam- ple obtained from the factors? 5. How are the signs determined which connect the terms? 81. Principle. — The product of two Mnomicd quantUiea having a common term is equal to the square of the common term, the (dgebraic sum of ihe other two mvMiplied by the comr man term, and the algebraie product of the unWce terms. Write out the products of the following : (3a; — 7) (3a; -f 5). (23,-3) (2y- 4). (4a + 6) (4a + e). (5a + 26) (5a — 2c). (3aa; + 4)(3aa; — 7). (2a2a; + 2) (2aH — 6). (2a;V+4)(2a;y-f 7). 37. (a;-f4)(a; + 3). 44. 38. (^-5) (a; + 3). 45. 39. (a;+3)(a;-4). 46. 40. (^_4)(a;-6). 47. 41. (a + 3) (a + 6). 48. 42. (a-\-m) (a + w). 49. 43. (2a; + 4)(2a;— 5). 50. DIVISION. INDITCTIVE EXERCISES. 82. 1. How many times are 2 bushels contained in 8 bushels? 26 in 86? 2a in 8a? 3 bushels in 12 bushels? 36 in 126? 2. How many times is 5d contained in IQdl %>x in 20a!? &y in 12i/? 3. How is the coefficient of the quotient determined? 4. What is the product of a^ X a^ ? 5. Since the product of a^ X «* is a'l if «* is divided by a* what will be the quotient? What will be the quo- tient if a^ is divided by a^ ? 6. What is the product when x^ is multiplied by a;*? 7. What is the exponent of the quotient when x^ is divided by x^ ? By »* ? a;' by a;* ? x^ by x^ ? a;' by x* ? K* by a;? 8. How is the exponent of a quantity in the quotient determined ? 9. When -|-5 is multiplied by +3, what is the product? 10. Since + 15 is the product of + 5 X + 3, if + 15 is divided by +3 what is the sign of the quotient? What when -|- 15 is divided by + 5 ? 11. What is the sign of the quotient when a positive quantity is divided by a positive quantity? 12. When +5 is multiplied by — 3, what is the product? 13. Since —15 is the product of + 5 X —3, if —15 (54) DIVISION. 55 is divided by + 5 what is the sign of the quofient ? What when it is divided by — 3? 14. What is the quotient of —24 divided by —4? By + 4? By +6? By— 3? By +3? By— 8? By +8? 15. What is the sign of the quotient when a negative quantity is divided by a positive quantity? 16. What is the sign of the quotient when a negative quantity is divided by a negative quantity? 17. What is the product of —4 by —3? 18. Since +12 is the product of —4 X —3, if + 12 is divided by — 3 what is the sign of the quotient? What when it is divided by — 4? 19. What is the quotient of +24 divided by —3? By — 4? By —6? By —8? By— 12? 20. What is the sign of the quotient when a positive quantity is divided by a negative quantity? DEFINITIONS. 83. Division is the process of iinding how many times one quantity is contained in another. Or, The process of separating a quantity into equal parts. 84. The Dividend is the quantity to be divided. 85. The Divisor is the quantity by -which we divide. It shows into how many equal parts the dividend is to be divided. 86. The Quotient is the result obtained by division. The part of the dividend remaining when the division is not exact is called the Bemainder. 87. The Signs of Division. (See Art. 14.) 56 ELEMENTS OF ALGEBRA. 88. Peinciples. — 1. The sign of any term of ike quotient is -\- when the dividend and divisor have like sigm, and — when they have unlike signs. 2. The coefficient of the quotient is equal to the coefficient of the dividend divided by that of the divisor. 3. The exponent of any quantity hi the quotient is equal tn its exponent in the dividend diminished by its exponent in Hie divisor. 89. The principle relating to the signs in division may be illustrated as follows: 4-aX+6 = +a6\ / + a6 -=- + 6 = + a + 6 = — a — 6 = +a — 6 ^= — a ■ — a X -\~b = — ab I ] — ah , > Hence, + a X —b = —ab [ — a X —6 = + ab ) V +' CASE I. 90. To divide when the divisor is a monomial. 1. Divide — ISx^j/^z* by ZxyH^. PEOCESs. Explanation. — Since the dividend and Zxy^z^) — Wx'^y^z* divisor have unlike signs, the sign of the 5j.„j,2 quotient ia — . (Prin. 1.) Then —15 divided by 3 is —5; I* divided by a; is a; ; y^ divided by y' is y; and z* divided by z' is z^ (Prin. 3). Therefore, the quotient is — 5xyz'. 2. Divide 12a^x^y^ by ba^x^z^. PROCESS. Explanation.— Since di- 12a^x^y^ 12a^x^y'' 12y' vision may be indicated by ba^x^z^ da^x^z^ ~5z^ writing the divisor under the dividend with a line between DIVISION. 57 them, we have -— And since the same factors are found in both dividend and divisor, they may be cancelled without changing the quotient. Hence, the quotient is — ^• 3. Divide 9a^x« — 12a»x^ + 6ax* by Box''. PROCESS. Explanation. — Dividing 3ax^) 9aH^ - 12a^x^ + 6a«=* 9«^-' ^y So^S the result is ■ Sax, dividing — 12o'a-* by 9/,™. 4«2«.8 I 9~2 " ■^ oax 'ia X -\- zx 3^2^ ^.j^^ ^^^jj -^ _4„2j.3 . dividing 6aa;* by Soi^, the result is 2x^. Therefore, the quotient is Zax — 4a^a;' + 21^. EuLE. — jy'mde each term of the dividend by the divisor, as follows : To the numerieal coefficient of the dividend divided by that of the divisor, annex each literal factor with an exponent equal to the exponent of that letter in the dividend minus its exponent in the divisor. Write the sign -\- before each term of the quotient when the tbrms of both dividend and divisor have like signs, and — wlien they have unlike signs. Proof. — The same as in Aritlmietic. 1. An equal factolr in both dividend and divisor is omitted from the quotient. 2. When the division is not exact, the common factors should be cancelled and the remaining factors written as a fraction. (4.) (50 (6.) (7.) (8.) Divide 6a — 12a2x 15a2y2 — 20a:V 242/223 By 3a 3a2a; — day — 5x^y -8y^z 9. Divide 2Sx^yH^ by 7ryz. 58 ELEMENTS OF ALGEBRA. Find the quotient in the following, and prove: 10. 11. 12. 13. 14. 15. 16. 17. 18. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 20a56«c -=- 10a6e. 30cd2/H- 15ed^. 3&ax^y -e- 18ay. — 18z^yz-i- 9xy. — 21vM»^ -i- 7vz^. — 33rZsz2 -^llrs. 20a;32/83» -^ lOx^yz^. 19. —14a'^x'^y*^7axy'. 20. 32r^s^q-i-8r^sq. 21. — ISv^x^y -7- v^xy. 22. 24a2"6c"-; a^'bc". 23. 36»i2''a»/»» -j 4n,"a»/2". 24. 25a«/82 -= 5x^yH. 25. — 282/2s2'»-^42/2z3:t;. 26. —SOn^x^ -H-6m2a;2. 27. Divide 2(a; + y) by 2. Divide 3a(x + y) by 3. Divide — 12c (a; -f 2) by 4c. Divide 20(1/ + «) by (y + z). Divide 18(a; + z)« by (» + «)«. Divide — 8a(a; — y)^ by — 8(a; — y)^. Divide 5a^(e + d)^ by — 2a(c + d)2. Divide lOx'^yiy — a)* by — 5y(y — z)*. Divide os^'y — 2xy^ by as/. Divide Sxy'^ — 3x^y by a^. Divide 4a;»2/* + 2a;2y« by 2x^y^. Divide 3a262_6a63 by 3a6. Divide abc^ — a^b^c by — abe. Divide dx^y^z -\- 3xyz^ by 3xy3. Divide Ibax^y' — Ba^x^y by — 5oa:y. Divide BOx^y^s'' — 20x'y^z^ by —5x^y^z^. Divide a* — 3ab-\-ao^ by a. Divide x^y — xy^ -^-x'^y^ by xy. Divide x^ — 2xy-\-y^ by x. Divide z^ — 3a» + 3z2 by a. Divide m''n + 2mn — Sm^ by mn. Divide c^^d — Scd^ + 4d3 by cd. Divide aH — 3a^y+3aH^ by o^a,. DIVISION. 59 50. Divide v''y — 2y^+Sv^y^ by vy. 51. Divide 3(a! + y)—2(jc + yyhy—(x + y). 52. Divide aQ) + c)2 + 6(6 + c)s by — (6 + c). 53. Divide 9(a — c) — 6(o — e)* by 3(a — c). 54. Divide 5(x + a)* — 10(a; + z)* by — b{x + as). CASE II. 91. To divide when the divisor is a polynomial. 1. Divide a* + Zx^y + Zxy^ + y* by a; + y. PEOCESS. »» + 3a;2y + 3*2/2 -|- yS g +y a;^ + 2an/ + y" 2a;2y + 3ot/2 2a;^y + 22^/" ExpiiANATroir. — For convenience, the divisor is written at the light of the dividend, both of which are arranged according to the descending powers of x. Since the first term of the dividend is equal to the product of the first term, of the divisor by the first terta of the quotient, if the first term of the dividend is divided by the first term of the divisor, the result will be the first term of the quotient, x is contained in x', x^ times; therefore, x^ is the first term of the quotient, x^ times the divisor equals a;' + x^y. Subtracting, there is a remainder of 2a;*y, to which the next term of the dividend is annexed for a new dividend. Since the first term of the new dividend is the product of the first term of the divisor by the second term of the quotient, if the first term of the new dividend is divided by the first term of the divisor, the result will be the second term of the quotient. X is contained in 2x^y, 2xy times; therefore, 2xy is the second term 60 ELEMENTS OF ALGEBRA. of the quotient. ^ times the divisor equals 2x'^y -\- 2xy^ . Sub- tracting, there is a remainder of xy'', to which the next term of the dividend is annexed for a new dividend. Reasoning as before, the third term of the quotient is found by- dividing the first term of the new dividend by the first term of tlie divisor, x is contained in xy'^, y' times, y' times the divisor is xy''-\-y^. Subtracting, there is no remainder. Hence, the quotient is x' -{-2xy-\-y^. Rule. — Write the divisor at the right of the dividend, ar- ranging the terms of each, according to the ascending or descend- ing powers of one of the literal quantities. Divide the first term of the dividend by the first term of the divisor, and vyriU the result for the first term of the quotient. Multiply the divisor by this term of the quotient, subtract the product from the dividend, and to the remainder annex as many terms of the dividend as are necessary to form a new dividend. Divide the new dividend as before, and continue to divide in this way uniU the first term of the divisor is not contained in the first term of the dividend. If there be a remainder after the last division, write it over the divisor in the form of a fraction, and annex it to the quO' tient vrith its proper sign. Proof. — To the product of the quotient and divisor add the remainder, if any, and the sum wUl be equal to the dividend, if the work is corred. 2. x* — aH^+2a«x — a^ x^-\-ax^ —a^x^ x^ -\-ax — a x^ — ax-i- a^ -ax^ + 2a3a; — a* -ax^ — g^' g- + a^x o^a;2 -\-a^x — o* a^x*-\-a^x — a* DIVISION. 61 3. x^ — 1 _a,8 X —1 a;« + »2 + K + 1 »«- -1 a;8 — aji* x2 — 1 x^—x x — 1 x — 1 4. a* a* — a^x a — a; a" + oo; + a:* oP'X _a;S a^x — ax'^ ax^ — w' aa;2 — a;S 5. 48a;» — TGckb^ — 64a^x + 105a« 48a;8 — 12ax^ — 4ax'^ — 64a^x — 4ax^ -\- Ga^x — lOa^x + 105a8 — TOa^a; + 105a» 2x —3a 24a;2— 2aa; — 35a2 6. Divide a^—2ab + b' hj a — b. 7. Divide a;* + 4a! +4 by a; + 2. 8. Divide 9 + 6a; + a;^ by 3 + a;. 9. Divide a;* -f- ^^2/ + ''^^ + y' ^7 a; + y. 10. Divide a* + a^y + ay^ -\-y* hy a +y. 11. Divide a:* + 3x2?, -}- 3aa/2 + yS by a; + j/. 12. Divide r* + Sr^s + 3rs^ + s^ by r' + 2rs + s^. 13. Divide x* + 4x»y + Gai^t/^ -f 4aa/« + ?/* ^7 »; + y. 14. Divide c*+4c3d + 6c2d2 + 4cc?3 + d* by o^+2ed+d'i. 15. Divide a;* — 3a;» — 36a;2 — 71a! — 21 by a;^ — 8a; — 3. 62 ELEMENTS OF ALGEBBA. 16. Divide a» + 5a^x + 5ax^ + x« by o^ + 4aa; + a;2. 17. Divide a2+26c — 62_c2 by a — 6 + c. 18. Divide a*—4a^y+6a^y^—4ay»+y* by a^— 2ay+2/^ 19. Divide ax^ —aV —bx^ +b^ by ax — 6. 20. Divide 20a^b — 25a^ — 186» + 27a62 by 6& — 5a. 21. Divide 3x* — 8x^y^+3x^s^ + 5y* — 3y''z^ by a;"— y^. 22. Divide 4a* — 9a2 + 6a— 1 by 2a2 + 3a— 1. 23. Divide 2aj^ + 36^ + 10a6 + 156^ hjy-f- 56. 24. Divide &— 668— 2a + 54a»—3a26 by 2a — 6. 25. Divide 25aS — a» — 8a — 2a2 by 5a2— 4a. 26. Divide a;^ + y* + a* — Sxyz by a; + j^ + a. 27. Divide 18a;*— 45a!«+82a;2— 67a;+40 by 3x^—4x+5. 28. Divide 16a;* — 72a2a;2 + 81a* by 2a; — 3a. 29. Divide a* + 4a2a;2 + 16a;* by a^ + 2aa; + 4x'^. 30. Divide a;* + x^z^ + a* by a;" — aai + a^. 31. Divide a;* — y* by a; — y. 32. Divide x^ +y^ by a; + y- 33. Divide a;^ + 1 by a; + 1. 34. Divide a;*— 81^/* by a; — 3?;. 35. Divide 81a* — 166* by 3a + 26. 36. Divide x' -\- y" hy x + y to S terms. ZERO AND NEGATIVE EXPONENTS. 92. 1. How many times is a* contained in a* ? o* in fflS? o* in a*? a' in a*? a" in a"? 2. When similar quantities have exponents, how may the division be performed? 3. What, then, will be the quotient when a^ is divided by a^ by subtracting exponents? a^ by a^ ? a* by a* ? (j2o by a2oy a" by a"? 4. Since a^-^-a^, a' -^-a^, a* -f-a*, and o^-na" are each DIVISION. 63 equal to a" and to 1, what is the value of a', or any quantity with for an exponent? 5. What wiU. be the quotient when a* is divided by a* by subtracting exponents? a^bya''? a^bya'? 6. What will be the quotient when a* is divided by a^ without subtracting exponents? a^ by a'' ? a^ by a' ? 7. Since a^-i-a^^ar^ and — , a^-f-a''=a~^ and — (j6 _!_ £j9 3— a-3 an,j , to what is any quantity with a negaiive exponent equal? The Reciprocal of a quantity is 1 divided by the quan- tity. Thus the reciprocal of a is —i of i + y is a x + y 93. PumcrPLES. — 1. Any quantity having for an expo- nent is equal to 1. 2. Any quantity having a negative exponent is equal to the reciprocal of the quantity vdth an equal positive exponent. 1. Divide abx by — ahx. 2. Divide &aH^ by Sa^a;*. 3. Divide 8a'x' by — Aax^. 4. Divide 21x^yH^ by —%x^y^z. 5. Divide ZQx^^-\-zy by 5a;% + a)2. 6. What is the reciprocal of ar"^ ? 7. What is the reciprocal of ar^y~^ 1 8. What is the reciprocal of a^ar^y"*? 9. Divide 12ar3y-* by sr^yT. 10. Divide 2Qa-*h^e-^ by —ba-^h^e-'^. 11. What is the reciprocal of 2!r^y~^ ? 12. What is the reciprocal of SaT"!/-"? 64 ELEMENTS OF ALGEBRA. EQUATIONS AND PROBLEMS. 94. 1. Find the value of x in the equation ax H- 4 = a2— 2a;. PROCESS. aa; -f 4 ^ o^ — 2x Transposing, ax -\-2x := a^ — 4 Then, (a + 2)a; = a^ — 4 And, x^a — 2 Explanation. — Transposing the known quantities to one mem- ber and the unknown to the other, the coefficient of a; is a + 2. Dividing both members of the equation by o + 2, the value of x is found to be a — 2. 2. Find the value of a; in the equation bx — 6*^ 4a; — 96 + 20. PROCESS. bx — b^=4x — % + 20 Transposing, bx — 4x:=b^ — 96 -|- 20 Then, (6 — 4)a; = 62—96 + 20 And, a; 3= 6 — 5 Explanation. — Transposing as before, it is seen that the coeffi- cient of X is 6 — 4. Dividing both members by b — 4, the coefficient of x, the value of X is found to be 5 — 5. Find the value of x in the following : 3. ca;— 9 = c2-(-6c — 3a;. 4. ax-{- 16 = a2 — 4a;. 5. 3a;— 12a = 4a2— 2aa;-|-9. 6. dx + 9a2 = d2 — Sax. 7. ax — a^ = 2ab + b'^—bx. 8. ax — 5ah^2a^ + 3b'^—bx. DIVISION. 66 9. ax — c* ^ a* -|- ac -j- a^c — ex. 10. 2ax — 6a2 = 13a6 + 6b^ — 35a;. 11. 2ax — lOab — 155 = 14a + 21 — Bx. 12. ax + bx = 5a2 + 7a5 + 26^ -j- 5ac + 2bo — ex. 13. 2ca! — 4c3 + d2 ^ 2c2d — 2c(i — da;. 14. 52a; + 35^0 + 6c3 ^b^ + 2bc^ — 2eH. 15. 4m.* — 2m2a; — Zmx =1 — 6m -f Qm^ — x. 16. as + 3a; — 9a2 = aa; — 27a + 27. 17. 2mH + 3m7i8 + Im'^n'^ — 4m* = 3m»ia;. 18. 5aa; == 15a3 — 5a5 + 5a62 + 262; — 6a25 + 25^ — 268. 19. A man being asked how much money he had, replied that if he had $25 more than 3 times what he then had, he would have $355. How much money had he? 20. A gentlenian divided $10500 among four sons, giving to the second twice as much as to the first, to the third twice as much as to the second, and to the fourth one-half as much as to the other three. How much did he give to each? 21. A man who met some beggars gave 3 cents to each and had 4 cents left, but found that he lacked 6 cents of having enough to give them 5 cents each. How many beg- gars were there? How much money did he have? 22. A man has six sons, each 4 years older than the one next to him. The eldest is 3 times as old as the youngest. What is the age of each? 23. A vessel containing some water was filled by pouring into it 42 gallons, and there was then in the vessel 7 times as much as at first. How many gallons did the vessel contain at first? 24. A man borrowed as much money as he had and spent a dollar; he then borrowed as much as he had left and spent a doUar; again he borrowed as much as he then had and spent a dollar, when he had nothing left. How much had he at first? 6 66 ELEMENTS OF ALGEBRA. KEVIEW EXERCISES. 95. 1. Add 6aa; — 140 + 3 j/ x, 5a;2 + 4aa; + 9a;2 , 7aa! + 4y^ x-\- 1'jO, and V x-}- Sax — 4x'. 2. Add 3am + 2a; — 3 |/"y — z, 2 Vy + Bs — 2x^+ Sam, 4ir2 — 3? + 2i/y + 3a;, and 2 [/y'—4am + 2z — 3x\ 8. From i/a^— fi^— 2 (a; + j/) — 6 subtract 4 (a; + 2/) — 4 From i/ x + 2 i/y — a + 6 subtract 3\/y — 2 i/i~— y + 2s — 16. 5. From a2a;2 _|_ 20?/ — Sy^-\-z^ subtract b^x'' -\-Bay — 6. From the sum of x^'^ + 3a;^y" — Sys -\- az and 4a;'" — 3yz + 2z+3x^y" subtract dx^" — 4z + Qx^y^ — Baz. 7. Multiply a;* + 2a;22/ -|- xy^ by a;* + 2xy — y''. 8. Multiply af + 2a;"2/» + /' by a;" + 2a;"2/" + y\ 9. Multiply 3ar» + 2ar 2»2/-2" — t/" by af — y^" + a;"^/". 10. Multiply 3af + 2 + 22/"+"' + a" by Sar^ — 2y-*-* + 11. Expand (x + y) (a; + y) (x + y)ix + y) (x + y). 12. Expand Ca + 2) (o + 2) (a — 2) (a — 2). 13. Expand (3a — 6) (3a — 6) (3a + 6) (3a -f 6). 14. Expand (a; + 2^) (a; + 2^;. 15. Square 2a; -)- by. 16. Square Bx'^ — 2y'^. 17. Square a:2» -|- 22/2". 18. Square a;-2» — 22r^". 19. Write out the product of (2x-\-y){2x — y). 20. "Write out the product of (Bx-\-ly) {Bx — 1y). 21. Write out the product of (4a;2 — 22/2) (4a;2 _^ 22/2), 22. Write out the product of (aa;" + j/") (oaf — 'f). 23. Write out the product of (aa;~" + a2/~") (aa;~" — ay-^). REVIEW. 67 24. "Write out the product of (a -|- a;) (a — x) (a* + *^) (a*+a:*). 25. Write out the product of (,x'^ -\-y^) (x^ — y^) 26. Write out the product of (2a; + 3^ (2a;— 3) (4a;2 +9) (16a;*— 81). 27. Divide 4a* — da'^b^ + b* by 2a'^ — 3ab+b^. 28. Divide Sai^y -(- a;* -|- 2/' + ^^^ by a:^ -|- y^ -(- 4a^. 29. Divide x^ — 5x*y + lOx^y^ — 10a;2i/3 + 5as/* — y^hj x^ -{-y"^ — 2a;y. 30. Divide 2a3» — 6a2"6" + 6a"62» — 263" by a^ — b". 31. Express in its equivalent without negative exponents a;~^2r*2~*. 32. Express in its equivalent without negative exponents 33. Express in its equivalent without negative exponents ar^y^z^. 34. Express in its equivalent without negative exponents 35. Divide by subtracting exponents a^^y by a%^y. 36. Divide by subtracting exponents a*x*y^ by a^x^y^. 37. Divide by subtracting exponents — ar^yg" by a;~2'> 38. Divide x'^y'>+Qx^y + 15x*y''+20x»y« + 15x^y*+6xy^ -\- x'^y^ by x^ -\- 2xy -\- x'^y^. 39. Divide x-^+dx-^y-^+lOx-^y-^+Wx-^y-^+dar^y-* 4-jr* by x-^+y-^. 40. Find the value of x in the equation 2aa; + 12a6 — 4a2=962+36a;. 41. Find the value of x in the equation 3a; — 9 — 3c = 12a — 2aa; + 4a2-j-2ac. 42. Find the value of x in the equation 2oa; -|- 9c* + 3ed = 4a^ +3ex + 2ad. FACTORING. 96. 1. What is the product of 4 times 5a? What are 4 and 5a of their product? 2. What is 3a of 9a? 5a of 15a? 3c of 15c? 3. What quantity will exactly divide 10c? 18d? 25a;? 302? 4. What are the exact divisors of 12xy^ 25x^y^^ 36x2/2? 5. What are the exact divisors of 24a;^y^1 30a^*? Ua^bcl DEFINITIONS. 97. An Exact Divisor of a quantity is a quantity that will divide it without a remainder. Thus, a, b, and x + y are exact divisors of a6(i + y). 98. The Factors of a quantity are the quantities which, being multiplied together, will produce the quantity. Tlius, o, 6, and x-\-y are the factors of ab(x + y). An exact divisor of a quantity is a factor of it. 99. A Prime Quantity is a quantity that has no exact divisor except itself and 1. 100. A Prime Factor is a factor that is a prime quan- tity. 101. Factoring is the process of separating a quantity into its factors. <68) FACTORING. 69 CASE I. 102. To separate a monomial into its factors. 1 What are the prime factors of 24x^y^z? PROCESS. Explanation.— The prime 24 = 2, 2, 2, 3 factors of 24 are 2, 2, 2, and 3 ; a!* = aa; of x', x and i; of y', j/, y, and y^ =^ ^^ y ; a"d z is a prime quantity. z = g Therefore, the prime factors 24a;2y3g = 2, 2, 2, 3, X, X, y, y, y, z are 2, 2, 2, 3, j;, x, 2/, y, y, yi. Rule. — Sq)arate the numerical coefficient into its prime factors. Separate the literal quantities into their prime factors by writing each quantity as a factor as many times as Hiere are units in its exponent. Find the prime factors of the following: 2. %a^h. 4. Iba^yH. 3. V)x^y^. 5. 20cKi;3y. 6. 42axy^. 7. Z&xyH^. 8. ^Sa^c^K. 9. Zbx^e^c^. CASE II. 103. To separate a polynomial into monomial and polynomial factors. 1. What are the factors of 5a^he + \0a'^c — 2Qa^hc'! PROCESS. Explanation. — By exam- ba^e)ba%c-{-10a'^c --20a'^'be i°™g the terms of the poly- J I 2 46 nomial it is found that 5oV _-,-,„ .,^ is a factor of every term. Di- ^ ' ■' Tiding by this common factor the other is found. Hence, the factors are ba^c and (6 + 2 — 46). 70 ELEMENTS OF ALGEBRA. HxTLE.— Divide the polynomial by the greatest factor common to aU the terms. The divisor, and tlie quotient, vM he the factors sought. Find the factors of the following polynomials: 2. ba^h + Ga^c. 3. '8*22/2 4- \2xH'^. 4. Qxyz + 12x^y^z. 5. 9x»y''z+18m/^z^. 6. a^x'^yH + a^xyz'^. 7. a''e + bH + c^d^. 8. 4x^y -\- cxy^ + Zxy^. .- 9. a^yz -\- a'^'xz -\- a'^x^y^z^. 10. b^x^y^ + b^xy^ + bx'^y^z. 11. a^x^y"z + ax"2/2" + a^mfyH^. CASE III. 104. To separate a trinomial into two equal binomial factors. 1. When a-{-b is multiplied by a-\-h, what is the product? 2. When x-\-y is multiplied by x -{- y, what is the product? 3. When c — d is multiplied by c — d, what is the product? 4. When x — 1 is multiplied by x — 1, what is the product? 5. When x-\-y is squared, what terms are squares? To what is the other term equal? 6. When x — 1 is squared, what terms are squares? To what is the other term equal? FACTORING, 71 7. When a trinomial consists of two terms that are squares, and the third term is twice the product of the square roots of the squares, how will the factors compare? 105. One of the two equal factors of a quantity is called its Square Root. 106. 1. Eesolve a* — 2xy -\-y^ into two equal binomial factors. PEOCESS. x^ — 23oy -\- y^ Vy' y (x — y)(x — y) Explanation. — Since the trinomial contains two terms that are squares, and the other term is twice the prod- uct of their square roots, the quantity may be separated into two binomial fac- tors. Since the terms that are squares are the squares of the two quantities, the square root of x^ and of y' gives us x and y, the two quantities; and since twice their product has the minus sign, the quantities must have had unlike signs. Therefore, the factors are x — y and x — y. Rule. — Find the square roots of the terms that are squares, and conned these roots by the sign of the other term. The result will he one of ihe equal factors. Find the equal factors of the following trinomials: 2. a^+2ah + b^. 3. x^+2Ky + y^. 4. 62— 26e + c2. 5. r2-f2rs + s2. 6. x^+2x-\-l. 7. a;2+4a; + 4. 8. 2/='-22/-fl. 9. 4:y^—4y+l. 10. 9a;2+6a;+l. 11. 9m2 -)- ISmn + 9n^. 12. 9 + Qx + x^. 13. l — 2x^+x*. 14. 16n^ — 8n + l. 15. 16 + 16a + 4a2. 16. 36 + 12a2-fo*. 17. 49 — 14x8-1- a;6. 18. Slx^ — lSax + a"^. 19. 4a2'' -I- 12a"6'' -I- 952- 72 ELEMENTS OF ALGEBRA. CASE IV. 107. To resolve a binomial into two binomial factors. 1. When a + 6 is multiplied by a — h, what is the product? 2. When a; + y is multiplied by x — y, what is the product? 3. When c -|- d is multiplied by c — d, what is the product? 4. When x-\-2 is multiplied by x — 2, what is the product? 5. When the sum of two quantities is multiplied by their difference, what is the product? 6. When a binomial consists of two terms that are squares, connected by the minus sign, into what factors may it be resolved? 1. Kesolve x^ — y^ into its factors. PBOCEss. Explanation. — Since the binomial >£2 yZ coneistB of two terms that are squares -/r2 __ „ connected by the minus sign, the bino- /-j- mjal may be separated into two bino- , ^ , X mial factors, which are respectively the \ 'vy) y yj g„m J^J]J t)jg (Jiflerence of the quantities. The square root of x* is x, and of y^ is y. Therefore, x-\-y and x — y are the factors. Rule. — Find the square root of eacfi term of iJie bino- mial, and malce the sum of tliese square roots one factor, and their difference the other. Binomials of the form of i* — y* may be resolved into the factors {x'-\-y') (x^ — y^), and x^ — y^ into (i + y) (x — y). Therefore, x* — y^ = {x'+y') (x + y) (x — y). FACTORING. 73 Resolve the following binomials into their factors: 2. a2 — 62. 3. c2— d2. 4. m^ — n^. 5. 4i;2 — 42/2. 6. 9x2 j2_ 7. a;2_9y2_ 8. 16x2 — 162/2. ij/'- 10. a;22^2 _ 42,2g2. 11. m* — n*. 12. a8— 68. 13. m2» — ?i2"'. 14. 9a2" — 46*". 15. a'" — 68. CASE V. 108. To resolve a quadratic trinomial into unequal factors. 1. What is the product of a; + 2 multiplied bya!-|-3? What is the first term of the product? What is the last term? Of what two numbers is it the product? What is the coefficient of the other term ? How does it compare in value with 3 and 2 ? 2. What is the product of a; + 3 multiplied by a; + 4? What is the first term ? Of what numbers is the last term the product ? How does the coefficient of the second term compare with 3 and 4? 3. What is the product of x — 10 multiplied by x — 2? How is each term of the product obtained from the quan- tities multiplied? 4. What is the product of a; + 2 multiplied by x — 5 ? How is each term of the product obtained from the quan- tities multiplied? 109. A trinomial of the form of x"^ ±ax±: b, in which 6 is the product of two quantities and a their algebraic sum, is called a Quadratic Trinomial. 74 ELEMENTS OF ALGEBRA. 1. Resolve x"^ — 9a; — 36 into two factors. Explanation. — The first term is evidently PROCESS. J,. Since 36 is the product of the two other a;2 9^; 3g quantities, 6 and 6, or 4 and 9, or 3 and 12 f f) V fi ^^^ ^^ other quantities. Since their sum is — 9, the quantities must be 3 and — 12, for the' other sets of 3 yc 12 _ factors of 36 can not be combined so as to produce this result. (a-f-S) (a;— 12) Therefore, (i + 3) and {x—\2) are the factors. 36= <^ 4X9 i 3xi: — 9 = 3 — 12 Rule. — For the first term of each factor take the square root of one term, of the trinomial, and for the second term such quantities that their product will be anotlier term ot tlie tri- nomial, and their sum multiplied by the first term of the factor will be equal to the remaining term of the trinomial. Separate into factors the following trinomials : 2. a;2 -I- Sx 4- 2. 3. a;2 + 7a; + 12. 4. a;2— 4.1;- 21. 5. a;2 — 7a;— 18. 6. a;2 + 6a; + 8. 7. a;2 + 12a; + 32. 8. a;2 — 10a; — 39. 9. a;2 — 12.r — 64. 10. 4a;2 — 10a; + 6. 11. 9.x2 — 27a;+18. 12. 4a;2 + 16aa; -f V2a^. 13. 9a2 + 30a6 -f 2462. CASE VI. 110. To resolve the difference of the same powers of two quantities into factors. By performing the operations indicated in the foUowing examples, it is found that, 1. {a-'—b-'')-^(a — h)^a-\-b. 2. {a^—b^)~{a — b)=a^-\-ab + b^. FACTORING. 75 3. (a*— 6*)H-(a — 6) = a3+a2& + a62+6s. 4. (aS —65) _=_ (a_6) = (j4 + a^ft + a^fi^ _|_ ajs _j. j4. 5. What will be the quotient when a^ — b^ is divided by a — 6? 6. What will be the quotient when a' — 6^ is divided by a — hi 7. How does the first term of the quotient compare with the first term of the dividend? What quantities does the second term of the quotient contain ? The third term ? The fourth term? 8. What is the sign of each term? . 9. What are the exponents of x and y, when the difier- euce of the same powers of two quantities is divided by the difierence of the quantities? 111. Principle. — The difference of the same powers of two quantities is always divisible by tlie difference of the quantities. 1. Write out the quotient of (a;* — t/*) 2. Write out the quotient of (.r" — 2/*) 3. Write out the quotient of (x* — 1) 4. AVrite out the quotient of (a;* — 16) 5. Write out the quotient of (x* — y^) (x—y)- ip—y)- (x-1). {x-2). (x^-y^). 115J. A course of reasoning which discloses the truth or falsity of a statement is a Demonstration. US. The following is a general demonstration of the principle given in Art. Ill : Let X and y represent any two quantities, and n the expo- nent of any power. Then, x" — y" will be the difierence of the same powers of two quantities, and x — y the difierence of the two quantities. 76 ELEMENTS OF ALGEBRA. PEOCES s. X" X" -r x — y z'-i + x'-^y 1st Rem., x^-^y — x^-^y^ 2d Bern., a;»-2y2 _^yn nth Rem., nT-^if' — y" sfy^ — 7/" r —y" = Demonstration. — By dividing x" — y" until several remainders are obtained, it is found that the first term of the first remainder is x"~'j/; of the second, x'^^y' ; of the third, if^'y' ; of the fourth, ^n-iyi^ and, consequently, of the mth x"^"y". But x"~" is x", which equals 1 (Art. 93, 1). Therefore, the first term of the reth remainder reduces to y". Since the second term in the mth remainder is — j^, the entire nth remainder is y^ — g", or 0; that is, there is no remainder, and the division is exact. Therefore, x" — y" is divisible by i — y when x and y represent any two quantities and n the exponent of any power; or, 7%e difference of the same powers of two quantities is always divisible by tlie difference of the quanlities. 114. By performing the operations indicated in the follow- ing examples, it is found that, 1. (x'^—y'i)^(x + y}=x — y. 2. (x» — y3) -^ (x + y) =x^ —xy + y^. Rem. — 2y^. 3. (x* — 2/*) -^ (x + y) =x^ — x'^y -\-xy^ — y^. 4. (_x^ — 2/') -r- (^ + 2/) ^ a;* — x^y -{- x^y^ — »«/' + «/*■ Rem. — 2y^. 5. What is the quotient of a;' — y^ divided by a; + y ? 6. What are the signs of the terms of the quotient when the difference of the same powers is divided by the sum of the quantities? What is the law of the exponents in the quotient? FACTORING. 77 7. What are the exponents of x and y when the difference of the same powers of two quantities is exactly divisible by the sum of the quantities? 115. Principle. — The difference of the same powers of two quantities is divisihle by the sum of ike quantities only when, tlie expmients are even. 1. Write out the quotient of («« —y^) -^ (» + y). 2. Write out the quotient of (a;'" — t/'") ^(x-\-y). 3. Write out the quotient of (»* — V)^(x-\- 1). 4. Write out the quotient of (a;* — 16) -^ (» + 2). 5. Write out the quotient of (a* — 2/*) ^ {^"^ + V'^)- 116. The following is a general demonstration of the principle in Art. 115: PEOCESS. af — 2/" iE" 4- 3^^y x-\-y 1st Rem., — og^^y — j/" 2d Eem., af^a^/^ — 2/" 3d Eem., _a^Sy3_2,» — 3^-^y^ — ai^*y* 4th Rem., af^^y^—y" Demonstration. — By dividing a" — y until several remainders are obtained, it is found that the first term of the first remainder is —jf'^^y; of the third, —x^^y^; of the fifth, —x'^^y^; and of the nth remainder, when n is an odd number, — ■jff^'^'f: But — i"-" is — x", which is equal to — 1. Therefore, the first term of the «th remainder, when n is odd, reduces to — y. Since the second term of the nth remainder is — y", the entire nth remainder, when % is an odd number, is — y" — y", or — 23/". Therefore, a;" — y" is not exactly divisible by r -}- y when n is an odd number. Hence, the 78 ELEMENTS OF ALGEBRA. difference of the same odd powers of two quantities is not divisible by the sum of the quantities. The first term of tlie second remainder is i"—^!/^; of tlie fourth, ^n-iyi.^ of the sixth, a;"-«j|6. ^nd of the 7ith remainder, when n is an even number, j."~"2^. But a:"-'y is ecjuai to a:''2/"> o"" !/"'• ■'^"'1 since the second term of tlie reth remainder is — y"; the entire Mth remainder, wlien n is an even number, is y"- — y", or 0. Hence, j:" — y" is exactly divisible hy x -\- y when n is an even number; or. The difference of tlie same powers of two quaniiiies is divisible by the sum of the (piantities only when the exponents are even. CASE VII. 117. To resolve the sum of the same powers of two quantities into factors. By performing the operations indicated in the following examples, it is found that, 1. (x''' -\-y^)^- {x-\-iJ)^x — y. 'Rem. 2y^. 2. {x^ -{- y^) ^ {x -\- y) = x'^ — xy -\- y"^ . 3. {x^-\-y*)-^(x-\-y)^x^—x'^y-\-xy'^—y». Rem. 2?/*. 4. (x^ -[- yS) ^_ (a; -|- 2/) = a;* — x^y -\- x^y^ — xy^ -\- «/*. 5. What are the signs of the terms of the quotient? What is the law of the exponents? 6. What are the exponents of x and y when the sura of the same powers of two quantities is exactly divisible by the sum of the quantities ? 118. Principle. — The sum of Vie same powers of two quan- tities is divisihle by the sum of the quantities only when tlie exponents are odd. 1. Write out the quotient of (x'' -f- 2/') 2. Write out the quotient of (x^ + y^) 3. Write out the quotient of (a;' +1) _=- (a; + 1) (a: + y)- (^ + y)- FACTORING. 79 119. The following is a general demonstration of the principle in Art. 118 : PROCESS. X" + y" oc" -f- x"~'^y •^ + 2/ a;"-i - -a;"- -^y + a;"- -32^2 — af- -4yS . a^-ly_a;n-2y2 ;«-V+2/" ..-2y2 _j_ jJ- 32,3 -af'-^i/* + j^" ■ af'~*2/* ■ - af'-*y* 1st Hem., 2d Eem., 3d Rem., 4th Rem., a;'-*?/* +2/" Demonstration. — Dividing and reasoning, as in the previous demonstrations, the mth remainder, wlien n is even, reduces to y + y, or 2!/". Hence, i" + y^ is not exactly divisible hy x-\-y when )i is even, or tlie sum of tlie same powers of two quantities is not exactly divisible by the sum of the quantities when llie exponents are even. The first term of the nth remainder, when n is odd, reduces to — y'\ and the entire remainder is — 2/" + ^", or 0. Hence, j:'' + y is exactly divisible \iy x-\-y when n is an odd number; or, The sum of the same powers of two quanlUies is dmisible by the sum of Ike quivniities only when llie erpone/nls are odd. 120. Perform the operations indicated in the following examples, and write down the quotients and remainders: 1. (^x^+y^^)^(x-y). 2. (a;3 4-2,3)_=_(^_2,). 3. {x* + y^)-^ix-y). 4. (x^+y^)^(x — y). 5. From the results, discover whether the sum of the same powers of two quantities is divisible by the difference of the quantities. 6. Demonstrate the truth of the following principle : 121. Pkinc'IPLE. — The sum of tlie same powers of two quantities is never divisiMe by the difference of the quantities. }J0 ELEMENTS OF ALGEBRA. COMMON DIVISOES. 132. 1. What number will exactly divide both 15 and 20? 2. What quantity will exactly divide both 3a and 2a? 3. What quantity will exactly divide both Sa^ and 2aa;? 4. Give all the exact divisors of 12a2a^ and ISax'^y. What is the greatest, or highest, of these common divisors? 5. What is the greatest, or highest, common divisor of 24a262e and 48a26c2? 6. What prime factors, or divisors, are common to 24a262c and 48a26e2? 7. How may the greatest, or highest, common divisor of 24a^b^c and 4SaHc^ be obtained from these factors? 8. How may the greatest, or highest, common divisor of Ibx^y^z and 20x^yH be obtained from their prime factors? DEFINITIONS. 123. A Common Divisor of two or more quantities is an exact divisor of each of them. Thus, 6a is a common di\isor of 12a, 24a^c, and SOa'^y. 124. The Greatest, or Highest,* Common Divisor of two or more quantities is the greatest or highest quantity that is an exact divisor of each of them. Thus, 4a^x is the greatest, or highest, common divisor of 12a'aa/, Sa'x'y, and Aa'xz. * Strictly speaking, Highest Common Divisor would be the appro- priate term to apply to literal quantities, because, although x' is a higher power than x, the valv£ of x^ may be less than the value of X, but common usage is followed in employing Greatest Common Dimsor to include both. COMMON DIVJESOBS. 81 125. When quantities have no common divisor they are prime to each other. Thus, 5x, 3y, and 8z are prime to each other. 126. Peinciple. — The greatest eommon divisor of two or more quantities is tlw product of all their eommon prinie fac- tors. CASE I. 127. To find the greatest eommon divisor of quan- tities that can be factored readily. 1. What is the greatest common divisor of Sa^h^c^ and 12a62e2 ? PROCESS. 8a262c3 =2x2x2XaXaX&X6XcXcXc 12ah^c^ =3x2x2XaX6XfcXcXc G. C. D. =2x2XaX&X&XcXc = 4aA2c2 Explanation. — Since the greatest common divisor is the product of all the common prime factors (Prin.), the quantities are separated into their prime factors. The only prime factors common to the given quantities are 2, 2, a, 6, 6, c, c; and their product, 4ab'c', is therefore the greatest common divisor. 2. What is the greatest common divisor of a(x^ — y^) and a(x^ +2xy + y^) ? PEOCESS. a(a;2— 2/2) =a{x-\- y) (x— y) a(a;^ + 2a!y + y^) = a(x + y) (x + y) G. C. D. =aX(x+y-)^a(x + y) Explanation. — Reasoning as in the preceding example, the quan- tities are separated into their prime factors, and the product of the common factors will he the greatest common divisor. The common factors are u and {x-\-y); therefore. a{x + y) is the greatest common divisor. 82 ELEMENTS OF ALGEBRA. Rule. — Separate ike quantities into their prime factors, and find the product of all the common factors. Find the greatest common divisor of the following : 3. 12m^n^x^ and ISm^nx^. 4 16r*s»x^ and 20r^s'^x^. 5. 21x*yH^ and lix^yH^. 6. Ibx^y'^z^ and 20x^yz^. 7. lla^xy, Sax^y, and 9axy. 8. 15a3a;2?/2, 9a^x^y», and Sa^ay^. 9. 1862c2ds, 862c2d2, and 12ab''e. 10. 10e3a;3y3, SaH^y^, and 12a2a^2. 11. 18r2s2t2, lOr^sSf^ and 16r^sV. 12. 20a3x32/3, Wa^x^y^, and lOa^a^^, 13. 12a;32/22;2, 18x42/323, and 15a; V^^- 14. a^ — b^ and a^— 2a6 + 62, 15. a;2 — 2a; and 2an/2 — 42/^. 16. 16a;2 — 2/2 and 16a;2 — 8w/ + 2/^. 17. a;2 — 2a; — 15 and a;^ + 9x + 18. 18. a;2 + 9a; + 20 and x^ + 2a; — 15. 19. x2 _|_ ^ _ 30 and x^ + 12x + 36. 20. a;2— X— 12 and x^— 4x — 21. 21. x2 + 9x + 14 and x^ + 2x — 35. 22. x^ -\-x — BO and a;^ + 9x + 18. 28. a(x* — 2/*) and x^ -j-2x^y + x''y''. CASE II. 128. To find the greatest common divisor of poly- nomials. 1. What are the exact divisors of 10? What are they of 2 times 10 or 20? Of 3 times 10 or 30? Of any number of times 10? COMMON DIVISORS. 83 2. What are the exact divisors of ax^ ? What are these also of 2 times ax^ or 2oa:2 ? Of c times ax^ or caa;^ ? 3. If a quantity is an exact divisor of some quantity, what will it also be of any number of times that quantity? 4. Since 15 and 20 are each divisible by 5, what must they each be of 5? 5. Since they are each some number of times 5, what will their sum be of 5? What will their difference be of 5? 6. Since a is a divisor of 2ah and 3ac, what will it be also of their sum? What of their difference? 7. If a quantity is an exact divisor of each of two quan- tities, what is it of their sum ? What of their difference ? 8. What is the greatest common divisor of 10 and 15? Of 2 times 10, or 20, and 15? Of 4 times 10, or 40, and 15? Of 10-^2, or 5 and 15? 9. What factors of these multipliers and divisors of 10 are factors of 15? 10. What is the greatest common divisor of 10 and 3 times 15, or 45? Of 10 and 7 times 15, or 105? Of 10 and 15 H- 3 or 5 ? 11. What factors of these multipliers and divisors of 15 are factors of 10? 12. By what quantities, then, may either quantity be multiplied or divided without changing their greatest com- mon divisor? 129. Principles. — 1. A divisor of any quantity is a divisor of any number of times thai quantify, 2. A divisor of two or more quantities is a divisor of their sum and of the difference between any two of them. 3. The greatest common divisor of two or more qtumtities is not affected by mvUvplying or dividing any of them by quantities which are not factors of the others. 84 ELEMENTS OF ALGEBRA. 130. 1. What is the greatest common divisor of Ga;^ + 37x + 35 and Sa;^ -)- 17a; + 10 ? 6a;2 + 37a; + 35 6x2 _f. 34a; ^ 20 PROCESS. 3a;2 + 17a; + 10 a;+5 3a; +2 3 )3a; + 15 a; + 5 3a;2 + 17a; + 10 33;" + 15x 2x + 10 2a; +10 Explanation. — The greatest common divisor of two quantities can not be greater than the smaller quantity; therefore, the greatest common divisor of these two quantities can not be greater than 3a;^ + Vlx + 10. Sx^ + 17a; + 10 will he the greatest common divisor if it is exactly contained in 63;^ + 37a; + 35. By trial, it is found that it is not an exact divisor of Qx^ -\- 37a; + 35, since there is a remainder of 3a; + 15. Therefore, 3a;^ + 17a; + 10 is not the greatest common divisor. Since 6a;2 + 37a; + 35 and 6a;2 + 34x + 20, which is 2 times Sa;^ + 17a; + 10, are each divisible by the greatest common divisor, their difference, 3a;+15, must contain the greatest common divisor (Prin. 2). Therefore, the greatest common divisor can not be greater than 3a; + 15. Since 3 is a factor of 3a; + 15, but not of the quantity whose greatest common divisor is sought, 3a; + 15 may be divided by 3 without changing the greatest common divisor (Prin. 3). Therefore, the greatest common divisor can not be greater than a; + 5. a; + 5 will be the greatest common divisor if it is exactly con- tained in 3a;^ + 17a; + 10, since if it is contained in Zx^ + 17a; + 10 it will be contained in ivnce 3a;^ + 17a; -\- 10, or 6a;^ + 34a; + 20, and in the sum of 3a; + 15 and 6x^ + 34a; + 20, or Qx^ + 37a; + 35. By trial it is found that a; + 5 is an exact divisor of 3a;^ + 17a; + 10. Therefore, a; + 5 is the greatest common divisor. COMMON DIVISORS. 85 2. What is the greatest common divisor of 3a;'' -f" H^ + 6 and 2a;2 -f Ha; + 15 ? PKOCESS. 3a;2 + 11a; + 6 2 6a;2 + 22a; + 12 6a;2 + 33a; + 45 — 11) — 11a; — 33 2a;2 + 11a; + 15 a; + 8 2a;2 + 6a; 2a; -1-5 5a; +15 5x-f 15 2a;2 -I- 11a; + 15 Explanation.— If 3x^ + Ux + 6 is divided by 21^ + Hi -|- 15, the quotient will be a fraction. To avoid the fractional quotient, we multiply Bx^ -|- lis -j- 6 by 2 without changing the greatest com- mon divisor, since 2 is not a factor of the quantities whose greatest common divisor is sought. (Prin. 3.) If the preceding divisor, 2i^ + 11j;-|- 15, is divided by — llj; — 33, the quotient will be a fraction. This result may be avoided by dividing — 11a; — 33 by the factor —11 without changing the greatest common divisor, since it is not a factor of the quantities whose greatest common divisor is sought. (Prin. 3.) Dividing by x-\-3, the division is exact. Therefore, a; -j- 3 is the greatest common divisor. Rule. — Divide ike greater quantity by the less, and if there be a remainder, divide the less quantity by it, then the preceding divisor by the last remainder, and so on until nothing remains. The last divisor vM be the greatest common divisor. If more than two quantities are given, find tlie greatest com- mon divisor of any two, then of tiiis divisor and another^ and so on. The last divisor will be the greatest common divisor. 86 ELEMENTS OF ALGEBRA. 1. If any quantity contains a factor not found in the other, the factor may be omitted before beginning the process. 2. When necessary, the dividend may be multiplied by any quan- tity not a factor of the divisor. 3. The signs of all the terms of either dividend or divisor, or both, may be changed without changing the greatest common divisor. Find the greatest common divisor of the following: 3. 2a;2 — 16a! + 14 and x^^—bx — 14. 4. 3a;2 + 14x + 8 and ix^ + 19x + 12. 5. 6a:2 _ 23a; _)_ 15 and 2x2 _ i2x + 18. 6. 4x2 _^ 21x - 18 and 2x2 ^ i^^ ^ ig. 7. 21x2— 26x + 8 and 6x2— x — 2. 8. x2_6m/ + 82/2 and x2 —8x2/ + 161/2. 9. x^ — y^ and x2 — 2xy-\-y'^. 10. X* —2x2 + 1 and x* — 4x3 _j_ 6x2 — 4x + 1. 11. 2x3 _j_ 6x2 + 6x + 2 and 6x3 _j_ 6x2 _ 6a; _ 6. 12. 3x3 + 3x2 — 15x + 9 and 3x* + 3x3 —21x2 — 9x. 13. 20x4+x2 — 1 and 25.x* + 5x3 —x—1. 14. x2— 9, x2 — 3x— 18, and x2+llx + 24. 15. x2— 3x — 28, x2— llx + 28, and x2 _ 15a; _j_ 56. 16. X24-6X + 9, x3— x2 — 12x, and x2— 4x — 21. 17. a*— 6*, a3+a2j_(jj2_j3^ and a^ —2a%^ -\-h^. 18. X* + 5x3 _j_ 6a;2, a;3 + 3x2 + 3x + 2, and 3x3 + 8x2 _|. 5x + 2. 19. a3+3a26-|-3aS2 -|-63_ 4a262 _|_ i2a63 + gji^ and a2 — 62. 20. 9x* + 12x3+ 10x2 +4x+l ^nd 3x* + 8x3 + 14x2 + 8x + 3. 21. x* + 3x3 + 9x2 + 12x+20 and x« + 6x3 + 6x2 + 8a; + 24. 22. 3a2a;2 + a-^x + 2a2 + 12x2 + 4x + 8 and o2x2 + 3a2a! + 4a2 + 4x2 ^ i2x + 16. COMMON MULTIPLES. 87 COMMON MULTIPLES. 131. 1. What is the least number that ■will contain 10 and 15 ? 2. What prime factors are common to 10 and 15 ? What factor occurs in 10 that does not in 15? What factor in 15 is not found in 10? What are all the factors of 15 and those in 10 not found in 15? What is their product? 3. What quantity will exactly contain 2, 3, a and 6? What will each of them be of their product ? 4. What is the Jeast or lowest quantity that will exactly contain 3o and 4a6? 5. What factor of 3a is not found in 4a6? What is the product of 3 multiplied by 4a&? 6. To what, then, is the least or lowest common multiple of several quantities equal? DEFINITIONS. 132. A Multiple of a quantity is a quantity that will exactly contain it. Thus, a^x is a multiple of a, a^, and x. 133. A Common Multiple of two or more quantities is a quantity that will exactly contain each of them. Thus, ib^c is a common multiple of 26 and c. 134. The Least, or Lowest,* Common Multiple of two or more quantities is the least or lowest quantity that will exactly contain each of them. Thus, 26e is the least common multiple of 25 and c. * Common usage is followed in employing the term Lmst Common Multiple, although Lowest Common Muitiple would be the appropriate term to apply to literal quantities. 88 ELEMENTS OP ALGEBRA. 135. Principle . — The least common multiple of two or more quantities is equal to the product of the highest quantity multi- plied by all Hie factors of tfie oHier quantities not contained in the highest quantity. 136. 1. What is the least common multiple of Sx^y^zv and dx^y^z"^ ? PROCESS. Sx^y'^zv — 3 X x^ X y'^ X ^ X.v dx'^y^z'^ = 5 X a:^ X y^ X z^ L. C. M.. = 5xSXx'' Xy^ X^^ X,v = Idx'^y^z^v Explanation. — Since the least common multiple is equal to the product of the highest quantity multiplied by the factors of the other quantity not found in the highest quantity (Prin.), for convenience in determining what factors of the other quantity are not found in the higher, the quantities are separated into their prime factors. Thus, the factors of the least common multiple are seen to be 5, 3, x', y^, z^, and v. Hence, their product, 15x''y^z'v, is their least common multiple. 2. What is the least common multiple of a^ — a — 12 and a2— 4a — 21? PBOCEss. Explanation. — Since the product of any two (a^—a — 12) (a^ — 4a — 21) ,.,. . ^, . ^^ ^_ ^ quantities is their com- - _l_ Q mon multiple, it follows r A-. /■ n A iti\ that if their common fac- Ca — 4) (a^ — 4a — 21)= . ,, , tors are omitted from the ' product, the result will be the feosi common multiple. Since their common factors or divisors will be the greatest common divisor of the quantities, the product of the two quantities divided by their greatest common divisor will be their least common multiple. Their greatest common divisor is a-\-S; omitting this factor from dividend and divisor, the result is (a — 4)(a2— 4a — 12), which is equal to a^ — 8a^ — 5a + 48, their least common multicle. COMMON MULTIPLES. 89 KuLB. — Separate the guantities into iheir prime factors. Multiply the factors of the highest quantity by the factors of the otiier qvmdities not found in it; or, divide the product of the quaniities by their greatest common divisor. Find the least common multiple of the following : 3. Sa^J^cS and lOa^fec. 4. lOx'^y'^z, 20x^y'^z, and 25x^y^z^. 5. 14a^6^c2, Ib^x^y, and Sbahcx. 6. 12m^n'^y'^, 18mny^, and lim^n^y. 7. ISr^s^gS, 9r3sz2, and B6rs^z*. 8. x^ — y^ and x^ — 2xy-\-y^. 9. x^ — y^ and x^ -\- 2xy + y^- 10. x^ — y^, x^ — 2xy -{-y"^, and x^ -|- 2xy + 2/*. 11. x"^ — y"^ and a;^ — y^. 12. a^(x — s) and y'^(x'^ — z^). 13. a;2 — 1, a;2 _|_ 1^ and x* — 1. 14. 2x(x — y'), 4xy(x'^ — j/^), and 6xy'^(x-\-y). 15. x^ — X, a;* — 1, and ic' + 1. 16. a;2 — 1, a;2— a;, andxi* — 1. 17. 4(l+a;), 4(1— a:), and 8(1 — x^). 18. a;2 + 5a; + 6 and a;^ + 6a; + 8. 19. a2— a— 20 and a^+a— 12. 20. a;2— 9a; — 22 and a;^— 13a; + 22. 21. a;2— 8a; + 15 and a;^ + 2a; — 5. 22. x» + x'^y + xy'^ + y^ and x^ —x^y + xy''—y^. 23. x^—x^y + xy^—y^ and x» +x^y — xy^ —y^. 24. a*— 2a2 4-4a — 8 and a3+2a2— 4a — 8. 25. a;2+2/2, a;8— a»/2, and a;^ +a;2/2 +a;22/ + y». 26. a;2— 4, a;^— a; — 6, and a;» — Sa;^ — 4a; + 12. 27. a; — 5, a;^— 2aa; + a2, a;2_i0a; + 25, and a;" + 5a — 5a; — ax. 28. a;* — 16, a;2+4a; + 4, and a;^— 4. 8 FRACTIONS. 137. 1. "When any thing is divided into two equal parts, what is one part called? How is it expressed? What does \ represent? |? |? 2. What does - represent ? — ? — ? — ? 2 o I y 3. How may one-fifth of x be expressed? Two-thirds of 6? Three-sevenths of 2/? Eight-elevenths of 2? DEFINITIONS. 138. A Fraction is one or more of the equal parts of a unit. 139. The Unit of a Fraction is the unit which is divided into equal parts. 14:0. A Fractional Unit is one of the equal parts into which a unit is divided. 141. Since a fraction is one or more of the equal parts of any thing, to express a fraction two numbers, or quantities, are necessary, one to express the number of equal parts into which the unit has been divided ; the other to express how many parts form the fraction. These numbers, or quantities, are written one above the other, with a horizontal line be- tween them. (90) FRACTIONS. 91 142. The Denominator is the number, or quantity, which shows into how many equal parts the unit is divided. It is written below the line. Thus, in the fraction — > 6 is the denominator. It shows that the unit of the fraction has been divided into 6 equal parts. 143. The Numerator is the number, or quantity, which shows how many fractional units form the fraction. It is written above the line. Thus, in the fraction — i o is the numerator. It shows how many fractional units form the fraction. 144. The numerator and denominator are called the Terms of a Fraction. 145. An indicated process in division may be written in the form of a fraction, the numerator being the dividend and the denominator the divisor. 146. A quantity, no part of which is in the form of a fraction, is called an Entire Quantity. Thus, 2a, 3c, 2x + y, etc., are entire quantities. 147. A Mixed Quantity is a quantity composed of an entire quantity and a fraction. Thus, 2a -| 1 2x-\-2y-\ 3^, are mixed quantities. 7 s; + 7 148. The Sign of a Fraction is the sign written before the dividing line. This sign belongs to the fraction as a whole, and not to either the numerator or denominator. Thus, in — ^LjlM the sign of the fraction is — , while the signs of the quantities x, y, and 2z are +. The sign before the dividing line shows whether the fraction is to be added or subtracted. 92 ELEMENTS OF ALGEBRA. REDUCTION OF FRACTIONS. CASE I. 149. To reduce fractions to higher or lower terms. 1. How many fourths are there in one-half? How many eighths ? 2. How many sixths are there in one-third? How many ninths ? How are tlie terms of the fraction f obtained from i? ffromi? 3. How many fourths are there in — ? — ? — ? •^ 2 2 2 4. How many sixths are there in — ? How many ninths ? o How are the terms of the fraction — obtained from its 6 equivalent — ? ■ — from — ? ^ 3 9 3 5. What, then, may be done to the terms of a fraction without changing the value of the fraction? 6. How many fourths are there in |-? In f ? In -f? 7. How many thirds are there in f? In f? In -j^? How are the terms of the fraction \ obtained from ■§■? Fromf? Fromj^? 8. How many thirds are there in — ? In — ? In — f 6 9 12' How are the terms of the equivalent fraction — obtained from these fractions? 9. What else may be done to the terms of a fraction, besides multiplying them by the same quantity, that will not change the value of the fraction? REDUCTION OF FRACTIONS. 93 150. Reduction of Fractions is the pwcess of changing their form without changing their value. A fraction is expressed in its Lowest Terms when its numerator and denominator have no common divisor. 151. PEraciPLE. — Multiplying or dividing both terms of a fradwn by the same quantity does not change the value of the fraction. EXAMPLES. 3 1. Change — to a fraction whose denominator is Gb"^. „, ,„ Explanation. — Since the fraction is- PROCESS. 3 to be changed to an equivalent fraction "oj expressed in higher terms, the terms of gjz _i_ 26 = 36 *^® fraction must be multiplied by the ^ "V ^h ^h same quantity, so that the value of the —' rr = ^t; fraction may not be changed (Prin.). In order to produce the required denomina- tor, the given denominator must be multiplied by 36 ; consequently, the numerator must be multiplied by 36 also. 2. Eeduce — to its lowest terms. 25x^y PROCESS Explanation. — Since the fraction is to „ be changed to an equivalent fraction ex- ^ = -^ pressed in its lowest terms, the terms of the 20X y ox fraction may be divided by any quantity that will exactly divide them (Prin.). Dividing by the factors 5, x*, and y, the expression is reduced to its lowest terms, for the terms are prime to each other ; or, The terms may be divided by their greatest common divisor. 152. To express a fraction in higher terms. EuLE. — Multiply the terms of the fraction by such a quantity as vMl change the given term to the required term. 94 ELEMENTS OF ALGEBRA. 153. To express a fraction in its lowest terms. Rule. — Divide the terms of the fraction by any common divisor, and continue to divide thus until they have no common divisor; or, Divide the terms of the fraction by their greatest common divisor. Q 3. Change — ^ to a fraction whose denominator is 28. 4. Change to a fraction whose denominator is 36. 5. Change — ^ — to a fraction whose denominator is 15. 3.1; 4- 7 6. Change to a fraction whose denominator is 30. 2x 7. Change ■ to a fraction whose numerator is 6a;. 6x -f 3 Sx 8. Change to a fraction whose numerator is 9.1; 6x — 8 2ax 9. Change to a fraction whose numerator is 4axK B + 2y 2a -\- X 10. Change —r- to a fraction whose denominator is a-{-b 11. Change ^ to a fraction whose denominator is a + a2+2a6 + 62. Efiduce the following to their lowest terms: j2 ^Sa^y'^ . IbxyH ^g 21sVa2_ ■ 2&x^yH^' ^ , lOabx'^ii 14. ^- 25abx^y^ , f. IGxyz^m 24a-2y2gj^8 16, REDUCTION OF FRACTIONS. 95 12mH* _ 24x^yHs 12x^y»z*' 49a;4y5g8 22a^x^yz* 18. 19. 20. 21. a2 4- 2a6 + 62 22. 23. 24. 25. 26. 27. a;2 4- 2a; + 1 a;2 — 1 2xy + 2y a;* — a^a; a;2— 2aa; + a2 j;5 j.32^2 x*-2/4 a;2 + 6a; + 9 a;3 _ a;2 _ i2x a;2-- 3a; — 28 a;2 — 11a; + 28 CASE II. 154. To reduce an entire or mixed quantity to a fraction. 1. How many fifths are there in 3? In 4? In 10? In a? In X? 2. How many sevenths are there in 2? In 4? In 6? In 62? In 2/2? 3. How many fourths are there in 2J? In 3|? In a-[-— ? EXAMPLES. 1. Reduce a + - to a fractional form. Explanation. — Since 1 is PBOCESS. ac equal to — , « is equal to —■ a = — ^ e c C b ac b b__M 6^ _ ae + b consequently, a + - = -^ + -^ c 96 ELEMENTS OF ALGEBRA. Rule. — Multiply the entire part by the denominator of the fraction ; to this product add the numerator when the sign of tlie fraction is plus, and svhtract it when it is minus, and write the residt over the denominator. If the sign of the fraction is — , all the signs of the numerator must be changed when it is subtracted. Reduce the following to fractional forms : 2. -+^ 3. -!• 4. 4.--. 5. 4 6. ..+-|^. 7. -+V- 8. 6 9. 10. 5a ^^ + ^- 5a 2 6a 'y + '. 4 11. 3c + 4'*+^ d 12. 4a -1 ; — 13. „ , 6a — X 3a; -| ■• O/JC 14. .+4+ ^ . 15. 2ac — c2 a 16. 2. 5 ^^+4. ic— 2 17. „,„,«'+«:' a 1 X 1 a — a; 18. , 2ac — c2 a + c- a — c 2 „2 19. a; — 2/ + ^ -^• 20. a; + 4- 21. a + x- 22. a — 6_ x' a; — 4 4aa; — 5a;^ a — a; 02+62 23. m-\-n- a — 6 2mn 4" w^ m — n REDUCTION OF FRACTIONS. 97 CASE III. 155. To reduce a fraction to an entire or a mixed quantity. 1. How many units are ttere in -J^? In -ij^-? In y? 2. How many units are there in -f? In -^P In AjJ-? 3. How many units in ■ ? In ? In ■ — ? EXAMFI^ES. bx -4- d 1. Keduce — — — to a mixed quantity. PROCESS. 6a; + d = x + - Explanation. — Since a fraction may be regarded as an expression of unex- ecuted division, by performing the di- vision indicated, the fraction is changed into the form of a mixed quantity. Reduce the following to entire or mixed quantities: 3. 4. 5. 6. 7. 8. a2 +c2 a bx + ed b 2o6 + 62 a + b a2— a;2 a — X a2— a;2 a + x a;8 + l a; + l x^+1 x—1 9 9. 10. 11. 12. 13. 14. 15. x^ -j- lax -f- x"^ a-\-x a8 -f 68 a — 6 5^2/ -\- ax-\-x ax 2ai — 262 a + b x^ + 2xy + 2y' + x x + y a — 6 x + y 98 ELEMENTS OF ALGEBRA. OASB IV. 156. To transfer a factor from one term of a frac- tion to the other. 1. To what is the reciprocal of any quantity equal? 2. To what is any quantity with a negative exponent equail ? 3. Change — to an equivalent expression which is not in the reciprocal form. 4. Change sr^ to the reciprocal form. 157. Peinciple. — Any quantity may be changed from one term of a fraction to the other by changing the sign of its exponent. EXAMPLES. 1. Change -r— -r to an equivalent expression in the form of an entire quantity. PROCESS. 5i^ = «'*' X 62^ 62c2 a^x"^ X -r—; = aH'^ X b'^ir"^ = aH'^b-^er^ ExPLAiTATiON. — Since is equal to aH^ V , and = h-'c-' ( Prin.), -— -^ equals a^i^ X 6-^c-^ which is a'^x'^b-^c-^. Rule. — Change the factors from one term of the fraction to the other and change the signs of the exponents. REDUCTION OF FRACTIONS. 99 Express the foUowing in the form of entire quantities: A 2. CUi^ 7. tt'^-X g cxy — z cxz g x^ — 2x!, + y" x^ a-\-x l_y2 10, n. a^'—x^ x^ — y^ x^y^ Change the following to equivalent quantities having posi- tive exponents : 12. 13. 14. 15. 3a;- ,. 4(a — a;)-" x-*y-' ID. — ^ = a — X 4ac-2 17 (?'''+ y^)(?o-yr^ 3a-i (a;2_2,2)-l(^^y) Baay 18. ^(^-^)-^ . (^ + 3) {a^— }>■>■) {a- -c)- -1 19 7(a; + 2/H-8)-V (a + c)(a2 + h^r -1 5(a;_y_g)-i CAS B V. 158. To reduce dissimilar firactions to similar frac- tions. 1. Into what parts may ^ of a dollar and ^ of a dollar be divided so that the parts may be of the same size? 2. Into what fractions having the same fractional unit may \ and \ be changed ? 100 ELEMENTS OF ALGEBBA. 3. Into what fractions having the same fractional unit may \, \, and ^ be changed? Express the resulting frac- tions in equivalent fractions having their least common denominator. 4. Into what fractions having the same fractional unit may — and — ■ be changed? 2a 5a 5. Into what fractions having the same fractional unit may — , — , and — be changed? Express the resulting 3a 4a 6a fractions in equivalent fractions having their least common denominator. 6. Express --> — , and in equivalent fractions 2a 5a 10a ^ having their least common denominator. DEFINITIONS. 159. Similar Fractions are those which have the same fractional unit. 160. Dissimilar Fractions are those which have not the same fractional unit. Similar fractions have, therefore, a common denominator. 161. When simUar fractions are expressed in their lowest terms, they have their Iicast Common Denominator. 162. Peinoiples. — 1. A common denominator of two or more fractions is a common multiple of their denominators. 2. 2%e least common denominator of two or more fractions is the least common multiple of their denominators. REDUCTION OF FRACTIONS. 101 EXAMPIiES. d 2e . . 1. Reduce — and to similar fractions having their 2ae SaH ^ least common denominator. PKOCESs. Explanation. — Since the d d X8ad 3ad^ ^^^* common denominator of seyeral fractions is the least 2ae 2ac. X 3ad Ga^cd 2c 2c X 2c 4c2 common multiple of their de- nominators (Prin. 2), the least Sl^d Sa^d X 2c Qa^cd common multiple of the de- nominators 2ac and Sa'd must be found, which is Ba^cd. The fractions are then reduced to frac- tions having the denominator Ga^cd, according to Case I, by multi- plying the numerator and denominator of each fraction by the quotient of 6a'cd, divided by the denominator of each of the given fractions. 6a^cd-i-2ac = 3ad, the multiplier of the terms of the first fraction. 6a^cd -H Sa^d = 2c, the multiplier of the terms of the second fraction. EuLE. — Find the hast common multiple of the denominators of the Jraetions for a least common denominator. Divide this denominator by the denominator of each fraction, and multiply the terms of the fraction by the quotient. 1. Any multiple of the least common denominator will be a com- mon multiple of the denominators. 2. All mixed quantities should be changed to the fractional form, and all fractions to their lowest terms before finding their least common denominator. Reduce the following to similar fractions having their least common denominator: 2. - and -• „ 7a J 5a 3. — and — • 8 6 102 3^^ ,^^ 2^ ELEMENTS OF ALGEBRA. 3 4as 5e 16 6. — and -^• 3a 6a fa. — and Zy 6?/ 2 „ 3ac - 26c? 7. :^^— and 2a22/ 3a;2a 8. ?^=i^and§^=^. 5a!2 10a; _ 4a + 56 , 3a + 46 9. — and 3a2 4a 11. 2a^ 4x^y Byx^z 12. ±, Jl, ^_. 4a; 4a^ 8x^y^ 13. A, ^, 4. 14. 15. 16. 10. ^^-^^^ and ^-^2'. 5ac lOa^c 17. 18 ^^-1,^^ + 1, ..d -* + l, ^2 + 1 a;2 — 1 a;* — 1 -in 1 — J 1 x + y^ a;— y x^ -\-y i 4 ' 2o ' 2a ' a; + 2 a;— 2 a; +3 a; — 1 a;+l a;^ — 1 x'^y xy xy^ a + b a—b a^—b^ x-\-y X — y x^-\-y^ X — y x-\-y x^ — y^ (a—b) (b—c) (a—b) (a—c) CLEARING EQUATIONS OF FRACTIONS. 163. 1. Ten is one-half of what number? 2. If one-third of a number is 12, what is the number? 3. If ^x equals 4, what is the value of a;? 4. If ^a; = 8, what is the value of a;? 5. If both members of an equation are multiplied by the same quantity, how is the equality of the members affected ? 6. When ^ = 6, what is the resulting equation when each FRACTIONAL EQUATIONS. 103 member is multiplied by 3? How is the equality of the members affected? 7. When— ^10, what is the resulting equation when o each member is multiplied by 5? How is the equality of the members affected? 8. Change into an equation without the fraction — =:= 5 ; 1=6; 1 = 12, 1 = 20; 1 = 10. 9. How may an equation containing fractions be changed into an equation without fractions? DEFINITIONS. 164. Clearing an equation of Fractions is changing it into another equation without the fractions. 165. Peinciple. — An equation may he cleared of fractions by multiplying both members by some multiple of the denominor tors of the fractions. (Art. 59, Ax. 3.) 1. Find the value of x in the following x-\—p = 12. PROCESS. Explanation. — Since y. the equation contains a * ~r "^ ^^ ■'••^ fraction, it may be cleared of fractions by multiply- Clearing of fractions, 5a; + a; = 60 jng both members by the Uniting terms, 6a; = 60 denominator of the frac- Therefore, a; = 10 tion(Prin.). The denom- inator is 5; therefore both members are multiplied by 5, giving as a resulting equation 5x + a; = 60. Uniting similar terms, 6a; = 60; therefore, 1 = 10. 104 ELEMENTS OF ALQEBBA. 2. Given x-\-~-\-—-\- -=—-pr, to find the value of x. 3 5 6 10 PEOCESS. ''+3+5+6~10 Clearing of fractions, 30a; -\- lOx -\- 6x -}- 5x ^= 459 Therefore 51a; = 459 And, a; = 9 Explanation. — Since the equation may be cleared of fractions by multiplying by some multiple of the denominators (Prin.), this equation may be cleared of fractions by multiplying both members by 3, 5, 6, and 10 successively, or by their product, or by any multiple of 3, 5, 6, and 10. Since the multiplier will be the smallest when we multiply by the least common multiple of the denominators, for convenience we mul- tiply both members by 30, the least common multiple of 3, 5, 6, and 10. Uniting terms, and dividing, the result is a; = 10. Rule. — Multiply both members of tlie equation by the least common multiple of the denominators. 1. An equation may also be cleared of fractions by multiplying each member by all the denominators. 2. If a fraction has the minus sign before it, the signs of all the terms of the numerator must be changed when the denominator is removed. 3. Multiplying a fraction by its denominator removes the denom- inator. Find the value of x, and verify the result in the following: 3. a; + - = 24. 5 4, - + a; = 21. 6 5. 2:6 + 1 = 28. 6. 4a; + - = 42. 5 FRACTIONAL EQUATIONS. 105 7. 8. 9. 10. 11. 12. 3a;-^ = 40. 7 x—- = 25. 6 — a; = —24. + 7a; = 38. 2^3^4 3^4^6 13. x+| + ^=:29. o 4 14. 2a; + |-| = 50. 15. 3a;-?^-^=18. 3 6 16. 4a; + - 74. 17. 3a; — - + — = 70. 6 ^12 18.^ + ^ + ^ 4^6 8 26. 19. 20. 21. 22. „. a; + 9 , 2a; 3a;— 6 , „ ^ „ , Cjriven \-—-^ \- 3, to find x. „. 3a; + 4 , 4a! — 51 „ , ^ , Ciriven • — 1 ;= 0, to find x. _,. 3a; , 180 — 5a; „„,£■, Given ■ = 29, to find x. 4 6 Given —x -\ x -A — x^ 19, to find x. 4 10 8 _,. a; + 3 , 3a; „ . 4a; — 5 . „ a Given — T7-+ — = 2H — , to find x. 2 5 o 24. Given 25 26, 27, a;+2 . a;— 1 _ a; — 2 , to find X. 7a; + 2 ^„ 3a; + 3 x ^ . , Given ^^ 12 = -—^ -, to find x. 10 ^ Given — +a — ^ = a + |+5i, to find x. 4 5 5 Gi,en 2£±i_3^ =.^ + 42^ to find .. 106 ELEMENTS OF ALGEBRA. 28. Griven = H , to find x. 2 8 ^ 16 29. Given ^^~^ + ^^L±l ^ 5a;— 17^, to find x. or, n- 5 — 3a; , 3 — 5a; 3 5a; , „ , 30. Griven — = , to and x. 4 3 2 3 31. Given ^ + ^ + ^ = 16, to find x. „_ „. 2a;— 1 , 6a; — 4 7a; + 12 ^ „ . 32. Given • — 1 — == ■ — + — -, to find x. o 7 11 33. Given ^i- — ?^ =a;_2, to find a;.* o 5 34. Given — — = , to find x. 4 9 6 35. Given ^ + 3 = ^ - ^=^, to find x. 2 4 5 oe n- * — 1 , a;— 3 a; — 2 2 „ , 36. Given — 1- — — = - to find x. „„ „. 1— 2a; 4 — 5a; 13 , ^ , 37. Given — — = — — , to find x. o 4 42 ^« rurnr, ^ + 3 a; — 2 1 3a; — 6 , ^ ^ 38. Given — _== ____^ to find x. .,„ P,. 4a; — 2 , . 3a; — 5 ^ „ , 39. Given — — [-4 — — = 5, to find x. XX lo .„ p,. 3a; — 3 3a; — 3 15 27+4a; 40. Given =r ' to find r 4 3 3 g , 10 nna a;. *Iii clearing this and the following equations of fractions, the signs should be changed, as indicated in Note 2, under the Kule. FRACTIONAL EQUATIONS. 107 41. A spent \ of his money, and then received $2. He then spent ^ of what he had, and had $7 remaining. How much had he at first? PEOCESS. Let X = the money he had at first. 3J Then, — = what he spent at first. 3a: (- 2 =: what he had after he received $2. 4 — 1 1-21= hi = what he spent the second time. 2\ 4 ^ / 8 ^ ^ Therefore, - + ^+l + 7 = a; + 2 4 8 Clearing of fractions, 2a; + 3a; + 8 + 56 = 8a; + 16 Transposing, 2a; -f 3a; — 8a; = — 48 — 3a; = —48 «= 16 42. What number is there to which, if J of it be added, the sum will be 15? 43. Find a number such that the sum of ^ of it and \ of it is 15. 44. One-third of A's age plus two-fifths of A's age equals 22 years. How old is he? 45. Three sons were left a legacy, of which the eldest received f , the second |, and the third the rest, which was $200. How much did each receive? 46. A's capital was | of B's. If A's had been $500 less, it would have been but ^ of B's. What was the capital of each? 47. A horse and carriage cost 1420. K the horse cost f as much as the carriage, what was the cost of each? 48. A had twice as much money as B, C 1\ times as 108 ELEMENTS OF ALGEBRA. much as A, D \ as much as A, and they all had $50. How much had each? 49. What number is there, \ of which is 3 greater than \ of it? 50. A clerk spent \ of his salary for board, \ of the rest for other expenses, and saved annually $280. What was his salary? 51. There is a number such that, if \ of it is subtracted from 50, and the remainder multiplied by 4, the result will be 70 less than the number. What is the number? 52. Divide 100 into two parts such that, if \ of one part be subtracted from \ of the other, the remainder will be 11. 53. There are two numbers whose difference is 1, such that ^ of the first plus \ of the first is equal to the sum of \ of the second and \ of the second. What are the numbers? 54. Five years ago A's age was 2^ times B's. One year hence it will be If times B's. How old is each now? 55. The difference between two numbers is 20, and -f of one is equal to \ of the other. What are the numbers? 56. When the sum of the fourth, fifth, and tenth parts of a certain number is taken from 33 the remainder is nothing. What is the number? 57. The difference between two numbers is 8, and the quotient arising from dividing the greater by the less is 3. What are the numbers? ADDITION OF FRACTIONS. 166. 1. What is the sum of — and -? Of — and— ? 9 9 11 11 Ofi^and-2?? 6 6 ADDITION OF FRACTIONS. 109 2. "What is the sum of - and -? — and -? — and -? 3 6 4 8 3 6 3. What kind of fractions can he added without changing their form? 4. What must be done to dissimilar fractions before they can be added? How are dissimilar fractions made similar? 5. What is the sum of — and A? ^ and ^? 3a 3a 2xy 2xy 6. What is the sum of ^^^ and ^^=^? Of -^ and 3c 3e ^-\-y 3«., Of -^ and ^ x + y a-\-b^ a4-62 3 3 5 7. What is the sum of — , — , and — ? 2x 4x 6x 167. Principles. — 1. Only dmilar fractions can be added. 2. Dissimilar fractions must be reduced to similar fractions b^ore adding. EXAMPI.ES. ^ -TTT, . ■, „ 5a 3a , 26 „ 1. What IS the sum oi — -, — , and — / PEOCESS. 5a , 3a , 26 30a , 27a , 86 57a-|-86 , 21a + 86 •"= = ; or, a-\ ■ 6 4 9 36 36 36 36 ^ 36 Explanation. — Since the fractions to be added are dissimilar, they must be made similar before adding. The least common denominator of the given fractions is 36. 5a 30a 3a 27o ^ 25 86 ~, „ ., . . 57a + 86 uu, ^>\ju, ou, ^ii^ J — __ Therefore their sum is — ; , 6 36' 4 36' 9 36 36 ' . . , 21a+86 which, expressed as a mixed quantity, is a-\ — 110 ELEMENTS OF ALGEBRA. 2aa; 3a; 2. Find the sum of a-\ — — and 3a -| 7 s PEOCESS. a -(- 3a =: 4a 2aa; , Zx 2axi , Zxy 2aaB-|-3a!M Entire sum ^4a- 2aoi»-\-3xy 7s Rule. — Reduce the given fractions to similar fractions. Add their numerators, and write the sum over the common denominator. When there are entire or mixed quantities, add ike entire and fractional parts separately, and then add their results. Find the sum of the following: 3. - and y- y 2 , 2d J 3a 4. — and — c ca 5. — and — • 22 3aa 6. ^ and ^. oa &ax „ 1 , 2x 7. - — and — -• oay oby Q 4a J 5a2 8. — — and Zxy 15. Add 3aa;^y 9. and a-\-x .„ 1 -l-a; 1- 10. and 11. 1+x 1—x^ and -X 2 12. T^^ and ^ l+a;2 2 -X' 13. 14. 1— K* a; x'^—y 2 l+a' 2 and y and a2+l 1+a a-{-a x'^ — 1 X — 1 and a; + l SUBTRACTION OF FRACTIONS. m 16. Add a + — and 2a — 6- a^ — h^ ' (a — 6)2 17. Add -^, -^, in^, and «^ a — b a-\-h a-\-h a^ — 6^ 18. Add y'-'^^-^' X 2 . and -^ ■xy X — y 19. Add , ^ , and 1. 2(x—l) 2(a; + l) x^ 20. Add . ^+'" and ^ ~ "^ 1+x + x^ 1—x + x^ SUBTRACTION OE FRACTIONS. 5 3 7 3 168. 1. From — subtract — From — subtract — 9 9 11 11 „ 3ffl , , , 2a _ 3a;2 , ^ ^ a^ irom — subtract — . From subtract 6 6 8 8 2. What is the difference between — and — ? Between 3 6 — ' and — ? Between — and — ? Between — and — ? 4 8 3 6 5 10 3. What kind of fractions can be subtracted without changing their form? 4. What must be done to dissimilar fractions before they can be subtracted? How are dissimilar fractions made similar? 5 2 5. What is the difference between -- and — ? Between 3a 3a 6a J 3a „ _ , lax . 3aa; , and — ? Between — — — - and 2xy 2xy 3(a + b) 3(a+b) 112 ELEMENTS OF ALGEBRA. 6. What is the difference between ^ ^ and "' ' ^ ? 3e 3c Between -^ — and —— — ? Between ■ and x+y ^+y Qm + yr ix + yy 3 8 7. What is the difference between — and — ? Between 4x 2x — and — ? Between — and — ■? 4x ox 4a 8a 169. Principles. — 1. Only similar fractions can be sulh traded. 2. Dissimilar fractions must be reduced to similar fractions before subtracting. EXAMPLES. 1. Subtract — from 76 11a PROCESS. 66 2a 4262 22a2 4262 — 22a2 11a 76 77a6 77a6 77a6 Explanation. — Since the fractions axe not similar, before sub- tracting they must be changed to similar fractions. The least common denominator of the fractions is 77 ah. Therefore. = '11a 426* 2a 22a* — — , and —- = rr-r' Subtracting the numerator of the subtrahend 77a6 76 77a5 426* 22a* from the numerator of the minuend, the remainder is 77a5 2.^ From 6a + ?^=^ take 2a + ^-^^ . SUBTRACTION OF FRACTIONS. 113 PROCESS. 6a — 2a^4a Sx — 2a 4a — 3x Sx^ —2ax 4a^ — Bax a X 3x^ ~\- ax — 4a^ aa; ax ax _1 , 3a;^ — 4a2 — ■■■ 1 aa; = 4a + 14- 3a;2 — 4a2 Explanation. — Since the quantities are mixed quantities, the entire quantities and the fractions may be subtracted separately, and the results united. Rule. — Reduce ihe given fractions to similar fractions. Sub- tract iJve numerator of the subtrahend from the numerator of ihe minuend, and place the resuit over the common denominator. When there are entire or mixed quantities, subtract the entire and fractional parts separatdy, and unite tlie results. Subtract: Subtract : „ 3a . 5a 3. — from — • 5 6 •9. 2a6 „ 5ad ■—- from - — 3a^ 2a^ >• 2a; „ 3a; 4. — from — • 7 5 10. Bmn „ 2mn , „ from , • 4y^ 4y _ 4a „ 3a 5. — from — -• 76 # 11. a-\-h „ a — 6 — — from ■ 3 2 „ 2x „ 5x 6. — from — • 4a 9a 12. 3 ^ 2 from —■ a -j- 6 a — „ Bd „ Id 7. — from - — ax 2ax 13. from x—1 x+1 8. from — • 2ad ad 10 14. x+1 ^ x—1 — ■ — from -• x—1 x + 1 114 ELEMENTS OF ALGEBRA. X X 15. From subtract » — 3 x-\-3 16. From ^^+^ subtract ^^^Ilf- ._ _ . , 2a — 36 ,^ ^ „ 3a + 26 17. From Gx -j subtract 3x 5a 6a 18. From 7x -\ subtract 3x y y Simplify the following expressions: j9 2a; + 5y 4an/ — 3y2 _ bmi — 2y^ „„ 3a6 — 4 6a2 — 1 56^+7 21.^ e!_+ - 1 — a; 1 — a;2 1 -\- x^ 22. -^ + 4a 5d2 a; — a (x — a) ^ (x — o) * 23 3 5 2a;-^7 ' X 2x—l 4a;2 — 1 24 x^ +y^ _ _£f y^ xy xy -\-y^ x^ ^ xy MULTIPLICATION OF FEACTIONS. CASE I. 170. When the multiplier is an entire quantity. 3 2a 1. How many fifths are 6 times -? 5 times — ? 5 5 MULTIPLICATION OF FRACTIONS. 115 2. How many times — is 5 times — ? 7 times — ? c c c 3a times — ? 3d times — ? c c 3. How may a fraction be multiplied by an entire quan- tity? 4. What effect upon a fraction has multiplying its nu- merator ? n 5. Express 2 times — in its lowest terms. How may 8 the result be obtained from the terms of ^? 8 6. In what other way, besides by multiplying the nu- merator, may a fraction be multiplied ? 7. How much is 3 times — ? 4 times — ? 6 times — ? 6 Oct 1-i 8. How much is 5 times -^ ? 6 times — -? 9 times — ? 7 5a 00 TmsciPi,E.— Multiplying the numerator, or dividing the de- nominator of a fraction by any quantity, mvMplies the fradim. by that quantity. EXAMPUSS. 1. Multiply - by m. a PKOCESS. Explanation. — Since a fraction is multiplied by multiplying its numerator n mn (Prin.), - is multiplied by m by multiply- - X m = — " d a a -TT 1 J i • ""* ing the n by m. Hence, the product is -— • 116 ELEMENTS OF ALGEBRA. \ 2a 2. Multiply by a;2. 2a X x^ = PEOCESS. 2a x^y- 2a y Explanation. — Since a fraction may also be multiplied by di- viding its denominator (Prin.), — may be multiplied by x' by x'y dividing the denominator by x', since it is a factor of the denominator. 2a Hence, the result is — y Rule. — Multiply the numerator, or divide the denominator by Hie multiplier. It is often best to indicate the multiplication, and then cancel equal factors from both numerator and denominator. Multiply: 3. - by z. 4. — by X. 2/2 5. — by a. a 6. 7. m^n , — — by n'a. ah 8. by a^lc. a^b 9. a,,„.,. Multiply: Zax 10. 11. 12. 13. 14. 15. x + y by 2aa;. 462/ 3(a -)- x) 5o'a;2 2(c + d)2 3a + & 3(0; + 2/) 202 4(m — n^). 4c2c?2 by 2ay. by (c + • -J 3aw+cjn , „ , „ 15. Divide ■ — — by x^+y^. x^ — y^ 16. Divide ^^^+i^ by a + 6. c^ + d^ ^ ^ 17. Divide ^'^ + ^'^ by 5(x + z). a-\- n DIVISION OF FBACUONS. 123 CASE II. 173. When the divisor is a fraction. 1. How many times is - contained in 1? -? -? -? 8 8 8 8 2. How many times is - contained in a? — ? — ? — ? 8 8 8 8 3. What is the value of 1-=-^? !-=---? l-~-~2 1-=--? 8 9 8 9 4. How many times is -■ contained in — ? - in — ? 4 4 4 4 a . 2a a . 2a„ 2a . 4a. 3a . 6a„ - m — ? - m — ? — m— ? — m — ? 33557 78 8 5. How many times is - contained in -? - in -? 4 2 8 2 a , a. a . a_ a . a„ — m -? -m-? - m -? 16 2 6 3 9 3 EXAMPIiES. A 1. What is the value of -^--? c d PEOCESS. _ 1 . . , . ExPLAiTATioN.— — IS Contained in 1, a_^6 *v- — 6 1 g ■ ^ „^ I "IT — c c times — ) or — — times. Or, — is equal to —-i and — is eqnal 6 6c c cd d be be . . . , . ad ad ^. to - - ; — - IS contained in — - > — — times. ed ed ad be 124 ELEMENTS OF ALGEBRA. Edie. — MtMiply the dividend by Ihe divisor inverted. 1. Change entire and mixed quantities to the fractional form. 2. When possible, use cancellation. Divide : 2. - by — 6 y „ cac . cd 3- ;— oy r— by 3y 4. ?^ by ^. 4a ^e Sac _ 4a'a; , 2a^a;^ 5. ^rt-r by &dy^ 6.1^ by Sad ^ Sa'^y 2xf_ Sa^d bx^y^z - lOxy^z^ 6a262 8a62c2 Divide 8, by &bcd ^ IGbcd Q ^ by ^^^ cmn dmn 10. 11. 13. mny' by ahe 5xy >>T7 a — X Dy 2aa;-f x^ as — »» m2 — n^ 12. r^:_^;i. by a^b^e^ lOxy a; a — X by.- 3ot -(- n 12 14. Divide ^^i^c^+e^d + c, 12 -^ 8 15. Divide 16. Divide 17. Divide 8a« fflS— 63 4 x^ + ^* 4_J4 2a6 by by by 4a2 8 -Z.2 4a262 18. Divide a -\ — by — d y 19. Divide c + dy by aa/. DIVISION OF FBACTIONS. 126 20. Divide ^ by ^ 21. Divide by * «* — 6a; — 6 «*+« 22. Divide -^^- by ^"^ a;2— 7» x^ — lBx-\-A2 23. Divide ^^^ by ^ x^ — 5x ■' x^ — llx + 30 24. Divide &i;i»— - by 2» + -- 4 4 25. Divide a^ + ^hj a + -- ^2 -^ ^2 26. Divide x^ + ^hj y + - X y 27. Divide -J— by ^^^^^ ^2 J.2 ^(, j.^2 6* + 1 1 28. Divide ^-^ by (6 + f-— 1). 29. Divide i— - by -^ 30. Divide ; by ■ — ^• 31. Divide 2_ by i -f!-- a-\-xff a^x^ — x*y' 174. ExpreBsions which have a fraction in either the numerator or denominator, or in both, are called Com- plex Fractional Forms. They are simply expressions of division. 126 ELEMENTS OF ALGEBRA. 32. Find the value of the expression — c 1 a e d PROCESS. a c a b c ad Explanation. — Fractional forms are simply expressions of diyision ; and, therefore, the given fractional form is the same as though it were writ- ten i — . Performing the division according to the principles b d ad already given the quotient is, be Find the value of the following: 33. 34. ^ + - x + - 35. 3«''-3y^ « + y 3 36. 37. 38. 4x — 4i/ bah 5x — By bxy 4ax x^ — y^ 4a2 — 4a;^ a-\-x -X 39. 40. 41. x + 2d Bac =^ + Bd 2ae x^-y- 2^. a: — 3y 2 Bx xy ao ■ + 2c REVIEW OP FRACTIONS. 176. Keduee to their lowest terms: a;3_6ii;i8-|-lla; — 6 a;3_2a;2_a;-f 2 m^ -\- m^ -\- m — 3 m» + 37n2-j-5,ft_(-3 x* — x^—4x'^—x + l. 'ix^ — Bx^ — Sx — l a8 — 7tt^-fl6a— 12 3a« — 14a2 + 16a REVIEW OF FRACTIONS. 127 Find the value of the following : X X , x^ 5. X-{-l 1 — X x^ — 1 3 + 2a; 2 — 3a; 16a; — a;^ 2 — a; 2+a;a;2— 4 1 , 1 Q>! — y)(y — e) (y — a;)(x — a) (» — 3)(» — 2/) 1,1,1 10. 11. 12. 13. a(a — 6) (a — c) 6(6 — a)(b — c) c(c — a)(^c — 6) i+^±^W(i-^V m — yl \ x + yf '-'+t+7)^('-'+ht)- .. + i+.)x(i-i + i)- a + 1 a — -^ \ . g a — 1 a -j-1 / a — 1 t)} 1 + ^ i + ^iz£ i + ^i=-^-i 14. — 1_£=^ 1_ a + as a^+a;* SIMPLE EQUAT10]N"S. 176. Review. — 1. Definition of an Equation. 2. Definition of Members of an Equation. 3. Definition of First Member; Second Member. 4. Definition of Clearing of Fractions. 5. Definition of Transposing. 6. Definition of an Axiom. 7. Definition of a Statement of a Problem. 8. Definition of a Solution of a Problem. DEFINITIONS. 177. The Degree of an equation is determined from the highest number of factors of unknown quantities contained in any term. Thus, i + 6 = c, 3ax-\-y = n, 46^a; + 3a'a; = a, are equations of the first degree. a;* + a^c, bx^ -\- Sy = d, x-\-xy = 7, axy-\-3y'=n, are equations of the second degree. x'^a, x'y = a, xy'^a, x -{■ x^ -\- x' =- a, are equations of the third degree. 178. An equation of the firnt degree is called a Simple Equation. 179. An equation of the second degree is called a Quad- ratic Equation. 180. An equation of the third degree is called a Cubic Equation. (128) SIMPLE EQUATIONS. 129 181. An equation in which all the known quantities are expressed by figures, is called a Numerical Equation. 182. An equation in which some or all of the known quantities are expressed by letters, is called a Literal Equation. « X:XAMFI.ES. Find the value of x in the following: 2 ^ 2. 3a; + 4 x , x — V2 3 9 ' 6 3. 2 ' 4 4. o « + 2 X x — o = — • 8 3 5. 15a! 3-« 4 -^ 2 6. "^ x-"-^ 9. 3 11 7. 9^ ^ + ^-2. 21. 7 5 8. ax — 6 , x-{-ae ^ + a — e e 9. 3a — 6a; 1 aa; == — ■ 2 4 10. a; , c a d n i30 ELEMENTS OF ALGEBRA. ., Bx — 5 ,„ 4-2a; 11.-^ 12=-^ .. 12. ^ ^=b. a — 1 a-|- 1 x^ + 2ax + a^ _4ab x+a ~lQb x+8 x+6 13. 14. 2 — 2x 4 3 ,^ 4a; , 36 a , 126 15. = — 5 2 6 2 16. a=- c — a 17. 2^:^+- = 30. ^ + ^2. 4 ' 3 2 18. 10— ?^+i=. 2^-34- 19. 4+10a; + 5 — 6a;/i — M = 27. oA » — 2 a; — 4 ^ , a; — 5 20.. _+__ = 7 + -^-^ a; a;2 — 5a! 2 22. 3 3a; — 7 3 3 a; + 1 _ a;" a;+l a; — 1 1 — x^ ^3_2x^+2a;3-9.^ + 12^^^,_^^_3. a;2 + 3a;— 4 „, 2 , 3a; — 3 3a; — 4 „ 27+ 4a; 24. — = D • 3^ 4 3 9 SIMPLE EQUATIONS. 131 2g 9a; + 20 x _ 4sc — 12 36 4 "" 5a;— 4 " ^ 9 5 ^2 27 6 a + 6 a 6 X — c X — a X- ■b 28 6^ + 13 9a;+15 g_ 2a! + 15 15 5a; — 25 5 - . X — a x-\- a 2ax a — b a + b a^ — b^ 7 2 2 PBOCESS. o/ r o\ o/ I o\ 1 Explanation. — When a; + o -j ^^ — ' — - = — ^^ the same expression is found in several terms, 7 2 2 .. i^y. z^ z. *l*^ process may be short- 7 2 2 ened by subslUvivm. Thus, 14« _|- 6u = 21« 7 y '^ substituted for 2+3. The value of y is found to be 7. Therefore, x-\-Z = 7, and a;=4. j, = 7 a; + 3 = 7 a;^4 31. ^ + 6-?^±^ = |(a;+6)-6.' 32.-.-7 + i=:-%5^==2i. 33 3(a;+4)_^_^+4_3(Hd)^ni. 132 ELEMENTS OF ALGEBRA. 3^ 2(^-3)^2(.-3) = 5-^-=^- „^ 21 — 3.1; 2(2a!+.3) „ 5a;+l 3 9 4 Qfi A 3.-«-5 2a;-4 36.0^-4 ^ = 8 —■ x — 7 x — 12 x—1 „„ 2a; — 10 x — % x — b ,. 38. =x — 14. 39. ?!=6_j--8_^_2_tli. ,, 3a; 3a; — 11 . 20a; + 13 41. — = 6a; 42. 1^+8- 8 = 27^^::^ + 4. 3a; — 1 3a; — 1 3a; — 3 3a; — 4 16 27 + 4a; 43. 4 3 3 .. 6a! + 18 11— 3a; _ .., 13 — a; 21 — 2a; 44. ■ =ox — 434 13 36 ' 12 18 .^ 4a; + 3 8a; + 19 7a; — 29 45. — 46. 9 18 5a;— 12 ab -\- x 6^ — X X — 6 ah — x 47. (a + a;)(6 + a>)-a(6 + c)=^ + a;2. 48. 49. 50. SIMPLE EQUATIONS. 133 Zax — 26 ax — a ax 2 36 26 ~"6~3' X 2-\-x c d a — h~ a-\-b~ a'^—h'^ a — h x — ab 562 , a; — 62 7^2 Hah x — a^ 6 ' 4 12 FBOBUBMS. 183. Directions foe Solving. — Repretent one of ike un- known quantities by x, and from the conditions of the problem find an expressixm for each of the other quantities given. Find from tiie problem turn expressions that are equal, and express them as an equation. Solve the equation. 51. When the half of a certain number is added to the number, the sum is as much more than 60 as the number is less than 65. What is the number? 52. The difference between two numbers is 8, and the quotient arising from dividing the greater by the less is 3. What are the numbers? 53. A man left one-half of his property to his wife, one- sixth to his children, a twelfth to his brother, and the rest, which was $600, to charitable purposes. How much property had he? 54. Find two numbers whose sum is 70, such that the first, divided by the second, gives a quotient of 2 and a remainder of 1. 55. Out of a cask of wine, one-fifth part had leaked away. Afterward, 10 gallons were drawn out, when the cask was found to be two-thirds full. How much did it hold? 134 ELEMENTS OP ALGEBRA. 56. A can do a piece of work in 5 days, and B can do the same work in 6 days. How long will it take both working together to do it? SOLXJTION. Let X represent the number of days it will take both to do it. — = the part both can do in a day. X — = the part of the work which A can do in a day. 5 — = the part of the work which B can do in a day. Therefore, — + — = — 5 6 a; 6w + 5a; = 30 110! = 30 x = 2^ 57. A can do a piece of work in 9 days, and B can do the same in 10 days. How long will it take both to do it? 58. A can do a piece of work in 5 days, B in 7 days, and C in 9 days. In how many days can they all together do it? 59. Two pipes empty into a cistern. One can fill it in 8 hours, and the other in 9 hours. How soon will it be filled, if both empty into it at the 'same time? 60. A cistern can be filled by a pipe in 3 hours, and emptied by another pipe in 4 hours. How much time will be required to fill the cistern if both are running? 61. A fish was caught whose tail weighed 9 pounds. His head weighed as much as his tail and half his body, and his body weighed as much as his hoad and tail. How much did the fish weigh? SIMPLE EQUATIONS. 135 62. Of a detachment of soldiers, -f are on duty, ^ of them sick, ^ of the remainder absent on leave, and the rest, 380, have deserted. How many were there in the detachment? 68. A person spends one-fourth of his annual income for his board, one-third for clothes, one-twelfth for other ex- penses, and saves $500. What is his income? Fractions may be avoided in this and similar examples, bj letting some number of times x, which is a multiple of the denominators, represent the number sought. Thus, in the above example, let 12a; represent the annual income. SOLUTION. Let 12a; = his annual income. 3x = what he paid for board. 4a! = what he paid for clothes. X = what he paid for his other expenses. 3x + 4a; + a; + 500 = 12a! 4a! = 500 a! = 125 12a! = 1500, his income 64. A farm of 392 acres was divided among four heirs, so that A had four-fifths as much as B, C as much as A and B, and D one-half as much as A and C. What was the share of each? 65. A farmer wishes to mix 300 bushels of provender, containing rye, com, and oats, so that the mixture may contain ■§■ as much oats as corn, and ^ as much rye as oats. How many bushels of each should he use? 66. Into what two parts can the number 204 be divided, such that I of the greater being taken from the less, the remainder will be equal to ^ of the less subtracted from the greater? 136 ELEMENTS OF ALGEBRA. 67. A man spent $14 more than ^ of his money, and had $6 more than \ of it left. How much had he at first? 68. A merchant lost ^ of his capital during the first year. The second year he gained f as much as he had left at the end of the first. The third year he gained -j^ of what he had at the close of the second, making his cap- ital $7000. What was his original capital? 69. An ofiicer wished to arrange his men in a solid square. He found by his first arrangement that he had 39 men over. He then increased the number on a side by 1 man, and found he needed 50 men to complete the square. How many men had he? SOLUTION. Let X = the number of men in each side in the first arrangement. Then x^ = the number of men in the first square. x-\-l = the number of men in each side in the second arrangement. (a; + 1) 2 1= the number of men in the second square. x^ -\-B9 = the entire number of men. (a; -)- 1) 2 — 50 = the entire number of men. Therefore, (a; + 1) « — 50 =: a;2 -)- 39 a;2 + 2» + 1 — 50 = a;2 + 39 2a; = 88 3!=: 44 a;2+39 = 1975 70. A regiment of troops was drawn up in a solid square with a certain number on a side, when it was found that there were 295 men left. Upon arranging them so that each rank contained 5 men more, it was found that there were none left. How many men were there in the regiment? SIMPLE EQUATIONS. 137 71. A colonel, upon attempting to draw up his troops in the form of a solid square, found that he had 31 men over. If he had increased the side of the square by 1 man there would have been a deficiency of 24 men. How many men were there in the regiment? 72. A person in purchasing sugar found that if he bought sugar at 11 cents he would lack 30 cents of having money enough to pay for it; so he bought sugar at 10^ cents, and had 15 cents left. How many pounds did he buy? 73. Into what two parts may the number 56 be divided, so that one may be to the other as 8 to 41 SOLUTION. Since one number is to the other as 3 to 4, one is f of the other. Therefore, to avoid fractions Let 4x = one part. Then 3x = the other part. 4a! + 3a; = 56 7a; = 56 X = 8 4x = 32, one part 8a; = 24, the other part 74. Find two numbers which are to each other as 5 to 7, and whose sum is 72. 75. A's age is to B's as 3 to 8, and the sum of their ages is 44 years. How old is each? 76. An estate of $15000 was divided between two sons, so that the elder's share was to the younger's as 8 to 7. What was the share of each? 77. A sum of money was divided between A and B, so that the share of A was to that of B as 5 to 3. The share of A also exceeded | of the whole sum by 850. What was the share of each ? 138 ELEMENTS OF ALGEBRA. 78. A and B began to play together with equal sums of money. A won $20, but afterward lost half of all he then had, when he found that he had just half as much as B. How much had each at first ? 79. A lady distributed $252 among some poor people, giving to the men $12 each, the women $6 each, and the children $3 each. The number of women was 2 less than twice the number of men, and the number of children was 4 less than 3 times the number of women. To how many persons did she give the money ? 80. A person bought a number of apples at the rate of 5 for 2 cents. He sold half of them at 2 for a cent, and the remainder at 3 for a cent, gaining 1 cent. How many did he buy? 81. A merchant engaged in business with a certain capital. His gain the first year lacked $1000 of being as much as his original capital. His gain the second year lacked $1000 of being as much as he had at the end of the first year, and the third year his gain lacked $1000 of being as much as he had at the end of the second year. He found that at the end of the third year his capital was 3 times his original capital. What was his original capital? 82. A and B began business with equal capital. The first year A gained a sum equal to \ of his capital, and B lost ^ of his. The second year A lost $72 and B gained $36, when it was found that B's capital was f of A's. What was the original capital of each? 83. A cistern, which held 648 gallons of water, was filled in 18 minutes by two pipes, one of which conveyed 6 gallons more per minute than the other. How much did each convey per minute? 84. A farmer has 90 sheep in four fields. If the number in the first be increased by 2, the number in the second SIMPLE EQUATIONS. 139 diminished by 2, the number in the third multiplied by 2, and the number in the fourth divided by two, the results wUl be equal. How many are there in each flock? 85. A gentleman who had $10000, used a portion of it in building a house, and put the rest out at interest for one year: ^ of it at 6% and f of it at 5%. The income from both investments was $320. What was the cost of the house? 86. Paving a square court with stone at 40 cents a square yard will cost as much as inclosing it with a fence at a dollar per yard. What is the length of a side of the court? 87. Two soldiers start together for a fort. One, who travels 12 mUes per day, after traveling 9 days, turns back as far as the other had traveled during those 9 days. He then turns and pursues his way toward the fort, where both arrive together 18 days from the time they set out. At what rate did the other travel? 88. A boy bought a certain number of apples at the rate of 4 for 5 cents, and sold them at the rate of 3 for 4 cents. He gained 60 cents. How many did he buy ? 89. A gentleman left $315 to be divided among four servants, as follows: B was to receive as much as A and ^ as much more; C was to receive as much as A and B and ^ as much more; D was to receive as much as the other three and \ as much more. What was the share of each? 90. Two numbers are to each other as 2 to 3; but if 50 be subtracted from each, one will be ^ the other. What are the numbers? 91. A woman sold eggs and apples. The eggs were worth 5 cents a dozen more than the apples; and 8 dozen eggs were worth as much as 13f dozen apples. What was the price of each per dozen? 92. Three men. A, B, and C, build 318 rods of wall. A builds 7 rods per day, B 6 rods, and C 5 rods. B 140 ELEMENTS OF ALGEBRA. works twice as many days as A, and C works ^ as many days as both A and B. How many days does each work? 93. A gentleman has two horses, and a carriage worth $150. The value of the poorer horse and carriage is twice the value of the better horse; and the value of the better horse and carriage is three times the value of the poorer horse. What is the value of each horse? 94. A man bought two pieces of cloth, one of which lacked 12 yards of being 4 times as long as the other. The longer cost $5 per yard, and the shorter $4 per yard. Twenty-three yards being cut off from the longer, and 5 from the shorter, and each remainder being sold for a dollar a yard more than it cost, he received $142. How many yards of each were there? 95. When, after 2 o'clock will the hour and minute hands of a clock be together? SOLUTION. Let X = the number of minute-spaces that the minute hand travels before they come together. Then, — == the number of minute-spaces that the hour hand travels. Then, since they were 10 minute-spaces apart at two o'clock, 12 lla; = 120 X = 10|^, the number of minutes after 2 96. When, after 5 o'clock, will the hour and minute hands of a clock be together? SIMPLE EQUATIONS. 141 97. When, after 8 o'clock, will the hour and minute hands of a clock be together? 98. When, after 4 o'clock, will the hour and minute hands of a clock make a straight line? 99. When, after 5 o'clock, will the hour and minute hands of a clock make a straight line ? 100. When, first, after 6 o'clock, will the hour and minute hands of a clock be 15 minute-spaces apart? 101. When, after half-past 8 o'clock, will the hour and minute hands of a clock be 15 minute-spaces apart? 102. After paying out — and — of my money, I had 6 m n dollars left. How much had I at first ?^ SOLUTION. Let X = the amount I had at first. Then, — -| — ^ the amount I spent. TO n Therefore, j \- b^x m n mx -\-nx-{- mnb = mnx mnx — mx — nxz= mnb (mn — -jji — n)x = mnb mnb x^ mn — m — n 184. A problem in which literal notaiion is used, is called a General Problem. Such problems give an infinite number of numerical re- sults, by assigning diflTerent numerical values to the literal quantities. Thus, in problem 102, given above, when m = 4, m = 5, and 6 = 66, the value of x is 120; when ot = 5, n = 8, and 6 = 54, the value of X -is 80. 142 ELEMENTS OF ALGEBRA. 103. A horse and saddle are worth m dollars, and the horse is worth n times as much as the saddle. What is the value of each when m = 200 and w := 9 ? 104. A gentleman gave two servants 6 dollars, giving A a times as much as B. How much did he give each? How much did he give each if 6 = 75 and a = 4? 105. Divide the number h into two such parts that one shall be a times the other. What will be the result when 6 = 24 and a = 7? 106. If A can do a piece of work in n days and B in m days, in what time can both do it working together? What will be the result when n is 5 and m is 7? What when n is 10 and m 8? 107. A pleasure party of a persons hired a coach. If there had been h persons more, it would have cost each d dollars less than it did. How much did each one pay? What is the result when a is 8, 6 4, and d $1? 108. A certain number divided by h gives a result such that the sum of the dividend, divisor, and quotient is c. What is the number? What is the number when 6 is 16 and c is 84? SIMULTANEOUS EQUATIONS. TWO UNKNOWN QUANTITIES. 185. 1. When a; = 2 and y^3, what is the value of a;+2/? What of 2a; + y? 2. When a; = 4 and y^3, what is the value of x-\-yt Of x—yl 3. When a; = 6 and y = 2, what is the value of 2x-{-yt Of x + 2y? SIMPLE EQUATIONS. 1^3 4. When a; = 10 and y = 3, what is the value ot 2x-\-yi Ot x + Sy? 5. Write down the results in each of the above in the form of equations. What is the value of x in each of the first two equations? Of y? Of a; in each of the second two? Of 2/? Of a; in the third two? Of y? 6. What are those equations called in which the same letter has the same value in each equation? (See Art. 185.) 7. What may be done to equations without destroying the equality ? (See Axioms, Art. 59.) 8. If the members of the equation x-\-y = 4 are multi- plied by 2, what is the resulting equation? What is the resulting equation, when the equation 2x-{-y=^8 is multi- plied by 3? 9. How can the equation 3a; + 6y == 18 be derived from x-\-2y=&^ How can 4x-\-2y = 8 be derived from the equation 2a; + 2/ = 4? 10. If x-\-y is added to x — y, what is the result? If x-\-2y is added to x — 2y, what is the result? 11. If the equation x-\-y^S is added to the equation X — y = 4, what is the resulting equation? How many un- known quantities does it contain? How many unknown quantities were there in the original equation? 12. If a; + y ^ 8 and x-\-2y^= 12, what equation wiU result by subtracting the first from the second? What is the value of j;? What is the value of a;? 13. If the sum of two numbers is 12, what are the num- bers? How many answers may be given to the question? 14. In the equation x + y==:12, how many values may .':; have? How many may y have? 15. What are those equations called in which the un- known quantities may have an infinite number of values? (See Art. 188.) 144 ELEMENTS OF ALGEBRA. DEFINITIONS. 186. Simultaneous Equations are those in which the same unknown quantity has the same value in every equa- tion. Thus, \ '"*'" [-are simultaneous equations in which a; = 7 ix—y= 2 ) and y = 5, 187. Derived Equations are those which are obtained by combining other equations or performing some opera- tion upon them. Thus, 2x-\-2y = 8, is an equation derived from x-\-y = i, and 2a; + % ^ 7, is derived by adding x-\-y = 3 and a; + 2y = 4. 188. Independent Equations are such as can not be derived from one another or reduced to the same form. Thus, 2x-\-y=5 and x-j-2y = 6, are independent equations. 189. An Indeterminate Equation is one in which the unknown quantities may have an infinite number of values. Thus, a; + y=12, is an indeterminate equation, because each of the unknown quantities may have an infinite number of values. Hence, 190. Pkinciples — 1. Every single equation containing two unknown quantities is indeterminate. Consequently, 2. In order to solve equations containing two unknown quantities, two independent equations, involving one or both of the quantities, must be given. 191. Elimination is the process of deducing from simul- taneous equations, equations containing a less number of unknown quantities than is found in the given equations. SIMULTANEOUS SIMPLE EQUATIONS. 145 CASE I. 193. Elimination by Addition and Subtraction. 1. When x-{-y = 8 and x — y = 2, how may the value of X be found? 2. When x-{-2y=10 and x — 2y = 6, how may the value of X be found ? 3. When 3a; -f ^j/ = 16 and 5x — 4y= 16, how may the value of X be found? 4. When may a quantity be eliminated by addition? 5. When x-\-2y = 6 and x-\-y = 4, how may the value of y be found ? 6. When 2x-\-3y^l0 and. x-\-Sy = 8, how may the value of X be found ? 7. When may a quantity be eliminated by subtraction? 193. Principle. — Qiiantities may be eliminated by addition or by subtraction when tiwy have the same coefficients. EXAMPLES. 1. Find the value of x and y in the equations 2x-}-Sy = 13 and 3a; + 2j? = 12. Explanation. — Since the quanti- ties in the given equations have not the same coefficients, the first equation is multiplied by 3 and the second by 2, producing equations (3) and (4) in which the coefficients of x are alike. Since the coefficients of x are alike, and they have the same sign, x may be eliminated by subtraction (Prin.). Subtracting (4) from (3), we obtain (5). Dividing equation (5) by the coefficient of y, we obtain (6). PROCESS. 2x + 3y-- = 13 (1) 3x + 2y-- = 12 (2) 6x + 9y-. = 39 (3) 6x + 4y-- = 24 C4) 5y- = 15 (5) y-- = 3 (6) 2a! + 9 : = 13 (7) 2x-- = 4 (8) x- = 2 13 (9) 146 ELEMENTS OF ALGEBRA. Substituting the value of y in equation (1), the resulting equa- tion is (7). Transposing and uniting, the value of a; = 2. Rule. — If necessary, multiply or divide one or both equa- tions so that one unknown quantity may have the sanie coeffi- cient in both. When the signs of the equal coefficients are the same, sub- tract the equations; when the signs are unlike, add the equa- tions. Find the values of the unknown quantities in the foUowing : 4. 6. 8. x + 2y^7.) 10 5x + 6y=40.) x+ y = 5.\ 8x-4y=4. j 4x + By = 7. ) 2a; — 32/=— l.j 11. 8a; + 62/=39. 1 5a; — 32/= 13.] 4a; — 52/ = 3. ) 3x + 5y=ll.] 2a; + 62/= 10. 1 3a; + 22/= 8. j 12. ^ '^ + ^ = 3. ~ 2^3 a; 2/ _ 23_ 5 "^ 2 ~ 10' 8a; + 32/ = 22.1 4x + 5y=18.\ Sx + 4y=25.) 4a; + 3y = 21.J 13. . \x y_ l^ 6 3 3' 2x 3y ' 3 4 5x + 6y = Gl.) 4b + 52/= 50. j 4a; + 32/= 32. 1 7a; — 6v=ll. 1 14. . 4 5 SIMULTANEOUS SIMPLE EQUATIONS. 147 15. 16. Bx y _B _ 5 2'~10' ■ 17. . 'l + T = '»-l 5 4 2£ 2/ _ 22_ 7 5 ~ 35' M--' !+!-* V ^ 18. . OAS B II. 194. Elimination by Comparison. 1. If, in the equation a; -|- y = 8, 1/ is transposed to the second member, what will be the form of the equation ? 2. If, in the equation x — y^4,y is transposed to the second member, what will be the form of the equation ? 8. If, in the simultaneous equations x-{-2y = 8 and x — y = 5,y in each is transposed to the second member, what wiU be the form of the equations? 4. Since the second members of these derived equations are each equal to x, how will they compare with each other? 5. If these second members are formed into an equation, how many unknown quantities will it contain? 6. How may an unknown quantity be eliminated from two simultaneous equations by comparison f EXAMPLES. 1. Find the value of x and of y in the equations « + 2v = 8 and Bx + 2y==12. 148 ELEMENTS OF ALGEBRA. Explanation. — Since, in elimination by comparison, the value of the same unknown quantity in each equation is to be found, and a new equation is to be formed from them, 2y in equation (1) is transposed, giving (3). Transposing 2i/ in (2) and dividing by 3, equa- tion (4) is obtained. Since these two values of x are equal, equation (5) is obtained. Clearing of fractions, we ob- tain (6). Transposing and uniting, (7) is obtained. Di- viding by 4, we obtain (8). Substituting this value of y in equation (1), we obtain (9). Uniting, we obtain (10). EuLE. — Find an expression for the value of the same un- knovm quantity in each equation. Place these values equal to each other, and solve the equa- tion. Solve the following equations by comparison: PROCESS. x + 2y= 8 (1) 3x + 2y = 12 (2) x= 8 — 2y (3) •-^ (4) i^= s-% (5) 12 — 22/ = 24 — 6y (6) 42/ =12 (7) y= 3 (8) x + Q= 8 (9) x= 2 (10) 4. 3a; + y=9. x + 2y=8. 2x— y = 3. a + 32/= 19, 4x + 2y=26. 3x + 4y=B9. 2a; — 32/ = — 14 3x+2y = U. 7. 9. 3a; + 42/ = 18. x + 2y = 8. a; + 62/ = 13. 5x + 2y—9. 4a; + 22/ = 26. 3a; — 4^ = 3. 2a; — 3y = —7 4a; — 5y^ — 9, SIMULTANEOUS SIMPLE EQUATIONS 149 10. 11. 12. 13. 14. 15. i Bx + 2y=33 •) \'' + 'y-i7.] ■ 9x — 4y=9. 5^3 16. < L Qx+ y = 45.) l+f="- 3x — 2y = 15.\ 4x — 5y = —B4:.] 'y+f-*-' 2x — 3y = —22.\ x + 4y = ll.) 17. . l+f-i-." 5x — 2y = ll.\ 5 4^ 2x — Sy = 3. 1 18. . ^a; 22/ - . 4x + 5y=S9.\ .7 5 -'"-J M-^-l 'f + f-^-~ „ 19. . J+! = 5. 3^ 52/ _ 3 2 7 6 CAS E in. 195. Iiliminationvby Substitution. 1. In the equation x -{-y^^5, if x = 2, what is the value of 3/? How is this value obtained? 2. In an equation containing two unknown quantities, if the value of one quantity is given, how may the value of the other be found? 3. If y is transposed to the second member in the equa- tion x-\-y = b, what will be the expression for the value of a;? 150 ELEMENTS OF ALGEEEA. 4. If X ia transposed to the second member in the equation x-\-y^^&, what will be the expression for the value of y'? 5. Express the value of x in the first of the simultane- ous equations x-\- y^d and x-\- 2y^=7. When the value of X is obtained, how may the value of y be obtained ? 6. How may an unknown quantity be eliminated from simultaneous equations by substitutionf EXAMPI,BS. 1. Find the value of x and of y in the equations 3x -\- 2y = 12 and 2x + 3y ^ 13. PBOCESS. 3x-\-2y = 12 (1) Explanation .— Since one unknown quantity can be elim- 2x-\-3y^ 13 (2) inated by finding its value in 12 — 2y ,- one of the given equations and 3 ^ ■' substituting this value in an- 24 — 4y other, we find the value of x ^ I- 31/= 13 (4) from (1) and obtain (3). Sub- stituting this value in (2), (4) 24 — 42/ + % = 39 (5) is obtained. Clearing of frac- 5y = 15 (6) tions, the resulting equation is y = 3 rj\ (5). Uniting terms we obtain j2_Q (6)- Dividing, y= 3. Substi- X=: =2 (8) tuting this value in (3), a;=2. KuLE. — Find an expreBswn for the value of one of the un- hMwn quantities in one of the equations. Svhstitute this value for the same unknown quantity in the other equation, and solve the equation. Solve the following by substitution: x-\-2y= 10. 2x~3y = —l. 3x — 2y=l. a; -f 42/= 19. SIMULTANEOUS SIMPLE EQUATIONS. 15] 4. 5. 6. 9. 10. x — 2y=^Q. 2x — y = 27. 9x—y = Q. X-\-y = 4:. 3x + 5y = 2 6x+5y = B 7x — 5y = lB. Sx + 3y = 21. 6x-\- y = 60. 3a; + 22/ = 39. 2a; + 52/ = 29. 2a; — 5y = — 21 ^ + ^ = 18. 5^6 ^-^ = 21. 2 4 11. 12. - a; 1 _ 1 m n 153 FROBI.E9IS. 196. 1. If 7 lb. of tea and 5 lb. of coffee cost $5.50, and 6 lb. of tea and 3 lb. of coffee cost $4.20, what was the price per pound of each ? PEOCESS. lb. Let X = the price of tea per i Let y = the price of coffee per lb. 7x + 5y = $5.50 (1) 6a! + 3y = $4.20 (2) a; = 8 .50 2/ = $ .40 (3) (4) Expi/ANATION. — Since there are two kinds of quantities involved, namely, tea and coffee, x may be used to represent the price per pound of the tea, and y the price per pound of the coffee. Then from the conditions of the problem we have equations (1) and (2). Solving them, the value of x is $.50 and y is $.40. 2. There is a fraction such that if 1 is added to the numerator the value of the fraction will be 1 ; and if 3 is added to the denominator the value will be -J. What is the fraction? 154 ELEMENTS OF ALGEBRA. SOLUTION. Let X = the numerator. y = the denominator. Then, - := the fraction. y X+1 J y (1) y + 3 * (2) x = 4 (3) 2/-5 (4) X _4 (5) 3. There is a number such that if it be divided by the sum of the digits which express it the quotient will be 4, and if 36 be added to it the sum will be expressed by the digits inverted. "What is the number? SOLUTION. Let X = the digit in tens' place. y = the digit in units' place. lOx -\-y = the number. lOt/ -{-X = the number when the digits are inverted. 10£±1^4 (1) x + y 10a; + y + 36 = lOy + x (2) X = 4: (3) 2/ = 8 (4) 10x + y = 48 (5) 4. The sum of two numbers is 24, and their difference is 8. What are the numbers? SIMULTANEOUS SIMPLE EQUATIONS. 155 5. The sum of two numbers is 29, and their difference is 5. What are the numbers? 6. The sum of two numbers divided by 2 gives a quo- tient of 24, and their difference divided by 2 gives a quo- tient of 17. What are the numbers? 7. A man hired for one day 6 men and 2 boys for |28, and afterward, at the same rate, 3 men and 4 boys for $20. What was paid each per day? 8. There is a fraction such that if 3 be added to the numerator its value will be \, and if 1 be subtracted from the denominator its value will be \. What is the fraction? 9. A man has two horses, and a saddle worth $10. The value of the saddle and the first horse is double that of the second horse, but the value of the saddle and the second horse lacks $13 of being equal to the value of the first horse. What is the value of each horse? 10. Two purses contain together $300. If $30 is taken from the first and put into the second, there wiU be the same amount in each. How much money is there in each? 11. A and B have $570. If A's money were three times, and B's were five times as great as it really is, they would have $2350. How much has each ? 12. What fraction is that to the numerator of which if 4 be added, the value will be \, and if 7 be added to the denominator the value will he. \1 13. There is a number of two digits, which is equal to 4 times the sum of the digits, and if 18 be added to the num- ber, the result will be expressed by the digits inverted. What is the number? 14. A person had two kinds of money, such that it took 10 pieces of one kind to make a dollar, and two pieces of the other to make a dollar. He paid a man a dollar, giving him 6 pieces. How many of each kind were used ? 156 ELEMENTS OF ALQEBBA. 15. A party which had hired a coach, found that if there had been three more persons, they would each have had to pay $1 less than they did ; and if there had been 2 less they would each have had to pay $1 more. How many persons were there ? How much did each pay? 16. A wine-merchant sold at one time 20 dozen of port wine and 30 dozen of sherry for £120. At another time he sold 30 dozen of port and 25 dozen of sherry for £140. What was the price per dozen of each ? 17. There is a number expressed by two figures. If to the sum of the digits 7 is added, the result will be 3 times the left-hand digit, and if 18 is subtracted from the number, the digits will be inverted. What is the number? 18. A and B had together a capital of S9800. A invested \ of his capital and B ^ of his, when each had the same sum left. How much had each before the in- vestment? 19. A farmer purchased 100 acres of land for $2450. For a part of it he paid 120 an acre and for the rest $30 an acre. How many acres were there in each part ? 20. The sum of the ages of a father and a son is 80 years. If the age of the son is doubled, it will exceed the age of the father by 10 years. What is the age of each? 21. A said to B: "Give me 20 cents of your money and I will have 4 times as much as you." B said to A: " Give me 20 cents of your money and I will have 1^ times as much as you." How much had each? 22. A farmer bought 100 acres of land, part at $37 and part at $45 an acre, paying for the whole $4220. How much land was there in each part? 23. A boy expended 30 cents for apples and pears, buying the apples at 4 for a cent and the pears at 5 for a cent. SIMULTANEOUS SIMPLE EQUATIONS. 157 He then sold ^ of his apples and \ of his pears for 13 cents, which was what they cost him. How many of each did he buy? 24. A railway train, after traveling an hour, is detained 30 minutes. It then proceeds at -f of its former rate, and arrives 10 minutes late. If the detention had occurred 12 miles further on the train would have arrived 4 minutes later than it did. At what rate did the train travel before the detention, and what was the whole distance traveled? THBBE OB MORE UNKNO^WlT QUANTITIES. 197. 1. In the equations x -\- 2y -{- z ^= ^ and 2a; + ^y -\-2z = 14, how may x be eliminated ? 2. In the equations 2a; + Sj/ + 42 = 26 and x -\- 4y -\- 23 = 18, how may z be eliminated? 3. If one of the quantities in the above equations is eliminated, how many quantities will be left? 4. How many independent equations are necessary before the values of two unknown quantities can be found? 5. How many independent equations containing the same two unknown quantities can be formed from the equations in (1)? From the equations in (2)? 6. Since there must be two independent equations given so that the values of two unknown quantities may be found, and since from the two equations given in (1) and (2) only one derived equation can be formed, how many independent equations must be given so that the value of any of the unknown quantities may be found? 7. When the values of two unknown quantities are known, how may the value of a third be found from an equation containing three unknown quantities? 158 ELEMENTS OF ALGEBRA. 198. Since it is necessary to have two independent equa- tions to find the values of two unknown quantities, and three independent equations to fiiid the values of three unknown quantities, etc., a general law may be expressed as follows: PmNCrPLE. — To find the values of unhwwn quantities, there must he as many independent equations as there are wnknovm quantities. EXAMPLES. 1. Given - ' x+2y + 3z = W 2x+ 2/ + 22 = 10 3x + 4y — 3z= 2 -, to find X, y, and z. PROCESS. 9! + 2y + 32 = 14 (1) 2a; + 2/ + 2s =10 (2) Zx-\-Ay — 3z= 2 (3) 2a; + % + 62 = 28 (4) 2a; + y + 22=10 3y + 42 = 18 (5) 3a; + 6y + 92 = 42 (6) 3a; + 4?/ — 32= 2 2y + 122 = 40 (7) % + 122 = 54 (8) 72/= 14 (9) y= 2 (10) 4 + 122 = 40 (11) 122 = 36 (12) 2= 3 (13) a; + 4 + 9 =14 (14) x= 1 (15) Explanation. — To elimi- nate X from the first two equa- tions, we multiply (1) by 2 and obtain equation (4). Subtracting equation (2) from (4), we have (5). Eliminating a by a similar process from (1) and (3), the resulting equa- tion is (7). From (5) and (7) we eliminate z and obtain (9), which contains only y. Equa- tion (10) gives us the value of y. Substituting this value in equation (7), we obtain (11), and the value of z is found in (l.S). Substituting the values of y and z in (1), we have equa- tion (14), from which we find the value of a; to be 1. SIMULTANEOUS SIMPLE EQUATIONS. 159 Rule. — Combine the equations so as to eliminate the same unknown quantity from each, obtaining a set of derived eqim- tions containing one less wnknown quantity. Combine these derived equations so as to eliminate a second unhnmon quantity, and tiivs conlmue until an equation is found containing but one unhnmon quantity. Then find the value of this unhnovm quantity. Substitute this value in one of the equations containing two unknown quantities, and obtain the value of a second quantity. Substitute the two values already fmrnd in an equation con- taining three unknown quantities, and find the value of a third quantity, and thus continue until the values of aU the unknown quantities are found. Find the value of each unknown quantity in the follow- ing: 3. ■{ 5. < x — 2y + 2z=^ 5. ^ 5x + 3y + 6B = 57. x + 2y + 2z^21. 7x — 4y-\-3e = 35. 4a; — 52/ + 2s= 6. 2x + 3y— 2 = 20. x-\- y-{- s^ 6. 5x + 4y-\-3z^22. 3x + 4y — 3z= 2. ^ x — 4y + 3e= 2. ' 4x — 3y+ z= 9. 2a; + 62/ — 42 = 14. ^ x+ y+ z = 35.^ x — 2y + 3z = 15, y— a;4-g = _5, ^ x + 22= y+ 2. ■^ y + 22=2x+2z. z+22^3x+3y. J (x+ y+ 2 = 12.] X— y =2. X — 2 ^4. M + y + a = 2a;. M + a! + 8 = 3y. u -\- x-\- y=4g. u-\- x= J -(- 36. 160 Elements of algebra. 10. »+ 2/ + 22 4- w = 18. a; + 22/+ z+ w=17. »+ 2/+ 8 + 2w = 19. 2a; + 2/ + ^ + w ^ 16. 11. ' tt + V + a; + 2/ = 14. "^ M+ i;+ «+ 2^15. M+ 'W+2/+ 3=16. M+a;+j/+ z = 17. {, V -\- X -\- y -\- s=18. By studying the equations a little before commencing the solution, the student will often discover modes of solution that ■will simplify the work very much. Thus, in example 12 the quantities may be eliminated vAthovi clearing of fractions. In example 8, by finding the sum of the three equations, the value of X may be found at once. In example 10, by finding the sum of the four given equations, and dividing by 5, the value of the sum of the unknown quantities is found. This, subtracted from each of the given equations succes- sively, gives the values of the unknown quantities. Find the value of each unknown quantity in the following: 12. I X y y a >■ 13. . X 2 x + -y = 5. a;+-z = 6. 3 2/+32 = 9- SIMPLE SIMULTANEOUS EQUATIONS. 161 14. 15. 16. J a b a e b c 17. < 18. ''3x + 4y+ 2 = 35. ^ 3g-^2y — 3t = 4. 2x— y + 2t = 17. 3z—2t-\- u= 9. t+ 2/ = 13. J xy x-^y _yz_ x-\-z xz X -\- z 1 PBOBIiEMS. 199. 1. Find three numbers such that their sum is 60; \ of the first plus \ of the second, and \ of the third is 19; and twice the first with three times the remainder, when the third is subtracted from the second, is 50. 2. Find three numbers such that the first with \ of the sum of the second and third is 119 ; the second with -1^ of the remainder, when the first is subtracted from the third, is 68 ; and \ the sum of the three numbers is 94. 3. A, B, and C together possess $1500. If B gives A $200 of his money, A will have $280 more than E; but if B should receive $180 from C, B and C would have equal amounts. How much has each? 162 ELEMENTS OF ALGEBRA. 4. Three persons purchased sugar, coffee, and tea at the same rates. A paid f4.20 for 7 pounds of sugar, 5 pounds of coffee, and 3 pounds of tea; B paid $3.40 for 9 pounds of sugar, 4 pounds of coffee, and 2 pounds of tea; C paid $3.25 for 5 pounds of sugar, 2 pounds of coffee, and 3 pounds of tea. What was the price of each per pound? 5. Divide 125 into four such parts that, if the first is increased by 4, the second diminished by 4, the third mul- tiplied by 4, and the fourth divided by 4, the sum, prod- uct, difference, and quotient shall all be equal. 6. A and B can perform a piece of work in 8 days; A and C can do it in 9 days, and B and C in 10 days. In how many days can each do the same work alone? 7. A certain number is expressed by three digits whose sum is 10. The sum of the first and last digits is f of the second digit; and, if 198 be subtracted from the number, the digits will be inverted. What is the number? Let x= the first digit, or hundreds; y, the second digit, or tens; z, the third digit, or units. Then, 100a; + lOy + 3 ^ the number. 8. There are two fractions which have the same denom- inator. If 1 be subtracted from the numerator of the smaller, its value will be \ of the larger fraction; but if 1 be subtracted from the numerator of the larger, its value wiU be twice that of the smaller. The difference between the fractions is ^. What are the fractions? 9. A man divided a sum of money among his four sons, so that the share of the eldest was ^ of the shares of the other three; the share of the second -^ of the shares of the other three, and the share of the third ^ of the shares of the other three. The eldest had $14 more than the young- est. What was the share of each ? ZERO AND INFINITY. 163 10. A farmer found that the number of his sheep was 26 more than the number of his cows and horses together; that ^ of the number of sheep was equal to the number of horses together with J of his cows; and that ^ of his cows, ^ of his horses, and ^ of his sheep amounted to 12. How many had he of each? 11. There are three purses such that if $20 is taken out of the first and put into the second, it will contain four times as much as remains in the first; if $60 is taken from the second and put into the third, the third will con- tain If times as much as remains in the second; if $40 is taken from the third and put into the first, the third will contain 2\ times as much as the first. How much is there in each purse? ZERO AND INFINITY. 200. How much is 2 times 0? 3 times 0? 500 times 0? a times 0? Any number of times 0? Principle 1. — When zero is muUiplied by a finite quantity the 'product is zero. 201. How much is divided by 2? divided by 6? divided by os ? divided by any number? Pbtnciple 2. — When zero is divided by any finite quantity the quotient is zero. 202. 1. Since 2 times = 0, 3 times = 0, 500 times = 0, and a times = 0, if both members of each equa- tion are divided by 0, one of the factors, what will be the results? 2. Since the value of % is found to be equal to 2, 3, 164 ELEMENTS OF ALGEBRA 600, and a, what may be said of the value of the expres- sion -I? PEmcrPLE 3. — When zero is divided by zero ihe quotient may he any finite qdhntity, or, it is indeterminate. 203. 1. What is the quotient of 2 divided by ^? By^? Bj i? By ,V? 2. When the divisor is diminished, while the dividend re- mains the same, what effect is produced upon the quotient? 3. If the divisor is made very small, what will be the effect upon the quotient? What, when the divisor becomes infinitely small or zero Pecnoiple 4. — When a finite quantity is divided by zero the quotient is infinitely large. 204. 1. What is the quotient when 4 is divided by 2? By 4? By 8? By 16? 2. When the divisor is increased, the dividend remain- ing the same, what is the effect upon the quotient? What, when the divisor becomes infinitely large? Peinciple 5. — When a finite quantity is divided by an infinitely large quantity ihe quotient is zero. 205. The preceding principles may be expressed by alge- braic formulas as follows, — the sign (oo) being used to indi- cate infinity : Principle 1, X a = 0. Principle 2, -^ = 0. a Principle 3, -— == indeter- minate result. Principle 4, — =: oo . Principle 5, — = 0. 00 PROCESS. Let X = the number. X 3 X 29a; 60 2a; 5 20a;- -15a;: = 29a;- -24a; 44a;- -44a!: = (44 — 44)a; = X-- = 44 — 44 ~" ZERO AND INFINITY. 165 EXAMPLES. 206. 1. What number is that whose third part exceeds its fourth part by as much as || of it exceeds f of it? Explanation. — Solving the example as shown, the value of ^ is found to be — : that is, it is indeterminate. This result may be inter- preted to mean that eeery num- ber will fulfill the conditions of the problem. 2. Find a number such that when 5 is added to 3 times the number, and the result is divided by the number increased by 2, the quotient wUl be 3. The solution of this example gives the number to be oo ; that is, there is no finite number which will fulfill the conditions, and consequently the problem is impossible. 3. What number is there such that when -f of it is diminished by 4, the result is 3 less than ^ of it plus -J of it? 4. I bought 400 sheep in two flocks, paying $1.50 per head for the first flock and $2 for the second. I lost 30 of the first and 56 of the second, and sold the rest of the first at |2 per head and the second for $2.50 without gain or loss. Eequired the number of each flock. GENERAL PROBLEMS. 207. 1. The sum of two numbers is a and their differ- ence is b. What are the numbers? 166 ELEMENTS OF ALGEBRA. SOLUTION. Let X = = the greater. Let y = = the less. X + y = = a (1) X -y = = 6 (2) 2a; = = a + b (3) x = a + b 2 (4) y = a — b (5) The general rule for the solution of problems when the sum and the difference of two quantities are given, may- be derived from the values of x and y obtained above. It is as follows: The greater is equal to one-half their sum and difference. Tlie less is equal to one-half the remainder when their differ- ence is subtracted from their sum. 2. A can do a piece of work in a days and B can do it in b days. In what time can both do it working together? Write a general rule for the solution of prob- lems like this. What will be the time if a ^ 10 and 6 = 12? 3. A is a times as old as B, and in b years he will be n times as old. What is the age of each? Write a gen- eral rule for the solution of problems like this. What will be the ages if a = 6, 6 := 3, and n. ^ 4? 4. A traveler sets out from a place traveling a miles per day; after n days another follows him at the rate of b miles per day. In how many days will the second over- take the first? Write a general rule from the values of the unknown quantities. INVOLUTION. 208. 1. What is the second power of a? Of 6? Of c? Of d? 2. What is the third power of a? Of 6? Of c? Of c?? 3. How many times is a used as a factor in producing a2? a3? a*? a^f a^2 o»? 4. How many times is a quantity used as a factor in pro- ducing the second power? Third power? Fourth power? Fifth power? The wth power? 5. What sign has the second power of -{- a? The third power? The fourth power? The fifth power? The sixth power ? 6. What sign has the second power of — a? The third power? The fourth power? 7. Which powers of a negative quantity are positive? Which negative? DEFINITIONS. 209. Involution is the process of finding the power of a quantity. 210. Power (Art. 18). Exponent (Art. 17). Names of Powers (Art. 19). 211. Peinciples. — 1. All powers of a positive quantity are positive. 2. All even powers of a negative quantity are positive, and aR odd powers are negative. 168 ELEMENTS OF ALGEBRA. CASE I. 212. Involution of monomials. 1. What is the third power of &a^hl Explanation. — Since in PROCESS. finding the third power of (6a26)3 = 6a26 X Sa^fe X 6^26= ^^ quantity the quantity is f .y o ■,, n •> iiii used three times as a fac- 6 X 6 X Ga^a^amb = , *u ■ tor, each factor of the given ■^•'^"* " quantity is used three times as a factor. The product of the factors is 216o*5', the third power of the quantity. EuLE. — Raise the numerical coefficient to the required power, and multiply the exponent of each literal quantity by the expo- nent of the power to which it is to be raised, prefixing the proper sign to the result. 2. Find the square of 6x^y. 3. Find the cube of — 4a^b^. 4. Find the third power of — Safe^. 5. Find the square of — Se^rf^. 6. Find the fifth power of 2aa;«y^. 7. Find the sixth power of 2x^yz^. 8. Find the fourth power of 4a6*d*. 9. Find the seventh power of — a^b^c^. 10. Find the fourth power of —ia^b^c*. 11. Find the fifth power of 2x^yz*. 12. Find the third power of — 5a^bc^. 13. Find the eighth power of a%~^e~^. 14. Find the -fourth power of — 4a^^(r*. 15. Find the ninth power of 2ar^b^c^. INVOLUTION. 169 Kaise to the required power 16. (2a;2j/3z)*. 17. (Zx-^^*y. 18. (— 4a3z22/)3. 19. (— 2a-V')^ 20. (2aa;22/-*)*. 21. (a;-*2/-2s")^ Eaise to the required power 22. (ai"2/-"a-")5. 23. (a-^'a"!))"*)*. 24. (a-*2r^"2~^'') ^• 25. (— s^^r^O"- 26. (— a*62c-"d-2")5". 27. (a36-4c3d-»-2)»-2. 28. What is the third power of -^ ? / 2a;^y \» _ \ 3a26 / ~ PKOCESS. 2x^y 2x^y 2x^y _ 8x^y^ 30^6 ^ 3a26 "^ 3a^b ~ 27a^b^ Explanation. — In raising a fraction to a power, both numera- tor and denominator must be raised to the required power. 2a 29. Find the square of — — ob 2x^ 30. Find the square of — — 31. Find the cube of 6xy_ bob 32. Find the fourth power of 8a2& Ix^y 33. Find the sixth power of ^ • 34. Find the seventh power oi — — — Cb % , a^b^e-^ 35. Find the »th power of -^^i^' 36. Find the 2»th power of ar*2/~*2* 170 ELEMENTS OF ALGEBRA. CASE II. 213. Involution of polynomials. EXAalPI-ES. I. Find the third power oi x-\-y. The process is simply to use x-{-y a,s a factor three times. Rule. — Use the given quantity as a factor as many times as there are units in the exponent of ihe required power. 2. Find the second power of (a-\-b). 3. Find the second power of (a + 1). 4. Find the third power of (x-\- y). 5. Find the third power of (2 -f- »). 6. Find the third power of (3 + ?/). 7. Find the second power of (2 -|- b^). 8. Find the second power of (a^ — 6). 9. Find the second power of (a + 6 -j- c)* 10. Find the fourth power of (x -J- 2y). II. Find the fourth power of (n — m). 12. Find the third power of (a -\- b). 13. Find the second power of (»" -j-y"). 14. Find the third power of (2a + 36). 15. Find the third power of (3x — 2y). 16. Find the second power of (n^ -|- m^). 17. Find the fifth power of (a + b). 18. Find the sixth power of (x-{-y). 19. Find the fourth power of (2a + 26). 20. Find the seventh power of (a-\-x). 21. Find the fifth power of (3x + 2z). 22. Find the fourth power of (2y -f z^). 23. Find the square of (a + 6 + c -|- d). INVOLUTION. 171 OASE III. 314. Special method of squaring a polynomial. 1. Examining the result obtained in the solution of the last example, which was squaring (a -\- b -j- c -]- d) , how many terms are squares? Of what are they the squares? 2. What is the coefficient of each of the other terms? 3. How many terms contain a ? By what quantities is a multiplied in the terms which contain it? 4. What other terms contain 6? By what is b multi- plied in the terms which contain it? 5. How many other terms contain c? By what is c mul- tiplied in the terms which contain it? 215. PEmcrpLE. — The square of a polynomial contains the square of each term and twice the product of each term multi- plied by eacJi of the terms following it. EXAMFIiBS. 1. Square (a -\- b -{- c -{- d). PBOCESS. (a + b + G + dy = a^+b^+c^+d^+2ab + 2ao + 2ad + 2bc + 2bd + 2cd Explanation. — We write down without multiplication the square of each term with twice the product of each term multiplied by each of the terms which follow it. Find the square of 2. a + b + c. 3. X — y -\- z- 4. a -{- c -\- d -{- e. 5. 1—a + a^—a^. Find the square of 6. a + 2b + Bc + d. 7. l-f2a~8a2-(-a» 8. l — 2x—y^+!mi. 9. 2a — b + c — d. 172 ELEMENTS OF ALGEBRA. CASE IV. 216. Involution of binomials by the Binomial The- orem. (a + 6)3 = aS + 3a26 + Sc*^ + 6«. (a + 6)4 = a* + 4a36 + %a%'^ + 4a63 + 6*. (a + 6)5 =«» + 5a*6 + lOaSft^ + IQa^h"^ + 5a6* + h^. (a — 6)2=a2 — 2a6 + 62. (a — 6')3 = a8 — 3a26 + 3a62 — 63. (a — 6)4 = o* — 4a^b 4- 6a262 — 4a63 + 6*. (a — 6) 5 = a5 _ 5a46 _|_ I0a862 — 10a263 + 5a6* — 65. Examine carefully the above powers of (a + 6) and (a — 6). 1. How many terms are there in the second power of (a + 6) ? Of (a — 6) ? How many terms in the third power? How many in the fourth power? How many in the fifth power? 2. How does the number of terms in any power compare with the exponent of the power? 3. In what terms is a found in the second power of (a + 6) ? Of (a — 6) ? In the third power? In the fourth power? In the fifth power? 4. In what terms of any power of a binomial is the first letter of a binomial found? 5. In what terms of any power of a binomial is the second letter of a binomial found? 6. What is the exponent of the first letter in each term of the second power of (a + 6)? Of (a — 6)? What in the third power? What in the fourth power? What in the fifth power? 7. What will be the exponent of the first letter of the INVOLUTION. 17JJ binomial in the first term of any power of a binomial? In the second ? In the third, etc.? 8. What will be the exponent of the second letter in the second term of any power? In the third term? In the fourth term, etc.? 9. What is the coefficient of the first and the last terms in any power? 10. How does the coefficient of the second term compare with the exponent of the power to which the binomial is to be raised? 11. If the coefficient of the second term is multiplied by the exponent of the first quantity in that term, and divided by the number of the term, or by the exponent of the second quantity in that term increased by 1, what is the result? Of what term is it the coefficient? 12. If the same thing is done to the coefficient of the third term, what coefficient is obtained? 13. What are the signs of the terms in aU the powers of (a + 6) ? 14. What terms have the plus sign in the second power of (a — 5)7 In the third power? In the fourth power? In the fifth power? 15. What terms have the minus sign in the second power of (a — 6)? In the third power? In the fourth power? In the fifth power? 16. What terms are positive and what negative in any power of (a — 6) ? 217. Peinciples. — 1. The number of terms in ihe power of any binomial is one more than the exponent of the required power. 2. The letter of the first term of Uie binomial is found in aJl the terms except the last; Ihe second letter in all the terms 174 ELEMENTS OF ALGEBRA. except the first, and both letters are found in all the terms es>- eept Hie first and last. 3. The expment of -the letter of the first term of the binomial in tlie first term of the power is the same as the index of the required power, and decreases by 1 in each term at the right. The exponent of the letter of the second term of the binomial is 1 in the second term, and increases by 1 in eaeh term at the right. 4. The coefficient of the first term is 1. The coefficient of tlie second term is the same as the index of the required power. The coefficient of any term, mvUiplied by the exponent of the first letter in that term, and divided by the number of the term, or by the exponent of the second letter increased by 1, mil be the coefficient of the next term. 5. If both terms of the binomial are positive, all ^ + 6» is the cube of (a -f- b), what is the cube root of a» + Sa^b + Bab'' + b^ ? 3. How may the first term of the root be found from 4. How may the second term of the root be found from the second term of the power, 3a^&? 5. What are the factors of Ba^'b + Bab'' + b^ ? 6. Since Ba^b + Bah^ f 6^ is equal to b(Ba^ + Bab + b''), what are the factors of the last three terms of the cube of a binomial ? SXAMPI.ES. 1. Find the cube root of a* + Ba^b + Bcd>^ + b^. PROCESS. a8 + Ba^b + Bab^ + &» | a + b Trial divisor, Ba^ Complete divisor, 3a^ -|- 3ab-\-b^ 3a26 + 3a62 -I- 63 Ba^ + Bah^ + b^ Explanation. — Since, if the quantity is a cube one of the terms is a cube, the first term of the root is the cube root of a', which is a. Subtracting o' from the entire quantity, there is left 3a ''6 + 3o62 + 6'. 188 ELEMENTS OF ALGEBRA. Since the second term of the root can be obtained from the first term of the remainder, by dividing it by three times the square of the root already found, the second term of the root of the quantity will be found by dividing Sa^b by Sa^, the trial divisor, which gives b for the second term of the root. And since the last three terms of the cube of a binomial consist of the product of the second term of the root with 3 times the square of the first, 3 times the product of the first and second, and the square of the second, 3a^ + Sab + b^, is the entire quantity, or complete divisor, which is to be multiplied by 6. Multiplying by 6, and subtracting, there is no remainder. There- fore, a + 6 is the cube root of a' + Za^b -\- 3a6* + 6*. Since, in cubing a-\-h-\-c, a-\-h may be expressed by X, the cube will be a;* + 3a;*c + Sac^ + c*. Hence, it is obvious that the cube root of a quantity, whose root con- sists of more than two terms, may be extracted in the same way as in example 1, by considering the terms already found as one term. 2. Find the cube root of x^ — 3x^ -f 5a;* — Sx — 1. PROCESS. a;6— 3a;5-f 5a;3— 3a; — 1 | a;^ — a; — 1 x^ Trial divisor, 3a;* Complete divisor, 3a;* — 3a;* + x -3a;5+5a;8 -3a;5+3a;* — a;* — 3a;*+6a;3— 3a;— 1 — 3a;* + 6a;3— 3a;— 1 Trial divisor, 3a;* — 6a;» + 3a;2 Complete divisor, 3a;* — 6a;* + 3a; + 1 Explanation. — The first two terms are found in the same manner as in the previous example. To find the next term x'' — a; is con- sidered as one quantity, which we square and multiply by 3 for a trial divisor. Dividing the remainder by this trial divisor, the next term of the root is found to be — 1. Adding to this trial divisor 3 times (j;^ — x), multiplied by — 1 and the square of — 1, we obtain the complete divisor. ThLs, multiplied by — 1, and subtracted, leaves no remainder. Hence, the cube root of the quantity is x' — x — 1. EVOLUTION. 189 Rule. — Arrange the polynomial with reference to the ■powers of some letter. Extract the cube root of the first term, write the result as the first term of the quotient, and subtract its cube from ihe given polyrurniial. Divide the first term of Hie remainder by S times ^ B2x*y» + ^ 162a:*y8 — f/ 512x*y« + ,^1250^5^ 22. SimpHfy S^1%+2^J^-^J^■ MULTIPLICATION OF RADICALS. 249. 1. When quantities have fractional exponents, what does the numerator of the exponent show? What the denominator? 2 s g 2 2. Express with the radical sign a^; a°; a^; (ab)^; 2 s 2 3. Express with the radical sign (6a)^; (3a)*; (5a)'^; (6a)^; (6a26«)f 4. How is a^ multiplied by a« ? a« by a^ ? a" by a"? a* by a*? a^ by a^? a^ by a^? 204 ELEMENTS OF ALGEBRA. 5. What is the product of a^ by a^? Of a^ by a^? Of a^Xo^l Of a*Xa*? 6. What must be done to the exponents \ and J before they can be added? 7. When quantities have fractional exponents with diflfer- ent denominators, what must be done to the fractional ex- ponents before multiplying? 1. Multiply 3ai/6^ by 26i/% PROCESS. 3ai/S^X 26i/% = 6a6vT8^= 6a6i/9p y^ y2x= UdbyV2x Explanation. — Since the radical quantities have the same index, their product may be found by multiplying together the various fac- tors. Multiplying the coefficients, we obtain 6a6; multiplying the radical parts, we obtain Vlixy^. Consequently, the entire product is 6aJbVUxy^, which, simplified, gives WahyV 'i'X. 2. Multiply -^a^ by \/ax^y^. PEOCESS. T 12 S '^axy X Voiic^y^ = (axy) X (ax^y^y = (aieyy^ X (aa;'y*)'^= l/a^x^y^ X va^x^y^ = y a°a;'y = xyya^x^y^ ExpliANATlON. — Inasmuch as the quantities have not fractional exponents with the same denominator, they must be changed to equivalent quantities whose fractional exponents have the same denominator. Thus, they become ^a^x^y^ and y'a'i'y'. Multi- plying as before, and simplifying, the result is xyy'a^x^y^. MULTIPLICATION OF "RADICALS. 205 EuLE. — Reduce the radical factors to the same degree, if necessary. Multiply the coefficients together fw /2 — i/l5 by i/3 — 1/5. 26. Multiply i/^" + Vy by l/x — l/y. 27. Multiply Vx — Vy by Vx — Vy. 28. Multiply x + "j/a^ +2/ by a; — \/xy + y- 29. Multiply a — \/a^~+b by a + i/a^-fft. 30. Multiply o^ + b^ by a^ + fti 31. Multiply a^ + ah^ + b^ by a* — 6^ 32. Multiply a^ — b^ by a^ — bk 33. Multiply a^ — a i/2 + 4 by a* + a i/2 + 4. 34. Multiply 3 + 61/5" by 4 + 5 v^ 8". DIVISION OF EADIOALS. 251. 1. How is a» divided by o^? as« by a"? dT by a" ? a^ by a*? a^ by a^? a^ by o^? 2. How are literal quantities divided? 8. What is the quotient of a^ -i-a^l Of a^-5-a^? 0/ a»-r-a^? Of a^ -=-«*? Ofa^^a^? 4. Since, in dividing literal quantities, the exponent of the divisor is subtracted from the exponent of the dividend, when quantities have fractional exponents with different denominators, what must be done to the exponents before dividing? DIVISION OF RADICALS. 207 EXAMFIiES. 1. Divide 3i/i5 by 2^/9, PKOCESS. 3v^-^-2i/9=:Hi/5 ExpiiAUATiON. — Since the radical quantities have the same index, the quotient may be found by dividing the various factors of the dividend by the corresponding factors of the divisor. Dividing the coefficients, the quotient is 1 J ; dividing the radical factors, the quo- tient is l/5. Consequently, the entire quotient is HV5. 2. Divide i/ax*y* by f^axy. PEOCESS. \/aafiy^ -=- yam) = {ax^y^')^ -—- (cm/)^ - 2 (ax^y^y-^ {axyY = yaH^y^ -=- ^/a'^x^y^ = l^^^=xy\/-^ Explanation. — Inasmuch as the quantities have not fractional exponents with the same denominator, . they must be changed to equivalent quantities whose fractional exponents have the same de- nominator. Thus, they become t^a^x^y^ and Va^x^y'^. Dividing, as before, and simplifying, the result is xyy/aiy. RtJi^. — Beduce the radical factors to &e same degree, if nec- essary. Divide iJie coefficient of the dividend by ^ "^ by 6^1^ ix^y 12. Divide i/a;^ — y^ by a; — y. 13. Divide in: by ^f^\. U. Divide (4a;2)* by (2aa;)^. 1.5. Divide i/'dx^y — 3xy^ by i/a; — y. 16. Divide 4(x2 — j/^)^ by 2(x — y)^. 252. Polynomials containing radical quantities, or quan- tities having fractional exponents, may be divided according to the rules already given. 17. Divide x^ + ^V — 6^/ by a; — 2i/y; also, x^ + 11 1 1 1 2x^y^ + y^ by x* + y*. PROCESS. x^ -\- x\/y - »2 — 2xv^y ■6y 5 — 21/2/ a; + 3/y 3xi/y- 3x\/y- ■6y 6y PEOCESS. 18. Divide a^ — y^hja^ y . 19. Divide a^—2aiy^ + y^ by a^ — y^. 20. Divide t/20 + t/T2 by \/5 + 1/3. 21. Divide 6 + lli/2 + 6 by 3 + 1/2. 22. Divide 8 + 10t/5 + 15 by 2 + v^5. 23. Divide 12 + 17i/6 + 36 by 3 + 2i/& a;* + 2a!i2/i + 2/* X^ + y^ x^+ x^y^ x^ + y^ X^yi + y^ xiy^ + yi INVOLUTION OF RADICALS. 209 INVOLUTION OF RADICALS. 253. 1. What is the cube of 3i/» ? PBOCESS. Explanation. — Expressing the ,n. /^ „ ,r. 4. » radical quantity with a. fractional ^ ' ■> ^ ■> exponent, and raising it to the 3^x = 27# = third power, we have 3» X a;^. or 27l/a;8 = 27a;i/x 27A Expressing this in the form of a radical, and simplifying, the result is 27x|/x. 2. What is the square of 2 + 3 l/S? PEOCESS. ExPLANATioii. — Since the (2 + 3\/xy—4:+12y'x + 9x ^^°^^ P"''^' °^ *« poly- nomial is sought, the power will be the square of the first term, plus twice the product of the first and second, plus the square of the second. RuiE. — liaise the rational foustor of a monomial radical to the required pressed under the radical sign or with fraelional exponents. Expand a polynomial which contains a radical part accord- ing to the binomial formula, or perform the involution by a^ucd mvUiplication. 3. Find the square of B\/x. 4. Find the square of 2#^2«2. ,5. Find the square of ^fW. 6. Find the cube of 3i/S. 7. Find the cube of 2aT/3a. 8. Find the cube of Zf^Viax. 9. Find the cube of (a + 6)^. 18 210 ELEMENTS OF ALGEBRA. 10. Find the cube of (2a + Se)^ 11. Find the square of l/3 + 2i/5. 12. Find the square of 2 + 4i/3". 13. Find the square of 2 + 3i/5. 14. Find the square of a;^ -)- y^. g 2 15. Find the square of a^ -\- y^. EYOLUTION OF RADICALS. 254. 1. What is the cube root of a" ? Of a9 ? Of x^y^ ? 2. What is done to the exponent of a literal quantity to extract the cube root of the quantity? To extract the fourth root? To extract the fifth root? To extract any root? EXAMPLES. 1. What is the cube root of Sa^j/aS? PEOCESS. Explanation. — Since any root 8a*l/a6 == Sa^a^b^ °^ ^ literal quantity may be found u by dividing the exponent of the \ Sa^aH'^ — 2aaH^ = literal quantity by the index of the root sought, we express all the Zay ao literal quantities with their ex- OB ponents, obtaining 8a'o*6 , and , 1- divide the exponents by 3, and 8a* j/aft = 2a\/ab extract the cube root of the nu- V merical factor. The result is 2aa*6*, which, expressed as a radical, is 2o^^ The same result might have been obtained by extracting the cube root of the coefficient and multiplying the index of the radical by 3. RATIONALIZATIOK 211 Rule. — Extract the required root of the coefficient and of the quantity under the radical sign, by extracting the root of the numerical quantities, when possible, and dividing the exponents of the literal quardities by the index of the required root. Or, Extract the required root of the coefficient, multiply tlie index of the radical by the index of the required root, and leave tlie quantity under the radical sign unchanged. 2. Find the square root of IGaj^i/oz. 3. Find the square root of SGa^x'^V'yz. 4. Find the cube root of 27a^b^\/xyz. 5. Find the fourth root of a^b^f^a^x^. 6. Find the cube root of d'a/q" 7. Find the square root of \&xf^2y. 8. Find the square root of (x + y)\/x -j- y- RATIONAIilZATION". CASE I. 255. To rationalize a monomial surd. 1. By what quantity may \/a be multiplied to produce a rational quantity? /a? i/^a? -^a^l 2. By what quantity may a^ be multiplied to produce a rational quantity or a quantity with an integral exponent? ah x^t x^i xh yh 356. Nationalization is the process of removing the radical sign, or fractional exponent, from a quantity. 212 ELEMENTS OF ALGEBRA. EXAMPLES. 1. Eationalize 'i/2a. PROCESS. /2a X l/2a — 2a Therefore, i/2a is the rationalizing factor. Explanation. — Since the square root of any quantity multiplied by itself will remove the radical sign, we multiply l/2a by V2a. 1 1 2. Eationalize a^x^. PROCESS. a^x^ X o^x^ = cix Therefore, a%^ jg tijg rationalizing factor. Explanation. — Since a radical quantity is rationalized by re- moving the radical sign or fractional exponent, we multiply a*a;' by such a factor as will make the exponents integral. Hence, to produce integral exponents in the product, we must multiply by a factor such that, when the exponent of each letter of the factor is added to the exponent of the corresponding letter of the surd, the sum of the exponents will be 1. Therefore, o*a;* is the rationalizing factor. Rule. — Multiply the surd by {he same quantity with an ex- ponent such that, when added to the fractional exponent of the surd, the sum of the exponents will he equal to 1, 3. "What factor will rationalize l/iax^ 4. What factor will rationalize ^Sa^y2 5. What factor will rationalize a\/Sx? 6. What factor will rationalize 2i^%2'? 7. What factor will rationalize 2\/aby? 8. What factor will rationalize 2axy^3ye? 9. What factor will rationalize 4x'^yf^3a^y? RATIONALIZATION. 213 CASE II. 257. To rationalize a binomial surd of the second degree. 1. When a-f-^ is multiplied by a — h, what is the prod- uct? 2. When i/a + V^is multiplied by l/a — Vh, what is the product? 3. When \/x-\-'\/y is multiplied by l/a; — Vy, what is the product? EXAMFIiES. 1. Eationalize i/2 — 1/3. Explanation. — Inasmuch as each of PBOCESS. the terms may be rationalized by sqnar- v ^ — V o ing it, we may obtain the i^quare of the ■|/2 -|- l/S terms by multiplying the binomial by the sum of the quantities. Hence, V 2 + V' 3 is the rationalizing factor. 2 — 3 = EuLE. — Multiply ihe binomial by anofher binomial having the same quantities connected vnth the opposite sign. 2. What factor will rationalize i/F — t/2? 3. What factor will rationalize l/9 — 1/6? 4. What factor will rationalize a;-|-i/3? 5. What factor will rationalize x — 3i/6? 6. What factor will rationalize i/o — l/a;? 7. What factor wiU rationalize 2i/a -j- 3\/y ? 8. What factor wiU rationalize i/4a — i/3a;? 9. What factor will rationalize x — l/y? 10. What factor will rationalize x^ — 1/3/3? 2a 1. Keduce to a fraction whose denominator is 214 ELEMENTS OF ALGEBRA. CASE III. 258. To rationalize either term of a fraction, 2a Vy rational. PBOCESS. Explanation. — Since - „ /— r. /— the denominator is to 2a 2a X VV "avy i, .• ,■ j — — z= — — — := be rationalized, we mul- l/2/ Vy X Vy y tiply the terms of the fraction by a quantity which will render the denominator a rational quantity. By Case I it is seen to be Vy. Therefore, the fraction, when its denominator is rationalized, is ^ . y Rule. — Multvply ilie terms of thefraetion by such a quantity as vnll render the reguired term rational. 3 2. Eationalize the denominator of 3. Rationalize the denominator of 4. Eationalize the denominator of 5. Rationalize the numerator of t/5 2 t/7 4 l/a 2i/3 1/5 6. Ra,tionalize the numerator of • ** 7. Rationalize the numerator of 2/5 3v/7 IMAGINARY QUANTITIES. 215 2 8. Eationalize the denominator of 9. Eationalize the denominator of 1/2 — 1/3 2x Va + Vh 10. nationalize the denominator of 'iVx 11. Eationalize the denominator of 12. nationalize the denominator of 2ah Vx—Vy 3 Vx — 1 — Vx-^l IMAGINARY QUANTITIES. 259. 1. What is the square root of a* ? Of a* ? Of — a2? Of— a*? 2. Into what factors may 1/ — a^ be separated so that one of them shaU be a perfect square ? 3. Into what fiictors may 1/ — 4 be separated so that one of them shall be a perfect square? 1/ — 9? 1/ — 16? 4. When the l/a is multiplied by the j/a, what is the product? What, when i/2a; is multiplied by i/2a;? '[/By byl/§^? 5. What is the square of any radical quantity of the second degree ? What is the square of l/F? Of 1/ — 5 ? Ofi/=^? Ofl/=^5a? 260. An Imaginary Quantity is an indicated even root of a negative quantity. Thus, V — 2(1, t^ — 3a!, ^ — a, are imaginary quantities. 216 ELEMENTS OF ALGEBRA. 261. Principle. — Every imaginary nwnomial can be re- duced to the form of a \/ — 1. 262. To add or subtract imaginary quantities. 1. Add i/=^ and 2^/=^^. PROCESS. Explanation.— Since the radi- x^ = xV — 1 -/ 2 __ -,/ -t cal expressions are dissimilar, they must be reduced to similar radicals before adding. Beducing and add- 5>ri/ ^ ing coefficients, the sum is hxV — 1. 2. Add V—a^ and i/=4^. 3. Add 2i/=4P' and 2iV^^. 4. Add 3i/— 16a2a;2 and aV—2bx^. 5. From V — Sa^ subtract a\/ — 16. 6. From i/— aw^a;^ subtract l/— 27m2x2. 263. To multiply imaginary quantities. 1. Multiply 2i/^3 by 2i/^^6". 2i/=3 = 2v'3Xl/=r 2v/^^=2)/6Xl/=i (2i/3 X ■l/=3) X (2l/6 X ■l/=i) = 4i/l8 X {V^^ = — 4i/T8 = — 12i/2 Explanation. — In order to determine the sign of the product, we separate the imaginary quantities into their surd and imaginary factors. "We then multiply, as in radical quantities, observing that W — 1)^ := — 1. V — 1 X V — 1 would, according to the ordinary rules for multiplication of radicals, give as a product T^+l, which is equal to itl; but, inasmuch as we know that l/+l was derived from the product of two negative factors, viz., V — 1 and V — 1, the root of V -\-l, in this case, is — 1, and not +1. Hence, IMAGINARY QUANTITIES. 217 364. Pkinciplb. — The product of two imaginary quantities i& real, and tiie sign before the radioed is determined by the ordinary rules reversed. 2. Multiply i/==4 by t/=5. 3. Multiply 2t/=6 by 3i/=?. 4. Multiply 2i/=4 by Sy'^a 5. Multiply 3i/— a* by 2V—aH. 6. Multiply l+l/=T by 1— i/^T, 7. Multiply 1— i/=i by 1—V—i. 265. To diride imaginary quantities. 1. Divide 6i/=6 by 3i/=2: ExpiitNATroif. — ^Division of PBOCESS. imaginary quantities is per- 6^/ — 6 -7- 3l/ — 2 = 2i/3 formed in the same manner as division of radicals. 2. Divide 3i/=8 by l/=?. 3. Divide 2i/=a2 by ZV^^aF. 4. Divide 2v' — o^a; by 3i/ — a;. 5. Divide 2 by l+l/=r. 6. Divide — 2i/=r by 1— t/=L BXrVZEW KXEBCISES. 266. 1. Kaise ax'^y^ to the m — 1 power. 2. Expand (a + 6)^. 3. Expand (2a — 36)*. 4 Expand (I -|)' 218 ELEMENTS OF ALOEBBA. 5. Expand (^+ I )'• 6. Expand (f + ff 7. Expand (a + 6)" to 6 terms. 8. Expand (a + 6)""^ to 5 terms. 9. Expand (a + y)' to 6 terms. 1 0. Expand (l-\-2x-\-^ + zy. 11. Extract the square root of 9a; — 2A^y^ -\- 12x^ + le/— 16^ + 4. 12. Extract the cube root of 5a;' — 1 — 3a;* + x^ — 3a;. 13. Divide m^-j-mw-f-n.* by m + i/mre -)- n. 14. Divide x*+y* by x''+xyi/2 + y''. 15. Multiply l/a + l/a — x-{- Vx by i/a — i/o — x-\-\/x. 16. Multiply i/a + i/a — a; by i/a — i/a — x. 17. Cube #'V X ^'^ 18. Square 2 4-aV2»+^. 19. Express the wth root of tfY s- l/a 20. Kationalize the denominator of 21. Bationalize the denominator of l/a — 1/6 4+3i/2 3 — 2i/2 22. nationalize the denominator of — !- — • a — l/a2— a;2 23. Rationalize the denominator of — • l/m2 + l + i/m2 — 1 RADICAL EQUATIONS. 267. 1. In the equation i/a! = 2, what is the value of a;? How is it obtained? 2. In the equation i/2a; = 4, how may the value of % be obtained? How in the equation i/a! + 2 = 4? 3. "What is the square of 2 + i/k? What of a; + l/a? Which of the results contains the highest powers of k? 4. What is the square of l/J+ v/2a;? What of i/k + 1/2? Which result contains the highest powers of «? 5. When the sum or difference of two radical expressions, one of which contains the unknown quantity in one term, and the other in both, is raised to a power corresponding to the index of the radical, which result contains the highest powers of the unknown quantity? 26S. A Radical Equation is an equation containing a radical quantity. Thus, 1/1=5, l/a;+3=:6, Vx-{-5 = V'3x, are radical equations. EXAMPLES. 1. Given \/x -|- 6 ^ 9, to find the value of x. PEOCESS. l/» 4- 6 = 9 Transposing and uniting, y^x^S Squaring, a; = 9 (219) 220 ELEMENTS OF ALGEBRA. 2. Given V'4 -|- a; = 4 — Vx, to find the value of a;. PROCESS. v/4-|-a; = 4 — l/a; Squaring, 4 -(- a; = 16 — Sj/a; -|- ai Transposing and uniting, 8l/a; = 12 Dividing by 4, 2Vx = 3 Squaring, 4x^9 3. Given ^— = , to find the value of x. PROCESS. X — ax l/a; l/a; » Clearing of fractions, x^ — ax^ = x Dividing by a;, x — aa; ^ 1 Factoring, (1 — a)x = 1 1 269. From an examination of the solutions of the above examples, some of the following suggestions will be seen to be of value. Experience will teach the student the value of the other suggestions, and his own ingenuity will devise elegant methods, not mentioned here. Suggestions. — 1. Transpose ike terms so that the radical qiumtity, if there be but one, — or the more complex radical, if tJiere be more than one, — mny constitute one member of the RADICAL EQUATIONS. 221 equation ; then raise each member to a power of the same degree as the radical. 2. When the equation is not freed from radicals by the first involviion, proceed again as indicated in Suggestion 1. 3. Simplify the equation as much as possible' before perform- ing the involution. 4. Sometimes it may be advantageous to dear a radical equation of fractions, either in whole or in part. EspemiRy will it be so when a radical denominator and anotJier numera- tor are similar. 5. It is sometimes convenient to ratimudige the denominator before clearing of fractions or involving. 4. Given t/« + 7 = 9, to find x. ' 5. Given \/2x — 5 = 3, to find x. 6. Given i/a; — 7 = 7, to find x. 7. Given #'4a; — 16 = 2, to find x. 8. Given Vx-\-9 = G, to find x. 9. Given ^2* + 3 + 4 = 7, to find x. 10. Given v^o;*— 9 + 9; = 9, to find x. 11. Given T/a;2 — ll + l=a;, to find x. 12. Given i/16 + a; + 1/» = 8, to find x. 13. Given i/o;— 16 = 8 — 1/«, to find x. 14. Given Vx — 21=i/a; — 1, to find x. 15. Given y^+i/a; — 9 = -^==-. to find x. yx — 9 __ 4 16. Given i/a; + i/x — 4= ^ to find x. yx — 4 17. Given — — = — zr^ , to find x. Vx + 4: i/x+12 18. Given — =: 4, to find x. j/3a+l 2 2222 ELEMENTS OF ALGEBRA. 19. Given w + v/jc^— i/l— a= 1, to find 20. Given -yjl -{-x\/x^ + 12 = 1 + », to find x. „. „. i/« — 8 i/« — 4 21. Given — ^ = — — , to find x. Vx — Q \/x + 2 a 22. Given i/a — x= — — x, to find x. ya — X 23. Given -^^=^ ^ ^^~^ +4, to find a:. Voa + 1 2 24. Given — ~ = — — \-l, to find x. l/a; + 4 l/o: + 6 25. Given — _ = — ^ , to find x. 3i/2a!— 10 V2x—A 26. Given \/x-\-& — Vx^^ = 2l/2, to find as. 27. Given 1 = — , to find x. X X 28. Given -v/— — Vx^=V'^ + », to find a. on r.- T^** + » + l/« I. * £ J 29. Given ——z=z — ' — = o, to find x. ya-\-x — Vx 30. Given V^+l + ^x ^^^ ^ ^^^ ^_ l/4a; + 1 — 2Vx 31. Given ^^"+^+"^^^ = 2. to find x. 1/5+0; — 1/5 — a; 32. Given V^±V^^^\ ^ g^^ ^_ l/a — \/a — X a 33. Given r^^^r:^ = a, to find x. 6 + 1/62— a; QUADRATIC EQUATIONS. 270. 1. How is the degree of an equation determined? (Art. 177.) 2. Wliat is the degree of the equation a; -f 2 = 7 ? Of a;2+4 = 8? Of a;2 + 4a;=15? Of x + xy = 122 3. What are equations of the second degree called? 4. When x^=4, what is the value of a? What, when a;2 z= 9 ? What, when a;^ = a^ ? 5. How many values has x in these equations? How do the numerical values of x compare? What are the signs of the values of a;? DEFINITIONS. 271. A Quadratic Equation is an equation of the second degree. Thus, a;^ = 9, and x^-{-bx = c, are quadratic equations. 272. A Pure Quadratic Equation is an equation which contains only the second power of the unknown quantity. Thus, 4x^ = 16, and ax^ = b, are pure quadratics. 273. A Pure Quadratic Equation is sometimes called an Incomplete Q^wdraiie Equation. 274. A Boot of an equation is the value of the unknown quantity. (223) 224 ELEMENTS OF ALGEBRA. PUEE QUADRATICS. 275. Since pure quadratic equations contain only the second power of the unknown quantity, they may always be reduced to the general form of ax^ = b, in which a represents the coefficient of x^, and b the other terms. 276. Pkinciple. — Every pure quadratie egvation has two roots numerieaUy eqwd, hit having opposite signs. EXAMPLES. 1. Given 3x^ + -— = 14, to find the value of x. PROCESS. 3*2 + — = 14 2 Clearing of fractions, 6a;2 -|- a!^ = 28 Uniting terms, Ta;^ = 28 a;2= 4 Extracting the square root, a; = ± 2 2. Given aa;^ _(- c = 6a;2 + d, to find x. PEOCESS. aa;2 -}-c = 6a;2 -\- d Transposing, ckb^ — i^i --^ — ^ Factoring, (a — b)x'^ = d — c „ d — c x^ = a — b Extracting the square root, x = ±-\/ , PUME QUADRATICS. 225 3. Given x^ — 3 = 46, to find the value of x. 4. Given 3x' + 7 = a* -f 15, to find the value of x. 5. Given 2x^—& = G6, to find the value of ». /gz 12 x^ 4 6. Given — = , to find the value of x. 3 4 7. Given 5x^—3^2x^ + 2^, to find the value of a;. 33-2 8. Given 3a;2 + 7 =^^ + 43, to find the value of x. 4 9. Given 2x^ = 1- 35, to find the value of x. 5 ^ 10. Given ^— i \- 8 =42, to find the value of x. 11. Given = -, to find the value of x. 4 a; + 3 j;2 3j.2 3 12. Given 1-9 =; •, to find the value of x. 3 5 13. Given ^ — 1-^^ = 3+, to find the value of x. X — 4 a;-)-4 14. Given -| ==24, to find the value of x. 1 — X 1 -\-x 15. Given = -^^ -, to find the value of x. 15 5 x'^ 16. Given (0:4 2) (a;— 2) = — — 1, to find the value of x. 17. Given \/a-\- x= , to find the value of x. yx — a 18. Given _ =«;-)- VW+W, to find the value of :. 1/3;=' + 5 19, . Given x + yjx^ — 2i/l — a; = 1, to find the value of x, 226 ELEMENTS OF ALGEBRA. 20. Given j/a+x —^|x+■\/x'^—b^,to find the value of x. 21. Given V^ '^' + l l/a;=' __i _ 1^ ^^ ^^ ^^^ ^^j^^ ^^ ^_ l/x^ + l + l/x^ — l 2 22. Given ^^ + ^= = «, to find x-\-y2—x'^ «— 1/2— a;2 x the value of ». 1 -|-a; 1 — K 23. Given =« — > to l+a;+l/l+a;2 ]_—xJ^\/\-\-x'^ find the value of a;. PKOBIBMS. 277. 1. What number is that, to the square of which, if f be added, the sum will be 1 ? 2. Find a number such that the square of f of it will be 7 leas than the square of it. 3. Find a number such that if 320 is divided by it, and the quotient added to the number itself, the sum will be equal to 6 times the number. 4. If a certain number is increased by 3 and also dimin- ished by 3, the product of the sum and diflTerence will be 55. What is the number? 5. Two numbers are to each other as 3 to 5, and the sum of their squares is 3400. What are the numbers? 6. A gentleman said that his son's age was \ of his own age, and that the difierence of the squares of the numbers which represent their ages was 960. What were their ages? 7. A man lent a sum of money at 6^ per annum, and found that, if he multiplied the principal by the number which expressed the interest for 8 months, the product would be $900. What was the principal? AFFECTED QUADRATICS. 211 8. A gentleman has two square rooms whose sides are to each other as 2 to 3. He finds that it will require 20 square yards more of carpeting to cover the floor of the larger than of the smaller room. What is the length of one side of each room? 9. The sum of two numbers is 12, and their product is 27. What are the numbers? Note. — Let 6 + a; = one number, and 6 — x= the other number. 10. The sum of two numbers is 15, and their product is 56. What are the numbers? 11. The sum of two numbers is 13, and their product is 42. What are the numbers? 12. Divide 20 into two such parts that their product will be 96. 13. Divide 32 into two such parts that their product will be 240. 14. A merchant bought a piece of cloth for $24, paying f as many dollars per yard as there were yards in the piece. How many yards were there? 15. A man purchased a rectangular field whose length was 1^ times its breadth. It contaiued 9 acres. What was the length of each side ? 16. Find two numbers which are to each other as 5 to 4, and the sum of whose squares is 164. AFFECTED QUADRATICS. 278. 1. How is a binomial squared? What is the square of a; + 2 ? 2. Since the second term of the square of a binomial con- tains twice the product of both terms, if the second term of the square is given, how may the second term of the binomial be found when the first term of the binomial is known ? 228 ELEMENTS OF ALGEBRA. 3. What term is it necessary to add to a; ^ -1- 4a; to make a perfect square? How is the term found? 4. What term must be added to x^ -{- 6x to make a per- fect square? How is it found? 5. What must be added to a;^ -|- 8a; to make it a perfect square ? 6. What is the square root of the completed square of which a;^ + 8a; are two terms ? 7. What is the square root of the completed square of which x^ -{- 12a; are two terms? 8. What is the square root of the completed square of which x^ + 6a; are two terms? 9. What is the square root of the completed square of which a; 2 4" 10a; are two terms ? 10. What is the square, root of the completed square of which x^ -{- 20a; are two terms ? 11. In the equation a;^ + 4a; -f 4 = 9, what is the square root of the first member ? What is the square root of the second member? 12. Since, in the solution of the equation x^ -\- 4x-\- 4 = 9, we obtain the result a; + 2 = ± 3, how many values has a;? 13. In the equation a;^ + 6a; -f 9 ^ 16, what is the square root of the first member ? What of the second member ? How many values has a;? DEFINIXIONS. 279. An Affected Quadratic Equation is an equation which contains both the first and second powers of the un- known quantity. Thus, x' + 2x = 4, 5i* -f- 6a; = 8, and ax^ + bx = c, are affected quadratic equations. AFFECTED QUABBATICS. 229 280. An Affected Quadratic Equation is sometimes called a Complete QuadraMc EquaJImn. 281. Since affected quadratic equations contain both the second and first powers of the unknown quantity, they may always be reduced to the general form of oa;^ + 6a; := c, in which a and h represent respectively the coefficients of a;^ and X, and c the other terms. 282. Principle. — Every affected quadratie equation has two roots, and only two. These roots are always numerically unequal, except when the second member of the equation reduces to 0. 283. First method of completing the square. EXAHIPrES. 1. Given a;2 + 4a; = 96, to find the values of a;. PKocEss. Explanation. — Com- a;2 _f- 4a; = 96 pleting the square in the ^. + 4^ + 4 = 96 + 4 *f "^"^"^"^ ^^ ^-^f ,^ the square of one-halt *' + 4^ + 4 = 100 the coefficient of x to *" "T -^ ^^ i -'^^ both members, we have a; = 10 — 2 = 8 x^ -\- ix + 4^ = 96 + A. X = — 10 — 2 = — 12 Extracting the square root of each member, we have x + 2 = ±10. Using, first, the positive value of 10, we obtain 8 for one value of x; using next the negative value of 10, we obtain — 12 for the other value of x. VERIFICATION. Substituting the values of x for (>A I on Of! (■\\ ^, ip the original equation, we see ^ J J i o nn /'o\ 'hat the values are correct. 144 — 4o = 96 (^Z) 230 ELEMENTS OF ALGEBRA. 2. Given x^ — 5a! = 24, to find the values of x. PKOCESS. a;2 — 5a; = 24 25 25 Completing the square, x^ — 5x -\ ^24-| 25 121 Uniting terms in second member, x^ — 5x-\ = 4 4 5 11 Extracting the square root, x ^ ± — 5 11 Transposing, a; = — | = 8 ^ z ^^5_11^_ "^ 2 2 Therefore, a; = 8 or — 3 3. Given 2*2 — 7a; = 30, to find the values of x. PROCESS. 2a;2 — 7a; = 30 7 Dividing by coefficient ofa;^, x^ a;=:15 Completing the square, a;^ — -a; + — = 15 -4- ^ = ?^ 2 ^16 ^16 16 7 17 Extracting the square root, x = ± — 4 4 7 17 Transposing, x = — | = 6 4 4 7 17 —10 Therefore, a; = 6 or — 2J AFFECTED QUADRATICS. 231 Ettle. — Bed/uee the eqyation to the form x^ ±bx= ±e, by dividing both members of the equation by the coefficient of ike highest power of the unknown quantity. Add the square of one-half the coefficient of the second term to both members of the equation, extract the square root of each member, and reduce the equaMon. The solution of the equation x^ -{-bx^^c gives » = yjc 62 /c -(- ^- Hence, the values of the unknown quantity may be found as follows : Write one-half the coefficient of the second term u/ith the con- trary sign, ± the square root of the sum of the square of half this coefficient and ^ second member of the equation. Inasmuch as it is impossible to extract the square root of a nega- tive quantity, it is necessary that the term containing the second power of the unknown quantity should have the positiioe sign. If it should be negative, change the signs of all the terms in both mem- bers. Find the values of x in the A. x^ -{• 4x — 45. 5. x'+ 6x = 27. 6. »2+ 8a! = 20. 7. a;«-f 10a; = ll. 8. a;2+20x = 21. 9. x^ + 18a; = 19. 10. a!2 + 24a; = 25. 11. a!2— 12a; = 45. 12. a;2— 8a; = 33. 13. fl;2— 14a; = 51. 14. a;2— 28a; = 60. 15. a;2 — 30x = 64. following equations: 16. 2a;2 + 3a; = 14. 17. 3a;2+4a; =39. 18. 2a;2 + 7a; =39. 19. 5a;2 + 15a; = 50. 20. 6a;2— 21a; = 12. 21. a,_5 a!+l 10 a; — 6 22. a; + 2 8 5 3a; + 4 95 „ 3a; — 3 3a;— 6 a;-3 232 ELEMENTS OF ALGEBRA. 384. Other methods of completing the square. When the second power of the unknown quantity has coefficients, the following method, sometimes called the Hindoo Method, will avoid fractions. Let it be required to find the values of a; in the equa- tion ax^ -j- 6a; = e. PROCESS. ax'' + bx = e (1) h G Dividing by the coefficient of x^, x^ -\ — a; = — (2) a a b b^ c b^ Completing the square, x^ -\ — x-] = " + ■ ■ (3) a 4a^ a 4a* Multiplying by 4a2, 4a2»2 _)_ 4ajbx -f &« = 4ae + 6^ (4) Explanation. — Solving this partially by the preceding method, we find, when the square Is completed, that equation (3) is obtained. By multiplying the members of this equation by 4a^, all fractions are removed, giving equation (4). That is, in compleling the sqvare, frac- tions may be avoided by multiplying both members by 4 times the coefficient of x^, and adding the square of the coefficient of x to both members. When the coefficient of x is an even number, the method may be modified as follows: Let it be required to find the values of x in the equat'on CKc^ + 2dx = c, in which 2d is an even number. PROCESS. ax^ + 2dx = e (1) a;2 -(- = — (2) a a x^-{ a;+ — = -+— ■ (3) a a' a a^ Multiplying by a*, a'^x^ + 2adx -\- d'' — ae -\- d'' (4) AFFECTED QUADRATICS. 233 Explanation. — Solving partially by the first method, equation (4) is obtained after clearing of fractions. That is, in completing the square, when the coefficient of x is even, fi-aetions may be avoided by mvMplying both members of the equMiim, by the coefficient of x^, and add- ing the square of ame-hdf of the coefficient of x to both members. EXAMPLES. 1. Given 2x^ -\-Zx = 14, to find the values of x. PKOCESS. Explanation. — ^Inasmuch o„2 _|_ o». ^ 1 4 as a;'' has a coefficient and the coefficient of a; is not even, to 16a;2 + 24a; + 9 = 112 + 9 avoid fractions we multiply 4a; -|- 3 = db 11 both members by 4 times this 4a; = 8, or — 14 coefficient, or 8, and add the x=i:2 or 3A square of the coefficient of x, or 9, to both members. Extracting the square root of each member, transposing, etc., the values of x are 3, and — 3J. 2. Given 5x^ + 4a; = 57, to find the values of x. PKOCESS. Explanation. — ^Inasmuch g„2 I 4™ ^ 57 as a;'' has a coefficient, and the coefficient of x is an 25a;2 + 20a; + 4 = 275 + 4 even number, to' avoid frac- 5at -|- 2 = dz 17 tions we multiply both mem- 5a; =15, or — 19 bers of the equation by the J — 3 Qj. 34 coefficient of x^, which is 5, and add the square of one- half the coefficient of x, or 4, to both members. Extracting the square root of each member, transposing, etc., the values of x are 3, and — 3f. Rule. — Jff the coefficient of the first power of the unhnown qviantUy be odd, multiply the equation by four times the coefficient of ihe second power of the unknovm quantity, and add the square of the coeffuA&ni of iJie first power to both members. 234 ELEMENTS OF ALGEBRA. If the coefficient of the first power of the unlmmmi quantity he even, imdtiply the equation by the coefficient of the second power, and add the square of one-half the coefficient of the first power to both members. When the coefficient of the second power of the wiknmvn quantity is a perfect square, divide the coeffi^cient of the first power by twice the square root of Hie coefficient of the second power, and add the square of the result to both members. Edract the square root of both members, and find the value of X in the resulting equation. The student may solve, in a manner similar to the above, a'x' -\- bx = c, and deduce a rule similar to that already given for com- pleting the square when the coefficient of x' is a square. Find the values of x in the 3. 3a;2-|-5a! = 8. 4. 2a!2+7a; = 22. 5. 4*2+ 5a; = 84. 6. 5a;2— 4a;=105. 7. 3a;2 — 16a;=140. 8. 4a;2— 7a; = 102. 9. 9a;2-+4a; = 44. 10. 8a;2 — 6a; = 464. 11. 5a;2— 6a;=144. 12. 3a;2+2aa; = 6. 13. a;2— 6a;— 14 = 2. 14 a;2 — 13a; — 6 = 8. 15. a;2 + 17a;— 18 = 0. 16. a;2— 11a;— 7 = 5. 17. 2a;2 — 18a; = — 40. 18. 2a;2H-5a; = 18. 19. 3a;2 4-2a; = 21. 20. 2a;2— 7a; = 34. following : 21. 5a;2 — 6a; = 41. 4a; X — 3 22. a; + 3 2a; + 5 = 2. 23. 3a;- — = 26. X 24. 7 2a; — 5 3a; — 7 4 a; + 5 2a; 25. 3a; — 5 6a; 1 9a; 3x— 25 3 26. 1 2 1 a; — 1 a; + 2 2 27. X 7 a; + 60 3a; — 5 28. a; +11 9 + 4a; X x^ 9Q 4c — 10 7 — 3a; 7 a; + 5 QUADRATIC FORMS. 235 EQUATIONS IN THE QUADEATIO FORM. 285. An equation which contains but two powers of an unknown quantity, the exponent of one power being twice that of the other power, is in the Quadratie Form. These equations in the quadratic form can be reduced to the general form ax^" -\- 6a;" = c, in which n represents any number. EXAMFI^ES. 1. Given k* + Sx^ = 28, to find the values of x. PROCESS. a;*4-8a;2 = 28 9 121 Completing the square, x* -\- Sx^ -\- — = — — 3 11 Extracting the square root, a^ + — = ± — a;2 == 4 or — 7 Extracting the square root, x= ±2 or i'j/ — 7 2. Given a^ + 3a;^=10, to find the values of k. FIEST PROCESS. »* + Bxi = 10 Completing the square, x ^ + 3.*+| = 4 _ 49 4 Extracting the square root. .* + - = "~ 2 xK = 2 or - -5 Raising to third power, X- = 8 or - -125 236 ELEMENTS OF ALGEBRA. SECOND PROCESS. Let ofi ^=^p Then, a;^=p2 Substituting in giver equation, p"^ -(- 3j3 = 10 ^. + 3p+ ^= ^ ■'^ ' 2 ~ 2 p ^ 2 or — -5 Hence, a;* = 2 or - -5 Cubing, a; = 8 or — -125 8. Given a;* — 2x'^ =: 8, to find the values of x. 4. Given x^ — 3a;' = 40, to find the values of x. 5. Given x^ — 4a;' =32, to find the values of x. 6. Given 2a;* — 4a;2 == 16, to find the values of x. 3 7. Given 2a;* — 1 = — , to find the values of x. x^ 8. Given a;* + 4a;* :=: 12, to find the values of a;. 1 1 9. Given x^ -\- 1x^ = 8, to find the values of x. 3 10. Given x* + 3a;^ = 88, to find the values of x. 11. Given x^ -\- 3a;' = 4, to find the values of x. 12. Given ^ — a;* = 20, to find the values of x. 13. Given aa;*" + 6af = c, to find the values of x. 286. Polynomials are sometimes affected with exponents, one of which is twice the other. When such expressions containing the unknown quantity are found in equations, the equations may be solved like the preceding. QUADRATIG FORMS. 237 14. Given (a; + 2) « -f (a; + 2) = 20, to find the values of x. FIRST PE0CES3. (a; + 2)2 + (a; + 2) = 20 1 81 Completing the square, (a; + 2)^ + (a; + 2) + — z= — - 1 9 Extracting the square root, (a; + 2) + — =f ± - a; + 2 = 4 or — 5 a; = 2 or — 7 SECOND PKOCESS. Let J) = (a; + 2) Then, p* = (a; + 2)^ Then, p" +p = 20 l>^+i'+ J = -^ ■^^2 2 p^4: or — 5 a;+2 = 4or— 5 a; = 2 or —7 15. Given (a;" + 1)* + (»'' + 1) = 30, to find the values of X. 16. Given (a;* +4)^ + (a;^ +4) =30, to find the values of X. 17. Given (a; — 1)* +5(a; — 1) =14, to find the values of X. 18. Given (aj^ — 9) ^ — 11 (as* — 9) = 80, to find the values of x. 19. Given (x* — xy — (a;* — a;) = 132, to find the values of X. 238 ELEMENTS OF ALGEBRA. 20. Given x-\-5 — \/x -)- 5 ^ 6, to find the values of x. 21. Given 3a; + 4 + 4 i/3x + 4 = 32, to find the values of X. 22 values of x. Given (— + xV + ( — + x\ = 42, to find the of X. 23. Given / — + a; V + / — + a; \ = 30, to find the values of x. 24. Given (2a;2 — 4a; + l)^ — (2x2 — 4a! + 1) = 42, to find the values of x. 25. Given a:2 — 7a! + 18 + \/x^ — 7x+18 = 42, to find the values of x. 26. Given 2a;2 + 3a; + 9 — 5l/2a;2 + 3a; + 9 = 6, to find the values of x. 27. Given 2(3a;2 + l)^ + 3a;2 + 1 = 63, to find the values of x. 28. Given i/a; + 12 + v^a; + 12 = 6, to find the values of X. PROBLEMS. 287. 1. Find two numbers whose sum is 12, and whose product is 35. SOLUTION. Let X = one. Then, 12 — a; = the other, 12a; — a;2=35 a;2 — 12a; = —35 a;2 — 12a; + 36 = 1 a; — 6=±1 a; = 7 or 5 12 — a; = 5 or 7 QUADRATIC EQUATIONS. 239 2. The sum of two numbers is 10, and their product is 21. What are the numbers? 3. Divide 27 into two such parts that their product may be 140. 4. A rectangular field is 12 rods longer than it is wide, and contains seven acres. What is the length of its sides? 5. A person purchased a flock of sheep for $100. If he had purchased 5 more for the same sum, they would have cost $1 less per head. How many did he buy? 6. An orchard containing 2000 trees had 10 rows more than it had trees in a row. How many i-ows were there? How many trees were there in each row? 7. The difference between two numbers is 2, and the sum of their squares is 244. What are the numbers ? 8. One hundred and ten dollars was divided among a certain number of persons. If each person had received $1 more, he would have received as many dollars as there were persons. How many persons were there? 9. A man worked a certain number of days, receiving for his pay $18. If he had received $1 per day less than he did he would have had to work 3 days longer to earn the same sum. How many days did hte work? 10. Find the price of eggs when 2 less for 12 cents raises the price 1 cent per dozen. 11. A person sold goods for $24, gaining a per cent, equal to the number of dollars which the goods cost him. What did they cost him? Let a;=the cost; then, —= the gain per cent. 12. The expenses of a party amount to $10. If each pays 30 cents more than there are persons, the bill will be settled. How many are there in the party? 13. A picture, which is 18 inches by 12, is to be 240 ELEMENTS OF ALGEBRA. surrounded with a frame of uniform width, whose area i» equal to that of the glass. What is the width of the frame ? 14. A man sold a quantity of goods for $39, and gained a per cent, equal to the number of dollars which the goods cost him? What did they cost him? 15. Two men dig a ditch 100 rods in length for $100, each receiving $50. A is to have 25 cents a rod more than B. How many rods does each dig? What is the price per rod? 16. A rectangular park, 60 rods long and 40 rods wide, is surrounded by a street of uniform width, containing 1344 square rods. How wide is the street? 17. A person purchased two pieces of cloth which to- gether measured 36 yards. Each cost as many shillings per yard as there were yards in the piece. If one piece cost 4 times as much as the other, how many yards were there in each? 18. A person drew a quantity of pure wine from a vessel which was full, holding 81 gallons, and then filled the vessel up with water. He then drew from the mixture as much as he drew before of pure wine, when it was found that the vessel contained 64 gallons of pure wine. How much did he draw each time? 19. Two persons start at the same time and travel toward a place 90 miles distant. A traveled one mile per hour faster than B, and reached the place one hour before him. At what rate did each travel? 20. A person found that he had in his purse, in silver and copper coins, just one doUar. Each copper coin was worth as many cents as there were silver coins, and each silver coin was worth as many cents as there were copper coins. There were in all 27 coins. How many were there of each ? FORMATION OF QUAVBATICS. 241 FORMATION OF QUADRATIC EQUATIONS. 288. 1. What are the factors of a;" + 5a; + 6? 2. If x^ -\- 4x — 5 ==0, to what is each factor equal? 3. If a; — 1 = 0, and a; + 5 = 0, what are the values of x, or the roots of the equation ? 4. If x^ -\- 4x = 5, what is the form when 5 is transposed to the first member? 5. How is the term that does not contain the unknown quantity formed from the roots? How is the coefficient of the first power of the unknown quantity formed from the roots? 289. When the unknown quantities are collected in the first member, and the known quantities united in the second member, the term of the second member is called the Absolute Term. 290. By the solution of the general equation x^ -\-bx=c, the facts developed may be shown in general: x^ -|- 6a; = c '= — t + a/' + t 6 l~b^ — -"2 -A/'+T The sum of the two roots gives — 6. Hence, Peinciples. — 1. The sum of the two roots of an affected quadratic is equal to the coefficient of the first power of the unknown quantity with the sign changed. 2. The product of the two roots is equal to the absolute term vriih the sign clumged. 21 242 ELEMENTS OF ALGEBRA. 1. Form a quadratic equation whose roots are 2 and 3. PROCESS. Explanation. — Since the coefficient 3-1-2=5 "^ *^6 fi""^* power is the sum of the Q w 2 = 6 roots, with the sign changed, and the ab- 2 K fl solute term is the product of the roots with the sign changed, x^ — 5x = — 6 is a quadratic fulfilling the required conditions. Form quadratic equations whose roots are as follows: 2. 3 and 4. 3. 2 and — 5. 4. 3 and 7. 5. — 4 and — 6. 6.-3 and 2. 7. —4 and —5. 8.-2 and 6. 9.-3 and — 7. 10. a and — 6. 11. b and — c. 12. i/5 and 2i/5. 13. 2 -(-t/7 and 2 — 1/7. SIMULTANEOUS QUADRATIC EQUATIONS. 291. A Homogeneous Equation is an equation in which the sum of the exponents of the unknown quantities in each term which contains unknown quantities, is the same. Thus, x^ + 2y' and m-{-y^ are homogeneous equations. 293. Simultaneous Quadratic Equations can usually be solved by the rules for quadratics, if they belong to one of the following classes: 1. When one is simple and the other quadratic. 2. When the unknown quantities in each equation are com- hir\.ed in a similar manner. 3. Wlien each equation is homogeneous and quadraiic. SIMULTANEOUS QUADRATICS. 243 293. The following solutions will illustrate the processes in many of the ordinary forms of simultaneous quadratics: (I.) Simple and Quadratic. 1. Given < ^ >• , to find the values of x and «. \ 2x2 + 2/2 ^ 17 j " SOLUTION. ^+2/ = 5 (1) 2a;2 + y2 = 17 (2) From (1), a =5— y (3) 2a;2 = 50 — 20y + 22/2 (4) Substituting in (2), 50 — 20y+ 2/2 + 2^2 ^ yj (5) Collecting terms, etc., 35/2 — 2O2/ = — 33 (6) Solving, y = ZoT^ (7) Substituting in (1), a; = 2 or 1^ (8) (II.) Unknown quantities similarly combined. 2. Given \ !■ , to find the values of x and y. SOLUTION. a; + j, = 5 (1) a!2/ = 6 (2) Squarmg (1), a;2 + 2*2/ + 1/2 = 25 (3) Multiplying (2) by 4, ^ = 24 (4) Subtracting (4) from (3), x^ — 2xy -\- y"" = 1 (5) Extracting square root of (5), x — 2/ = ± 1 (6) From (1), a; + 2/ = 5 (7) Adding (7) and (6), 2a; = 6 or 4 (8) a; = 3 or 2 (9) Subtracting (6) from (7), 22/ = 4 or 6 (10) 3/ = 2 or 3 (11) 244 ELEMENTS OF ALGEBRA. 3. Given < „ , „ „„ V , to find the values of a; and «. t »2y + a«/2 = 30 J SOLUTION. x-\-y = b (1) x^y + as/2 = 30 (2) Factoring (2), ocy{x + y) = 30 (3) Dividing (3) by (1), xy = % (4) Squaring (1), a;^ + 2as/ + 2/^ = 25 (5) Multiplying (4) by 4, Axy = 24 (6) Subtracting (6) from (5), x"^ — 2xy -\- y"^ := 1 (7) Extracting square root of (7), x — y = + 1 (8) Adding (8) and (1), 2x = 6 or 4 (9) X = 3 or 2 (10) Subtracting (8) from (1), 2j^ = 4 or 6 (11) 2/ = 2 or 3 (12) 4. Given \ ,T , , ^„ )■ , to find tlie values of a; and «• \ a;* 4- y» = 152 J " SOLUTION. a; + y = 8 (1) a;3 4- 2/8 = 152 (2) Dividing (2) by (1), a;^ — a;j/ + i/^ = 19 (3) Squaring (1), x^ -\- 2xy -\- y-^ = 64 (4) Subtracting (3) from (4), 3a;?/ = 45 (5) a^ = 15 (6) Subtracting (6) from (3), x"^ — 2xy -\- y^ = A (7) . Extracting square root of (7) x — y= ±2 (8) Adding (8) and (1), 2a; = 10 or 6 (9) a; = 5 or 3 (10) Subtracting (8) from (1), 2j/ = 6 or 10 (11) 2/ = 3 or 5 (12) SIMULTANEOUS QUADRATICS. 245 (III.) Homogeneous equations. 5. Given < "^ >■ , to find the values of x and y. \2aa/— y2=16j " SOLUTION. x^ — xy=15 (1) 2xy-y^=16 (2) Assume, x = vy (3) Substituting i;^ in (1), v^y^ — vy^ = 15 (4) Substituting in/ in (2), 2vy^ — y^—l& (5) From (4), 2/'="^ (6) From (5), 2/' = ^^^ ^^^ Equating (6) and (7), ^^=^^ (8) 2t; — 1 v^ — V Clearing of fractions and reducing, 1 61; ^ — 46w = — 15 (9) 5 3 Whence, v^— or — (10) 2 8 1 ft Substituting the value of v in (7), y'' =: or 4 (11) 5 — 1 And «2,^-^or— 64 (12) y 1—1 ^ y=±2or± 8-|/^^ K= 5orzt20i/=^ Find the values of the unknown quantities in the following: jx — y=^4.\ \(x + y = 5.\ •\ a;2_ yi = 32.i I ■ I a;2+ 1/2 = 13. J 246 ELEMENTS OF ALGEBRA. 2x + y xy =W.\ = 18. J s.{ 9 f ^ = 6. 1 ■ l2a; —Sy = 9. J \2x^— y^ = U.) ^^ (2x +By =22.1 ■ I 2ot =40. J 3 ^ 12. 13, I +1 =-. X y 10 A + l=i- a;2 ^ ^2 20 f a^=24.) \ a;2_22/2= 4. J 14. { ^=\n \ a; — 2/ = 3. J 15. { ^-^H \a; + 2/= 7./ (■« — 2/ = 1. "I (x +y = 4. I \a;3+2/*=28. J f a;3+ 2/3=28. \ ■ I a!^?/ + 01:^2 = 12. J ( a;3— 2/« =26. \ I OT^ — a;2« = — 6. J 16 17 18. 20, 21 f a;2 — as/= 6. \ U^ +2/2 = 61./ f3x2+ a^ = 18. "1 I 4^2+ 3x2/ = 54. j ra=^+an/ = 70. ) • \a:2/ — y''=12./ r2a;+ 2/ = 22.) ■ I as/ + 22/2 = 120. J (x — ■u = 15. J 25. { a; -|- 42/ = 14. 2/2 — 2y-\-4x = 14. 1 = 11. J 26 f 4»2/+*'^2'^=96. I 1 a; -{- 2/ = 6. J 27/ a;2 +2/2 = 52.1 ' \ 0! +2/ + OT = 34. J x+y + xy-- a;2 +2/^: .aa/ +2/2: 23 ra,2 +2/2 = 13.1 laa/ +«2^15.J (rx+ / = 72. J 30. x + a;3 — 2/' =^ 5^- 16 -y =--- a^ 31. (^ +2/ = 3.1 la;4+2/*=17. J SIMULTANEOUS QUADRATICS. 247 32. Given {(^-2')(^^+^^)=13 I to find . and ,. 33. Given{^'+^ + 2' = 18-y^ | ^ ^^ g^^ ^ ^^^ I ^ a^ = 6. J 34 Given *<> find x and y. 36. Given {'^ + 2' + ^! + ^= M , to find :. and y. 37. Givenf'^'+^'-f + 2') = ^« } , to find . and ,. I xy +(x + y) = S9 i ' ^ 38. Given(^^+2'^+^^^7+"^:-= ^H , to find . and ,. I «* + 2/* = 337 i PROBIiEMS. 294. 1. The sum of two numbers is 8, and their prod- uct 12. What are the numbers? 2. The sum of two numbers is 12, and the sum of their squares 104. What are the numbers? 3. Divide 13 into two such parts that the sum of their square roots is 5. 4. The product of two numbers is 99, and their sum 20. What are the numbers? 5. The sum of two numbers is 100, and the difference of their square roots is 2. What are the numbers? 6.- The difference of two numbers is 2, and the diflTerence of their cubes is 56. What are the numbers? 248 ELEMENTS OF ALOEBBA. 7. Find two numbers whose sum multiplied by the second is 84, and whose difference multiplied by the first is 16. 8. The product of two numbers is 48, and the difierence of their cubes is 37 times the cube of their difference. What are the numbers? 9. The sum of two numbers is a, and the sum of their squares is b. What are the numbers? 10. What two numbers are there such that their sum in- creased by their product is 34, and the sum of their squares diminished by their sum is 42? 11. There is a number expressed by two digits, such that the sum of the squares of the digits is equal to the number in- creased by the product of its digits, and if 36 is added to the number the digits wiU be reversed. What is the number? 12. From two places, distant 720 miles, A and B set out to meet each other. A traveled 12 miles a day more than B, and the number of days before they met was equal to one-half the number of miles B went per day. How many miles did each travel per day? 13. A merchant received^l2 for a quantity of linen, and an equal sum, at 50 cents a yard less, for a quantity of cotton. The cotton exceeded the linen by 32 yards. How many yards did he sell of each? 14. A farmer has a field 18 rods long and 12 rods wide, which he wishes to enlarge so that it may contain twice its former area by making a uniform addition on all sides. What will be the sides of the field when it is enlarged? 15. A merchant bought a piece of cloth for $147, from which he cut off 12 yards which were damaged, and sold the remainder for $120.25, gaining 25 cents on each yard sold. How many yards did he buy? How much did it cost per yard? 16. The fore wheel of a carriage makes 6 revolutions SIMULTANEOUS QUADRATICS. 249 more than tLe hind wheel, in going 360 feet. If the circum- ference of each wheel had been 3 feet greater, the fore wheel would have made only 4 revolutions more than the hind wheel in going that distance. What is the circum- ference of each wheel? 17. Find two numbers such that their sum, product, and difference of their squares shall all be equal. 18. The joint capital of A and B was $416. A's money- was in trade 9 months, and B's 6 months. When they shared stock and gain, A received $228 and B $252. What was the capital of each? 19. A rectangular piece of ground has a perimeter of 100 rods, and its area is 589 square rods. What are its length and breadth? 20. Twenty persons sent together $48 to a benevolent society. One-half the amount was contributed by women, and the other half by men; but each man gave a dollar more than each woman. How many women contributed? How many men? What was the contribution of each? 21. In a purse containing 9 coins, some are of gold, others of silver. Each gold coin is worth as many doUars as there are silver coins, and each silver coin is worth as many cents as there are gold coins, and the value of the whole is $20.20. How many are there of each? 22. The difference of two numbers is 15, and half their prod- uct equals the cube of the smaller. What are the numbers ? 23. A and B set out from two places, C and D, at the same time. A started from C and traveled through D in the same direction in which B traveled. When A overtook B, it was found that they had together traveled 60 miles, that A had passed through D 5 hours before, and that it would have required 20 hours for B to return to C at the rate he had been traveling. What was the distance from C to D? RATIO. 295. 1. How does $2 compare with $6? 3 pounds with 9 pounds? 4 tons with 8 tons? 2. How does 3a compare with 6a? 5a with 15a? 4x with 12a;? 5aa; with lOa^a;? Bax^ with 6a^x^? 3. What relation has 2a to 4a ? 3a to 6a ? 3a2 to 15a2 ? 4. What is the relation of 8a; to 16a;? 5y to lOy? 4axy to Sa^x^y^ ? 5. How does 8a compare with 2a? What is the relation of 8a to 2a? 6. How does 9xy compare with 3xy^ What is the rela- tion of 9xy to Sxy^ 7. What is the relation of 2a to 4a? What is the rela- tion between 2a and 4a? 8. What is the relation of Sa;^ to 9a;2 ? What is the relation between Sx^ and 9a;^ ? 9. What is the relation of (x -\- y) to 2(a! + 1/) ? Be- tween (x + y) and 2(a; + 2/) ? 10. What is the relation of (a + x) to b(a -\- x)7 Be- tween (a -|- x) and b(a + a;) ? 11. What is the relation of Sx'^y to 9x^y? Between 3x^y and 9x^y? DEFttaXIONS. 296. Ratio is the relation of one quantity to another of the same kind. 1. This relation is expressed either as the guotient of one quantity (250) RATIO. 251 divided by the other, and is called Geometrical Eaiio or simply Baiio, or as the difference between two quantitiee, and is called Arithmetical Satio. 2. When it is required to determine whai the rekUicn of one quantity to another is, it is evident that the/rs< is the dividend and the second the divisor. Thus, when the question is, " What is the relation of 5o to 10a?" the answer is J. 3. When it is required to determine the relation between two quan- tities, eiiker may be regarded as dividend or divisor. Thus, when the question is, "What is the relation between 5a and 10a?" the answer is h or 2. 4. The first quantity is commonly regarded as the dividend, al- though whether it should be such or not depends upon the question asked, as shown in Notes 2 and 3. 297. The Terms of a Batio are the quantities compared. 298. The Sign of ratio is a colon ( : ). Thus, the ratio between 12a and 6o is expressed 12a : 6a. The colon is sometimes regarded as derived from the sign of division, by omitting the line. 299. The Antecedent is the first term of the ratio. Thus, in the expression 5a : 3o, 5a is the antecedent. 300. The Consequent is the second term of the ratio. Thus, in the expression 5o : 3a, 3a is the consequent. 301. A Couplet is the antecedent and consequent taken together. 302. A Simple Batio is the ratio of two quantities. Thus, (2a + 6) : 3x, and 3x : 4y, are simple ratios. 303. Katies are compounded by multiplying the ante- cedents of the ratios together, for the antecedent of the 252 ELEMENTS OF ALOEBBA. new ratio, and the consequents for the consequent of the new ratio. Thus, if the ratios a : c and a : b are compounded, the resulting ratio is a' : be. 304. The ratio of the squares of two quantities is called the Duplicate ratio of the quantities ; the ratio of their cubes, their Triplicate ratio. Thus, a' : b' and a^ : b^ are respectively the duplicate and trip- licate ratios of a and 6. 305. Since the ratio of two quantities, as the ratio of a to b, may be expressed by a fraction, as — , it follows that the changes which may be made upon a fraction without altering its value, may be made upon the terms of a ratio without changing the ratio of the terms, since the numerator is the antecedent and the denominator the consequent. Hence — Principle. — Multiplying or dividing both terms of a ratio by the same quantity does not change the ratio of the terms. EXAMPLES. 1. What is the ratio of 3a to 6a? 5a to 10a? 2. What is the ratio of Ix to 35a;? 12ay to 13a? 3. If the antecedent is 15a, and the consequent 20a, what is the ratio? 4. What is the ratio of I to 4? J to ^? f to |? When fractions are reduced to similar fractions they have the ratio of their numerators. 5. When the antecedent is 2a, and the ratio is ^, what is the consequent? PROPORTION. 306. 1. What two numbers have the same relation to each other as 3 to 6? As 2 to 8? As 5 to 15? As 8 to 24? 2. What two quantities have the same relation to each other as 2a to 4a? As 36 to 66? As 86 to 166? 3. What quantity has the same relation to 6a that 26 has to 46? 4. What quantity has the same relation to 10a; that Zy has to %yl 5. What quantity has the same relation to 4a; ^ that 5a has to lOo? 6. What quantity has the same relation to 5a that 56 has to 5a6? 7. What quantity has the same relation to 4aa; that 3a; has to Qxy'i 8. What two quantities have the same ratio to each other that bay has to lOay^ ? 9. What two quantities have the same ratio to each other that 8aa; has to Aax'^t 1 0. How have the two ratios in each of the several exam- ples given above compared in value? DEFINITIONS. 307. A Proportion is an equality of ratios. Thus, 5 : 6 = 10 : 12, and bxy : lOaJw = 4(iK : 802, are proportions. (263) 254 ELEMENTS OF ALGEBRA. 308. The Sign of proportion is a double colon ( = 0- This sign has been supposed to be the extremities of the lines which form the sign of equality. It is written between the ratios thus: X : y :: 2a : 26. The sign of equality is frequently used instead of the double colon. 309. The Antecedents of a proportion are the anteced- ents of the ratios which form the proportion. Thus, in the proportion a:h::c:d,a and c are the antecedents. 310. The Consequents of a proportion are the conse- quents of the ratios which form the proportion. Thus, in the proportion a:b::c:d,h and d are the ccmsequents. 311. The Extremes of a proportion are the first and fourth terms of the proportion. Thus, in the proportion a : b :: e : d, a and d are the extremes. 312. The Means of a proportion are the second and third terms of the proportion. Thus, in the proportion a : b -.-.c : d, b and c are the meam. 313. A Mean Proportional is a quantity which serves as both means of a proportion. Thus, in the proportion a : 6 : : 6 : c, 6 is a mean proportional. 314. Since a proportion is an equality of ratios, and the ratio of two quantities is found by dividing the antecedent by the consequent, it foUows that — Peinciple. — A proportion may be expressed as an eqtudion in which both members are fractions. Thus, the proportion a:b::c:d may be expressed as — = — . h d PRINCIPLES OF PROPORTION. 255 315. Since a proportion may be regarded as an equation in which both members are fractions, it follows that — Pelnciple. — T/ie changes that may be made upon a pro- portion wiOiout destroying the proportion, are based upon the changes that may be made upon an equation without destroying the equality, and upon a fraetimi without aMering its value. PEINCIPLES OF PROPORTION. 316. 1. Let any four quantities form a proportion ; as, a:b::e:d. 2. In what other manner may this proportion be ex- pressed? See Art. 314. Express it in that manner. 3. Clear the equation of fractions. 4. What does each member of the resulting equation contain ? 5. How are the members of the equation produced from the terms of the proportion ? Peincipije 1. — In any proportion the product of the extremes is equal to the product of the means. Thus, when a: b :: c: d, cui = bc. Since a mean proportional serves as both means of a pro- portion, as a : 6 : : 6 : c, it follows that The product of tlw extremes is equal to the square of the mean proportional. DEMONSTRATION OF PRINCIPLE I. Let a: b :: c: d represent any proportion. Then, J=i. a Clearing of fractions, ad = be. Therefore, etc. 256 ELEMENTS OF ALGEBRA. NUMERICAL ILLUSTEATION. 3: 6 :: 8: 16 3X16 = 8X6 48 = 48 317. 1. Change the proportion a:h::c:d into an equa- tion, according to Principle 1. 2. Since, then, ad==^bc, how may the value of a be found? How the value of cZ? What are a and d of the proportion? 3. How, then, may either extreme of a proportion be found? How may either mean be found? Peinciplb 2. — Either extreme is equal to the product of the means divided by the other extreme. Either mean is equal to the product of the extremes divided by the other mean. mi, I, J. J be , be , ad ad Thus, when a:b :: c:a, a=: —-, d:= — , 6= — , c= —-. a a e b Demonstrate Prin. 2, and illustrate its truth with numbers. 318. 1. If ad = be, what will be the resulting equation when both members are divided by bdl 2. Express the resulting equation as a proportion. 3. What does ad, the first member of the equation, form in the proportion? What 6c? Principle 3. — if the product of two quantities is equal to the product of two other quantities, two of them may be made the extremes, and the other two the means, of a proportion. Thu3, when ad = bc, a:b::c:d. Demonstrate Prin. 3, and illustrate its truth with numbers. PRINCIPLES OF PROPORTION. 257 319. 1. Change the general proportion a:h::c:d into an equation, according to Principle 1. 2. Divide the members of the equation by cd. 3. Express the result as a proportion. 4. What change has taken place in the order of an- tecedeuts and consequents, compared with the original proportion ? Principle 4. — If four qvantities are in proportion, the antecedents wUl have the same ratio to each other as the consequents. Thus, when a :b :: c : d, a: c •.•.b : d. When the antecedents have the Bame ratio as the consequents, the quantities are said to be in proportion by AUemation. Demonstrate Prin. 4, and illustrate its truth with numbers. 320. 1. Change the general proportion a:b::c:d into an equation, according to Principle 1. 2. Divide the members of the equation by ac. 3. Express the result as a proportion. 4. What change has taken place in the order of the terms in each couplet, compared with the original pro- portion ? Prcncipm; 5. — Jf four qvmitities are in proportion, the sec- ond will he to the first as iJie fourth to the third. Thus, when a:b::c:d, b:a\:d:e. "When the second is to the first as the fourth is to the third, or when the terms of each ratio are written in the inverse order, the quantities are said to be in proportion by Inversion. Demonstrate Prin. 5, and illustrate its truth with numbers. 22 258 ELEMENTS OF ALGEBRA. 321, 1. Express the proportion a : 6 : : c : c? as a fractional equation. 2. Add 1 to each member of the equation. 3. Reduce each of the mixed quantities to the fractional form. 4. Express the result as a proportion. 5. How are the terms of this proportion formed from the terms of the original proportion? 6. Since, when a : h : : c : d, b : a : : d : e (Prin. 5), if the changes just indicated are made in the second pro- portion, how may the terms of the resulting proportion be obtained from the terms of the original proportion? Peinciple 6. — If four quantities are in proportion, the sum of the terms of the first ratio is to either term of the first ratio as the sum of the terms of the second ratio is to tlie corre- sponding term of the second ratio. Thus, when a:b :: c:d, a-\-b:b :: c-\-d:d and a-\-b:a :: c-\-d:e. When the sum of the terms of a ratio is to one of the terms as the sum of the terms of another ratio is to its corresponding term, the quantities are said to be in proportion by Cam/position. Demonstrate Prin. 6, and illustrate its truth with numbers. 323. 1. Express the proportion a : 6 : : c : d as a fractional equation. 2. Subtract 1 from each member of the equation. 3. Reduce each of the mixed quantities to the fractional form. 4. Express the result as a proportion. 5. How are the terms of this proportion formed from the terms of the original proportion? 6. Since, when a : h : : c : d, b : a : : d : c (Prin. 5), if PRINCIPLES OF PROPORTION. 259 the changes just indicated are made in the second pro- portion, how may the terms of the resulting proportion be obtained from the terms of the original proportion? Peinciple 7. — J^ four quantities are in proportion, the dif- ference between the terms of the first ratio is to eiiher term of the first ratio as the difference between the terms of the second raMo is to the cm responding term of the second ratio. Thus, when a:b :: c:d, a — 6:6 :: c — did and a — b:a :: c — d:c. When the difference of the terms of a ratio is to one of the terms as the difference of the terms of another ratio is to its corresponding term, the quantities are said to be in proportion by Division. Demonstrate Principle 7, and illustrate by numerical examples. 323. 1. Change the proportion a:b::c:d, according to Principle 6. Express the resulting proportion. 2. Change the same proportion according to Principle 7. Express the resulting proportion. 3. Change these proportions to fractional equations. 4. Divide the first equation by the second. 5. Express the result as a proportion. 6. How are the terms of this proportion formed from the terms of the original proportion? Peinciple 8. — Jff four quantities are in proportion, the sum of the quantities which form tJie first couplet is to their difference as the sum of the quantities which form the second couplet is to their difference. Thus, when a: b ■.: c : d, a-\-b : a — 6::c-fd:c — d. Demonstrate Principle 8, and illustrate by numerical examples. 260 ELEMENTS OF ALGEBRA. 324. 1. Express the proportion a:h::c:d as a fractional equation. 2. Eaise both members to the nth power. 3. Express the nth root of both members. 4. Express each of the equations as a proportion. 5. How may these proportions be formed from the original proportion ? PiUNCiPLE 9. — If four quantities are in proportion, the same powers of those quantities, or the same roots, vnll he in pro- portion. i. i- i J. Thus, when a : 6 : : c ; d, a" : 6» : : c" : d" , and a" : 6" : : c" : d" . Demonstrate Principle 9, and illustrate by numerical examples. 325. 1. Express the proportion a : 6 : : c : d as a fractional equation. 2. What may be done to a fraction without changing its value ? 3. Multiply the terms of the first fraction by m, and the terms of the second by n. 4. Express the result as a proportion. 5. How are the terms of this proportion formed from the original proportion? PEmciPLE 10. — If four quantiiies are in proportion, any equi-multiple of the terms of tiie first couplet will he proportional to any equir^multiple of the terms of the second couplet. Thus, when a:b :: c:d, ma : mb ■.: nc: nd. Demonstrate Principle 10, and illustrate by numerical examples. PRINCIPLES OF PROPORTION. 261 326. 1 . Express the proportions a:6::c:tZ aiida;:i/::z:w as fractional equations. 2. Multiply the resulting equations together. 3. Express the resulting equation as a proportion. 4. How are the terms of this proportion formed from the terms of the original proportions a : h :: e : d and xiy.-.sxvi^ PEmcrPLE 11. — If Jour quantities in proportion are multi- plied term by term by four other quantities in proportion, the products vnll be in proportion. Thus, when a:b : : c:d and x:y :: z:w, ax-.by :: ex: dw. Demonstrate Principle 11, and illustrate by numerical examples. Prove that the quotients will be in proportion if the proportions are divided term by term. 327. 1. Express the proportions a:b::c:d and a:b::e:f as fractional equations. 2. Since the first members of^ the equations are equal, what will the second members form? 3. Express the resulting equation as a proportion. 4. How are the terms of this proportion formed from the terms of the original proportions a:b :: e-.d and a : 6:: e:/? Prtnciplb 12. — ^ two proportions have a couplet in each, the same, the other couplets will form a proportion. Thus, when a:h : : c:d, and a:h : : e :/, then c:d::e:f. Demonstrate Principle 12, and illustrate by numerical examples. 262 ELEMENTS OF ALGEBBA. EXAMPLES. 328. 1. la 5 : 8 : : 4 : a;, find the value of x. SOLUTION. 5 : 8 : : 4 : a; By Prin. 1, 5a; = 32 Therefore, a; = 6|- In Rolving examples like the following, the student should employ as many of the Principles of Proportion as are applicable. 6, find the value of x. 10, find the value of x. 4, find the value of x. 12, find the value of x. : 6, find the value of x. — 2 : : 2x -\- 1 : X -\- 2, find the value between two men so that their shares shall be in the proportion of 3 to 7. 9. There are two numbers in the ratio of 2 to 3, and if 8 is added to each, the ratio of the resulting numbers wiU be 5 to 7. What are the numbers ? 10. There are two numbers which have to each other the ratio of 3 to 5, and if 4 is added to each, the results will have the ratio of 2 to 3. What are the numbers ? 11. Mr. A's crop of wheat was to his crop of oats as 2 to 3. If he had raised 50 bushels more of each, the quantity of ■wheat would have been to the quantity of oats as 5 to 7. How many bushels of each kind of grain did he raise? 12. Find two numbers such that the greater is to the less as their sum is to 6, and the greater is to the less as their diflference is to 2. 2. In 3 : a; : ; : 4: 3. In a; : 5 : : 3: 4. In 4 : 6 : : ; X : 5. In 3 : a; : : X : 6. In a; : 4 : : :x^ 7. In x—1 : X of X. 8. Divide $4 [0 1 PRINCIPLES OF PROPORTION. 263 SOLUTION. Let a; = the greater; y=:the less. By the conditions, x : y :: x-\-y X : y :: x — y By Prin. 12, x-\-y : 6 :: x — y By Prin. 4, x^y : x — y : : 6 By Prin. 8, 2x : 2y: i 8 By Prin., Art. 305, x: y :: 2 From (1) and (6), Prin. 12, x + y : 6 :: 2 From (2) and (6), Prin. 12, x — y:2::2 From (7), From (8), 6 (1) (2) (3) (4) (5) (6) (7) (8) x + y = x—y = 12 (9) Whence, 4 (10) 2a; = 16 (11), a = 8 (12), y = 4 (13) 13. The product of two numbers is 20, and the diiference of their squares is to the square of their difference as 9 :' 1. By the conditions, SOLUTION. Let a; = the greater; y^the xy = 20 U2 — 2/2 : (x — yy Dividing first couplet by (x — y), Prin., Art. 305, x-\-y : x- By Prin. 8, 2x : 2y : : 10 By Prin., Art. 305, X : y : : 5 By Prin. 1, 4x = 5y by Substituting in (1) ^^^=20 4 y2 = 16 y =±4 a; =±5 HI) l/(2) 1 (3) (4) (5) C6) (7) (8) (9) (10) (11) 264 ELEMENTS OF ALGEBRA. 14. Find two numbers such that their sum is 8 and their product is to the sum of their squares as 15 to 34. 15. Find two numbers whose difference is 3, and whose product is to the sum of their squares as 10 is to 29. 16. What two numbers are those whose sum is to their dif- ference as 7 to 1, and whose product is to the sum as 24 to 7? 17. The sum of two numbers is 12, and their product is to the sum of their squares as 2 to 5. What are the numbers? 18. The sum of two numbers is 6, and the sum of their squares is to the square of their sum as 5 to 9. What are the numbers? 19. What two numbers are those whose product is 12, and the difference of whose cubes is to the cube of their difference as 37 to 1? FRACTIONAL EQUATIONS SOLVED BY THE PRINCIPLES OP PROPORTION. 329. Since a proportion is an equality of ratios, and the ratio of two quantities may be expressed as a fraction, it is evident that the Principles of Proportion are applicable to equations which have both members fractions. Eegarding the numerator of each fraction as an anteced- ent, and the denominator as a consequent, the terms of each fraction as a couplet, and the equation as a proportion, the Principles of Proportion may be easily applied. 1. Given =— , to find x. a; + i/a; + 1 H SOLUTION. X — Vx - fl _ 5 aj + i/^+T ~ 11 (1) PRINCIPLES OF PROPORTION. 265 By Principle 8, the sum of the numerator and denomina- tor of each member, divided by their difference, will form an equation. Hence, _Jl^ = J^ (2) By Prin., Art. 305, ■ " = — (3) l/a+l 3 Squaring, ^ = — (4) Clearing of fractions, etc.j 9a;'' — 64iB = 64 (5) Whence a; ^ 8 or — — y 2. Given — ' z=::- = — . to find x. V'x-\-a-\-'[/x — o 2o SOLUTION. l^x^a — ti/x — a x \/x + « + V^* — » 2a (1) ■D T> • a 2t/a!. + « x + 2a By Prm. 8, „ . = "T (2) ^y X — a 2a — X By Prin., Art. 305, J^ = ^±^ (3) yx — a 2a — x . x-\-a a;^ -f 4aa; 4" 4a* . Squaring, -^^ = ,^-4ax + 4a^ ^^^ ^ ^ . o 2a; 2a;2 + 8a2 ..^ By Pnn. 8, - = -^ (5) ~ „ . „«_ a' a;*+4a* .„, By Prin., Art. 305, — = —j^ (6) Dividing denominators by a, a; = — - — — (7) Whence a: = ± 2al/4 266 ELEMENTS OF ALGEBRA. Solve the following by applying the Principles of Proportion when possible: „ „. i/6i— 2 Vto — 9 ^ „ , 3. Given — ^^ = n: , to nnd x. l/6a; + 2 4:y^6x + 6 . _,. VHk — h S-\/ax — 26 , ~ , 4. Given — ^:z = , to iind x. Vax -\- b ^yax + 56 _ „. l/4c + l + i/4» Q X fl J 5. Given — z== — ■ ;;rr = 9, to nnd x. -\/4x + 1 — y4x In the solution the second member may be written as |. ^. l/a + x + Va — x J. ^ £ J 6. Given — ^i:;;^ — rrr: = 6, to find x. ■j/a + a — V « — X 7 P- v^M^t — ^^^^^^ 1 * fl J 7. Given — rr::::i:rr =;rr=:r = — > to find x. \/x' + l + Vx' — l 2 - „. 3t/J— 4 3i/J+15 , „ , 8. Given — *^— = — ^ , to find x. l/F+2 i/a; + 40 Multiply the denominators by 3, and apply Prin. 7. 9. Given ■ = — ^r^ . to find x. i/k + 4 Vx + &' in n- a + a; -f l/2aie + x "^ i . n j 10. Given ■ — . = 6, to find a;. a-\-x — y2ax -|- x^ 11. Given — zrrrzzr: = — , to find x. Va' -{-x'—x c „. i/a + j/a — a;2 . „ , 12. Given — — — "^ = a, to find a;. ya— va —x^ PROGRESSIONS. 330. 1. How does each of the numbers 2, 4, 6, 8, 10, 12, compare with the number that follows it? 2. How may each of the numbers 4, 6, 8, etc., be ob- tained from the one that precedes it? 3. Write five numbers in succession, beginning with 2 and increasing regularly by 3. 4. Write five quantities in succession, beginning with x and increasing regularly by 2«. 5. Write a series of five quantities, beginning with a and increasing regularly by d. 6. How does each of the quantities 2x, 4k, Sw, 16a!, com- pare with the one that precedes it ? 7. Write a series of six quantities, beginning with 2a and increasing by a constant multiplier 3a. 8. Write a series of six quantities, beginning with a and increasing by a constant multiplier r. DEFrNITIONS. 331. A Series of quantities is quantities in succession, each derived from the preceding according to some fixed law. 332. The first and last terms of a series are called the extremes, the intervening terms the means. Thus, in the Reries o, a -)- rf, a + 2rf, a -|- 3d, the quantities o and a4-Zd are extremes, and the others are meam. ' (267) 268 ELEMENTS OF ALGEBRA. 333. An Ascending Series is one in -which the quanti- ties irufrease regularly from the first term. Thus, 2, 4, 8, 16, and a, a + d, a + 2d, etc., are ascending series. 334. A Descending Series is one in which the quanti- ties decrease regularly from the first term. Tlius, 24, 12, 6, 3, and a^'a — d, a — 2d, a — 3d, are descending ARITHMETICAL PROGRESSION. 335. Au Arithmetical Progression is a series of quan- tities which increase or decrease by the addition or subtrac- tion of a constant quantity. Thus, 4, 6, 8, 10, 12, 14, and a, a — d, a — 2d, a— 3d, are arith- metical progressions. 336. The constant quantity which is added or subtracted is called the Common Difference. Thus, in the progression 2, 4, 6, 8, 10, the common difference is 2. CASE I. 337. To find the last term. 1. In the arithmetical progression 2, 4, 6, 8, 10, what is the common difierence? How is the second term obtained from the first? How is the third term obtained from the first? How is the fourth? 2 . In the arithmetical progression x, x-\-2, x-\-4, a; + 6, what is the common diflference? How many times does the common diflference enter into the second term? How many times into the third term ? How many times into the fourth term? ARITHMETICAL PMOGRESSIOK 269 3. In the series a, a -\- d, a -\- 2d, a-\-M, how is the second term formed from the first term? The third term? The fourth term? 4. Since, in an arithmetical series in which a is the first term and d the common difierence, the first three terms are a, a-\-d, a-^2d, what is the fourth term? The seventh term? The eleventh term? Any term? 5. If the above series were descending, the first three terms would be a, a — d, a — 2d. What would be the fifth term? The seventh term? The eleventh term? Any term? 338. Let a represent the first term, d the common dif- ference, I the last term, and n the number of terms. Then, since each term contains the first term, increased or dimin- ished by the common difierence multiplied by the num- ber of terms less 1, according as the series is ascending or descending, the rule for finding the last term may be expressed by the following formula: l = a±(n — T)d. That is, The. last term is equal to the first term increased by the com- mon difference multiplied by the number of terms less 1, when the series is ascending, or decreased by the common difference multiplied by the number of terms less 1, when the series is KXAMFIiES. 1. Find the 15th term of the series 1, 3, 5, 7, etc. ExPLAJJATiON. — ^InthisexEimplea = l, l^a -{-(n — V)d d =: 2, and n = 15. Substituting these l^\ -\- (15 — 1)2 values in the formula, the value of I, or I __ 29 the last term, is 29. 270 ELEMENTS OF ALGEBBA. 2. Find the 18th term of the series 4, 7, 10, 13, etc. 3. Find the 12th term of the series 3, 7, 11, 15, etc. 4. Find the 10th term of the series \\, 2, 2^, 3, etc. 5. Find the 13th term of the series 2J, 3, 3|, i\, etc. 6. Find the 12th term of the series 25, 23, 21, 19, etc. 7. Find the 20th term of the series 8, 4, 0, — 4, etc. 8. Find the 30th term of the series a, 2a, 3a, 4a, etc. 9. Find the 18th term of the series x, Bx, 5a;, 7x, etc. 10. Find the 15th term of the series |, ||, f , ^, etc. 11. Find the wth term of the series 1, 3, 5, 7, etc. 12. A boy agreed to -work for 50 days, at 25 cents for the first day, and an increase of 3 cents per day. What were his wages the last day? 13. A body fells 16^ feet the first second, 3 times as far the second second, 5 times as far the third second. How far will it fall the seventh second? CASE n. 339. To find the sum of the terms. 1. What is the sum of the terms of the series 2, 4, 6, 8, 10? Since there are five terms, what is the average terra? How does it compare with the sum of tho first and last terms? 2. What is the sum of the terms of the series 3, 6, 9, 12, 15? Since there are five terms, what is the average term? How does it compare with the sum of the first and last terms ? 3. What is the sum of the series 1, 3, 5, 7, 9, 11, 13? Since there are seven terms, what is the average term? How does it compare with the sum of the first and last terms? ARITHMETICAL PBOQBESSTON. 271 4. How does the average term ia any arithmetical series compare with the sum of the first and last terms? 340. The formula for the sum of an arithmetical series may be deduced as foUows: Let a represent the first term ; d, the common difference ; I, the last term ; to, the number of terms ; and s, the sum of the terms. Writing the sum of a series of four terms, we have 8= a +(a+d) + C a+2d) + ( a+3(f) Inverting, g = ( a+3d) + ( a+2d) + (_ a+ d)+ a Adding, 2s = (2a+Sd) + (2a+Zd) + (2a+3d) + (2a+3d) Whence, 28:=4(2a+3d) or 4 times the sum of the first and last terms. And in general, 2s=n(a-\-t) Whence, s=—(a-\-l) or nl "^ |. That is. The sum of any arUhnetical progression is equal to one-half the sum of the extremes multiplied by the number of terms. EXAMPLES. 1. What is the sum of the series 2, 4, 6, 8, etc., contain- ing 12 terms? PBOCESS. l^^a-\-(n — T)d Explanation.— Since the last i ^ 2 + (11 X 2) ^ 24 term is not given, it is found by jj I ^ \ the previous case to be 24. Then, (jj I ^ \ the previous case to oe zt. inen, X I by the formula given for obtaining the sum, it is found to be 156. 272 ELEMENTS OF ALGEBRA. 2. What is the sum of 12 terms of the series 1, 3, 5, 7, etc.? 3. What is the sum of 9 terms of the series 4, 6, 8, 10, etc.? 4. What is the sum of 8 terms of the series 5, 8, 11, 14, etc.? 5. What is the sum of 7 terms of the series 3, 4|-, 6, 7|, etc.? 6. What is the sum of 8 terms of the series 3a, 5a, 7a, 9a, etc.? 7. What is the sum of 9 terms of the series a-\-b, a-\-h -\-e, a-\-h-\-2c, etc.? 8. What is the sum of n terms of the series x, 3a;, 5a;, 7a;, etc.? 9. What is the sum of 8 terms of the series 2, 1, 0, —1, —2, etc.? 10. A man walked 15 miles the first day, and increased his rate 3 miles per day. How far did he walk in 11 days? 11. How many strokes does a common clock strike in 12 hours? 12. A person received a gift of 1100 per year from his birth until he was 21 years old. These sums were de- posited in a bank, and drew simple interest at Q%. How much was due him when he became of age ? 341. Formulas for finding any element. By combining the fundamental formulas given in the pre- vious cases, all problems which may arise in Arithmetical Progression may be solved. When any three of the elements are given, the other two may be found. ARITHMETICAL PBOORESSION. Deduce the following formulas: 273 K FOBMUIiAS. I, d, n, Cj U, If d, n, g, a, n, s, 1, n, s, lij d, /, a, I, s, Uf dj 8, k d, 8, I, s. a, s. d, s. a, I. d, I. d, a. n, s. n,d. I, n. a, n. I =a+{n—l)d. a=Z — (ji — l)d, J I — a a= • n— 1 2g — n(n — l)d d= 2n 2(g — an) n{n — 1) i{nl—s) n{n — 1) I— a , d 2s s=^{2a+{ri^l)d). g=j7i(2Z— (ji— l)d). , 2s+nin—l)d ~ 2» , 2s I = a. n 2s , n Jjl+a) (l—a+d) *"" 2d (l+a) {I— a) ' 2a—(l+a) ' a=ld±.V {l+\d)^—'Mi. %+d±V (2l+dY—%ds 2d SPECIAL APPLICATIONS. 343. In solving some of the problems in Arithmetical Progression there are several ways of representing the un- known terms of the series. 1. When X represents the first term of a series, and y the common difference, the series is represented by X, x-\-y, X +.22/, X + 3^, etc. 274 ELEMENTS OF ALOEBRA. 2. When there are three terms in the series, the middle term may be represented by x, and the common difference by y; as, x — y, x, x -\- y. 3. When there are five terms in the series, the middle term may be represented by x, and the common difference by y; as, x — 2y, x — y, x, x-\-y, x-\-2y. 4. When there a,re four terms in the series, x — By may represent the first term, and 2y the common difference; as, x — 3y, x — y, x + y, x + 3y. It is obvious that by this notation the mm of the quantities contains but one unknown quantity. probi:b:ms. 1. There are three numbers in arithmetical progression, whose sum is 18 and the sum of whose squares is 116. What are the numbers? SOLUTION. Let X — y==the first term. X = the second term. x-\-y^ the third term. y = the common difference. By the conditions, < > ^ ^ 1 3a;2 + 22/2 = 116 J (2) From (1), x = 6 (3) Substituting in (2), 108 + 2y^ = 116 (4) Whence, y = 2 (5) 'x — 2/ = 4, 1st term^ (6) Therefore, \ a; = 6, 2d term • (7) jr-f 2/ = 8, 3d term! (8) ARITHMETICAL PROORESSION. 275 2. The first term of an arithmetical series is 5, the last term 92, and the sum of the terms 1455. What is the num- ber of terms ? 3. The first term of an arithmetical series is 2, the lasj term 30, and the sum of the terms 160. What is the num- ber of terms ? 4. The first term of an arithmetical series is 16, the com- mon difierence —3^, and the sum of the terms 30. What is the number of terms ? 5. The sum of three numbers in arithmetical progression is 15, and the product of the second and third is 35. What are the numbers? 6. The sum of three numbers in arithmetical progres- sion is 9, and their product is 15. What are the num- bers? 7. The sum of three numbers in arithmetical progression is 18, and the sum of their squares is 126. What are the numbers ? 8. There are three numbers in arithmetical progression such that the product of the first and third is 16, and the sum of the squares of the numbers is 93. What are the numbers? 9. There are three numbers in arithmetical progression such that the first is 3, and the product of the first and third is 21. What are the numbers? 10. The sum of four numbers in arithmetical progression is 10, and their product is 24. What are the numbers? 11. There are four numbers in arithmetical progression such that the product of the first and fourth is 27, and the product of the second and third is 35. What are the numbers? 12. There are four numbers in arithmetical progression such that the product of the fourth number by the common 276 ELEMENTS OP ALGEBRA. difference is 16, and the product of the second and third is 24. What are the numbers? 13. There are five numbers in arithmetical progression such that their sum is 40, and the sum of their squares 410. What are the numbers? 14. The sum of five numbers in arithmetical progression is 25, and their product is 945. What are the numbers? 15. The product of four numbers in arithmetical progres- sion is 280, and the sum of their squares is 166. What are the numbers? 16. A number is expressed by three digits which are in arithmetical progression. If the number is divided by the sum of the digits, the quotient will be 26, and if 198 be added to the number, the digits will be inverted. What is the number? GEOMETRICAL PROGRESSION. 343. A Geometrical Progression is a series of quanti- ties which increase or decrease by a constant multiplier or divisor. Thus, 2, 4, 8, 16, 32, and a5', ab^, ah, a, are geometrical pro- greesions. 344. The constant multiplier or divisor is called the Ratio. Thus, in the progression 2, 4, 8, 16, 32, the ratio is 2. CASE I. 345. To find the last term. 1. In the geometrical progression 2, 4, 8, 16, 32, what is the ratio? How is the second term obtained from the first? GEOMETRICAL PROGRESSION. 277 How is the third term obtained from the first? How is the fourth term obtained from the first? How is the fifth term obtained from the first? 2. In the geometrical progression x, xy, xy^, xy^, osy*, what is the ratio? How many times does the ratio enter as a factor into the second term? How many times into the third term? How many times into the fourth term? How many times into the fifth? 3. Since, in a geometrical series in which a is the first term and r the ratio, the first four terms are a, ar, ar'^, esr*, what is the fifth term? The seventh term? The eleventh term? Any term? 346. Let a represent the first term, r the ratio, I the last term, and n the number of terms. Since each term contains the first term multiplied by the ratio used as a factor 1 less time than the number of terms, the rule for find- ing the last term may be expressed by the following formula : 1 = 0^-^. That is, The last term is equal to the first term, multiplied hy the rcdio raised to a power whose index is 1 less than the number of terms. EXAMPLES. 1. Find the 8th term of the series 2, 4, 8, etc. PEOCESS. Explanation. — In this example a = 2, ?• = 2, I = af^^ and Ji := 8. ; _- 2 X 2'' Substituting these values in the formula, the I =z 256 value of I, or the last term, is 256. 2. Find the 6th term of the series 5, 10, 20, etc. 3. Find the 9th term of the series 2, 4, 8, etc. 4. Find the 7th term of the series 3, 9, 27, etc. 5. Find the 10th term of the series 1, 2, 4, 8, etc. 278 ELEMENTS OF ALGEBRA. 6. Find the 7th term of the series 2a, 4a^, 8a*, etc. 7. Find the 9th term of the series 3, &ax, 12a''a;2, etc. 8. Find the nth term of the series 1, 2, 4, 8, etc. 9. Find the nth. term of the series 3, 12, 36, etc. 10. Find the, 8th term of the series 3, 1, J, etc. 11. If a person should be hired for 8 days for $1 the first day, 13 for the next day, $9 for the third day, and so on, what would be his wages for the last day? 12. If a man begins business with a capital of $1000, and doubles' it every three years, how much will he have at the end of 15 years? CASE n. 347. To find the sum of a series. 1. In the geometrical series 5, 15, 45, 135, 405, what is the ratio? 2. If each term of this series is multiplied by the ratio, how will the terms of the product compare with the terms in the given series? 3. Since all the terms of both series except two are alike, if the sum of the terms of the given series is sub- tracted from the sum of the terms of the derived series, what terms wiU the remainder contain? 4. Since the sum of the given series was subtracted from the same series multiplied by the ratio, when the subtraction is performed, how many times the sum of the given series remains ? 5. Since the sum, multiplied by the ratio — 1, is equal to the first term multiplied by the ratio raised to a power equal to the number of terms, and the product dimin- ished by the first term, how may the sum of a geometrical series be found? GEOMETRICAL PROGRESSION. 279 348. The formula for the sum of a geometrical series may he deduced as foUows; Let a represent the first term ; r, the ratio ; I, the last term ; n, the number of terms ; and s, the sum of the terms. Then, s^=a-{-ar -\- ar^ -\- ar^ . . . . -f ar'^'^ (1) rs= ar -\- ar"^ -\- ar^ .... + «^"~'+ t**"" (2) Subtracting (1) from (2), rs — s = ar" — a (3) Whence, (r — l)s = ar" — a, or s^ — (4) r — 1 By formula. Case 1, 1 = ar'^'^ ; therefore, rl = ar". Substituting rl for ar" in the formula for s, the following formula is obtained* • rl — a 8 = r EXAMPLES. 1. Find the sum of 10 terms of the series 2, 4, 8, etc. PROCESS. „ -r 1 . Explanation. — ^In tnis ex- _ ar'' — a pie, a = 2, r = 2, » = 10. Sub- r — 1 stituting in the first formula O \/ 910 9 obtained for the sum, the sum s ^ iySiA ZlA ^ 2046 is 2046. 2 — 1 2. Find the sum of 11 terms of the series 1, 2, 4, 8, etc. 3. Find the sum of 9 terms of the geometrical series 1, 3, 9, 27, etc. 4. Find the sum of 12 terms of the geometrical series 4, 8, 16, 32, 64, etc. 5. Find the sum of 11 terms of the geometrical series 3, 9, 27, 81, 243, etc. 280 ELEMENTS OF ALGEBRA. 6. Find the sum of 10 terms of the geometrical series 2a, 4a, 8a, etc. 7. Find the sum of 10 terms of the geometrical series 2a;2, 6a;S ISa;^, etc. 8. Find the sum of ?i terms of the series 2, 4, 8, 16, etc. 9. Find the sum of 10 terms of the series 2, 1, ^, \, etc. 10. Find the sum of 8 terms of the series 8, 2, ^, |, etc. 11. The extremes of a geometrical progression are 4 and 1024, and the ratio 4. AVhat is the sum of the series? 12. The extremes of a geometrical progression are 2 and 512, and the ratio 2. What is the sum of the series? 13. What is the sum of a series in which the first term is 2, the last term 0, and the ratio \; or what is the sum of the infinite series 2, 1, ^, \., etc. ? , 14. What is the sum of the infinite series 6, 3, \\, etc. ; or what is the sum of a series in which the first term is 6, the last term 0, and the ratio ■!■? 15. What is the sum of the infinite series 2, |, ^, etc,? 16. What is the sum of the infinite series 1 -\ -{• X 2 \-\-\, etc.? 17. What is the sum of the infinite series x — y -{• X x^ x^ 18. A man engaged to work for 8 months, upon condition that he should receive f 2 for the first month, $4 for the second, 18 for the third, and so on. How much did he earn in the time? 19. A man rented a ferm of 500 acres for 20 years, agree- ing to pay $1 for the first year, $2 for the second year, $4 for the third year, and so on. What was the entire amount of rent paid for the farm? GEOMETRICAL PBOGBESSION. 281 349. Formulas for finding any element. By combining the fundamental formulas given in the previous cases, all problems which may arise in Geometrical Progression may be solved. When any three of the elements are given, the other two can be found. Deduce the following formulas: H « FORMULAS. I, s. l=ar'-'. ai^ — a a, r, n, ' r-1 a,s. I li"—l l,r,n, ' ^_^i n, r, s, a, I. «=fc?- (r-l)8r-> a, I, n, T, s. ^ = -^1- a, n, s, r, I. ai" — re = a —s. i{s— «)"-• = a(s—o)"-\ I, n, «, r, u.. (s-r)r' + l-- = SI^\ a(g_„)'>-> = i(g_i)-^. a, r, I, 8, ra. rl — o log. i— log. a , ,'• log.r r, n. « — fl log. i— log. o 1 1 a, I, s, ""log.fg— a)— log.(! -I) ' a, /, s, I, n. r 1)3 loK.(o+(r-l)3)- n — , log.r -log. a ;, r, s, a,n. a=Tl—{r- -l)s. loK.l— log. {lr—(r— n= -— ; log.r !)') + !. The values of n are given here to complete the Bcheme. They may be found by the student after studying Logaiithms. 282 ELEMENTS OF ALGEBRA. SPECIAL APPLICATIONS. 350. 1. When x represents the first term and y the ratio, the series may be represented by X, xy, xy^, xy^, ocy^, etc. 2. When there are ana lution by logarithms we Therefore, (25)2=625. « j *i, i vi. may find the logarithm of the given quantity, and multiply it by the exponent of the power to which it is to be raised; the number corresponding to the resulting logarithm will be the power sought. 2. "What is the second power of 19? 3. What is the second power of 35? 4. What is the second power of 45? 5. What is the second power of 29? 6. What is the third power of 32? 7. What is the third power of 25? 8. What is the third power of 14? 365. Evolution by Logarithms. Since in involution the logarithms, or exponents, are multiplied to produce the power, what must be done when roots are to be extracted? 294 ELEMENTS OF ALGEBBA. 1. What is the square root of 625? PEOOESS. Explanation. — Since in evo- T f es>K ' 'y 7qF;8S l"*ioi "^^ divide the exponent °' ' of the quantity by the number Dividing by 2, 1.39794 corresponding to the root to be 1.39794 is the log. of 25. extracted, in evolution by loga- Therefore, (625)* = 25. rithms we find the logarithm of the given number, and divide it by the index of the required root; the number corresponding to the resulting logarithm will be the root sought. 2. What is the square root of 196? 3. What is the square root of 256? 4. What is the square root of 4096? . 5. What is the square root of 1296? 6. What is the cube root of 4096 ? 7. What is the cube root of 13824? 8. What is the cube root of 74088? MISCELLANEOUS EXAMPLES. 366. 1. Add Z:^y— 4x VT+ 5, V^ -\- 2xy^ -\- i, QyVx'— Vxy — 7, 4yj/x~— Zy^x — 6, and 2 + bxy^ — 3yx^. 2. Add 2x*y'^ +2ar^y^—Sx'^, 2b^ar"y^~a + Qx^, Bx*y^— 5ex*y^ — 262a;-»yT -f 3a, and 2x^ + ey. 3. Add 4b — 2ey~^ + m to 7c2/~* + 8aa; — 56+ lOoa;-^ 26 -f 8m — 3, and subtract from the result the sum of 5ax — 4m -j- 3, 5cy ^ — Sax — 6, and 3m — lOej/ ^ — 2m. 4. From 17an/2 + 3x3 + lOa subtract lny^ -\-4xz-\- 12a;— 26a!. MISCELLANEOUS EXAMPLES. 295 5. Multiply a* -|- a^b + a^b"^ + ab^ + 6* by a — 6. 6. Multiply Sa;"-! —2^-^ by 2x — ^y'^. 7. Multiply 3a; ""^ — 2/ by Zx~^-^2y^. _ wi m — n m m — n 8. Multiply 2a;2» — ^'~^ by 2a;2^ -|- Sy^z". 9. Expand (a;^" + j/Z") (a;2" -f yZ-") 10. Divide a;* — y* by a; -(- y. 11. Divide x'' -\-y'' by a; + y- 12. Divide a;" — 1/" by x — y to 6 terms. 13. Divide — + a!^ + — + - by - + 1. 2 8 4 2 14. Factor 4a;'* + 'kny -f- 2/^- 15. Factor a;* — y*. 16. Factor a;^— 2a; — 35. 17. Factor a;^ — 6a; — 27. 18. Factor x^—y^. 19. Find the greatest common divisor of x^ — y"^, x"^ — 2as/ + y^, and xy — y'^. 20. Find the greatest common divisor of 6a;* -f- Ha;^ +3 and 2a;* — 5a;2 — 12. 21. Find the greatest common divisor of 4a;* — 24a;* -)- 34a;2 + 12a; — 18 and 4a;3 — 18a;2 + 19a; — 8. 22. Find the greatest common^|rvisor of a;* — 4a;' — . 16a;2 + 7a; H- 24 and 2a;3 —l^x^'^x + 40. 23. Find the least common multiple of 2a2a;, 3aa^, and 4a^a;*2/*. 24. Find the least common multiple of a;^ — y^, x -\- y, X — y, and xy — y'^. j;2 7a; + 10 25. Reduce — to its lowest terms. 2a;2 — a; — 6 ^„ „ ^ a;3— 4a;2 + 9a;— 10 ^ .^ , 26. Reduce ■ to its lowest terms. a;3_|.2a;2— 3a; + 20 296 ELEMENTS OF ALGEBRA. 27. Eeduce to its lowest ten a2 +2a6 + 62_c2 28. ^InnnUft- "" X X^ Simphfy^^^ l-o; ^^-1 29. -in...Hfv^ + 2^ 2-^^ 16a;-:«^ uimphfy 2_^ 2+. ' x^-4 30. Bimphfy ^__ ,, ^, __^ 1 ,, ^, , - (c — a) (c — 6) l + a; , 1—x 31. Simplify -_^^_^^^ , i_^^^, 1 + x^+x^ 32. Simplify (. + i)(.^ + ^)(.-i). 33. Simplify «-+^ + ^-^^?:=^. 35. Divide \ a; + 1- / ^ X — . 1 _. - \'-^ '^._ a; y .34. Simplify I ^ _i_l * + ^ "a;— 1 y^i^. y x + yl \x^+y^ x^—y 36. Multiply ";~!^ + ^!~^; by ^+i::i-^. x^ -\- 2xy -{- y^ — z^ x — y -\- s 2x — 3+- 37. Simplify—- -^■ 2x — 1 MISCELLANEOUS EXAMPLES. 297 38. Divide 2-l^-i5 by Vx-V^ ^ V^y Vx i/xy 39. Eaise x -\-y to the seventh power. 40. Expand (2a + 36)5. 41. Expand (a + by. 42. Kaise \/x -\-y to the sixth power. 43. Find the sum of f¥J, V8x*y\ fxf'. 44. From VZx'^z + 6aa/s + ^yH subtract i/l2^. 45. From 6i^32 subtract 6-^27 46. Multiply -\/x — i/jT by \/x-\- \/y. 47. Multiply i/a + i/Fby \/a-\-Vh- 48. Given 7 — (7 + 7 — (7 + a;) ) = 7, to find the value of X. 49. Given a — — ^ & , to find the value X X of X. 1 a — h 1 ,0 + 6,,. , 50. Given -^ H = — — + -^, to find a — 6 a; a + o x the value of a;. 51. Given — = - — — , to find the value of x. a c b d 4a;+3 , 7a; — 29 8a; + 19 , » , ., 52. Given — ~-+- rr = — 75 — , *» find the 9 5a; — 12 18 value of X. 53. Given (x +i){x — i) — (x + 5) (a; — 3) + | = 0, to find the value of x. 54 Gi^,^^^^(^±^+I^(^=m + n, to find the x + b x + a value of X. 55 Given '^ ~ = 6 — a;, to find the value of x. x + 4 298 ELEMENTS OF ALGEBRA. 56. Given (1 +x)^ + (1 —x)^ = 242, to find x. 57. Given i/4 -{-x = — . to find the value of x. l/4 — X 58. Given \/x — 32 = 16 — \/x, to find the value of x. 59. Given i/4a; -f- 21 = 2i/a; + 1, to find the value of x. l/9i— 4 15 + 3l/x" 60. Given l/o; + 2 i/a; + 40 61. Given i/2 + a; + V a; = of X. to find the value of x. to find the value 62. Given V2 + X . , to find the values of x, y, and s. 63. Given ^+-2/ + ^==22 2 ^3^ ^4 -x+ y + -z = 33 4 ^ ^^2 » +-y + -3 = 19 to find the values of X, y, and z. '7a; — 23 +3w=17 4y — 2e + v = ll 64. Given -j 5^ — 3a; — 2w= 8 42/ — 3m; + 2i; = 9 ^3z+8w =33 ' ex -\- y -{- az = a -\- ae -{■ e 65. Given - c'^x-\- y -{- aH = Boo ^ OCX -\- 2y -\- acz = a^ -{- 2ac -\- c^ ^ to find the values of the unknown quantities. to find the values of the unknown quantities. MISCELLANEOUS EXAMPLES. 299 {a;2 4- an/ ^ 12") > , to find the values of x and «. 2/^+a^=:24j f 3a; + 3« = 15 ) 67. Given ^ > , to find the values of a; and v. .0 ^. K^y + ™^ z= 180 I „ , , 68. (jriven ^ > , to find the values of x ^3 + 2,3=189 ^^^ 2/- 69. Given f«'-2/ = 8(l/.-l/y) | ^^ ^^^ ^^^ ^^^^^^ [l/on/ =15 j of a; and y. X»—yS= 56 70. Given { 16 ^ > to find the values of x xy y ~.. 1 and y. !j;4 ^4 = 369 I I, to find the values of x x^—y^= 9 1 ^^^^_ {a;2 _|_ 2'U2 — 41 "I V, to find the values of x x^+2xy = Z^^ andy. 73. Given I '^''+*' + 2/= ~2/ I ^^ g^^ ^.jjg ^^j^gg \ »2/ = 6 j of a; and y. 74. Given (^ + '^^=^2 1 to find the values of x I a; + a^s = 18 j and y. 75. Given ] * 2/ = ~^ ( ^ ^^ fin^ the values of x ] a; + y = 6 j and y. 300 ELEMENTS OF ALGEBRA. 76. Given I - + 2/ + 1^- + 2/= 12 |^ , to find the values \ a;3 + 2/3 = 189 77. Given x^ -\-y == 3a; « + 2/^ ^ a; I and y. of a; and y. , to find the values of x 78. Given 79. Given 2xy — 24:(x + yy. •240 to find the val- X -\-y 100 j ygg Qf J, g^jj(J y_ (a; -J,) (0,2 +2/2) 3= 13 a;2?/ — an/2 __ g . , to find the values of X and y. 80. What two numbers, which are to each other as 3 to 4 have a product which is equal to twelve times their sum? 81. A person being asked the time of day, replied that the time past noon was equal to -f of the time to midnight. What was the time of day? 82. Find a number which being increased by 4 and the sum multiphed by 3, gives the same result as if half the number were multiplied by 8 and the product were dimin- ished by 8. 83. Find two numbers in the proportion of 3 to 4 such that if 9 be added to each the sums will be as 6 to 7. 84. The sum of two numbers is 12, and the difierence of their squares is 72. What are the numbers? 85. It is required to divide 99 into five such parts that the first may exceed the second by 3, may be less than the third by 10, greater than the fourth by 9, and less than the fifth by 16. 86. There are two numbers whose product is 6, and whose sum added to the sum of their squares is 18. What are- the numbers? MISCELLANEOUS EXAMPLES 301 87. What number is that to which if 12 be added, and from 1^ of the sum 12 be subtracted, the remainder will be 12? 88. A boy paid 20 cents for 200 apples and pears together, buying 25 apples for a cent and 25 pears for 3 cents^ How many of each did he buy ? 89. A steamboat, whose rate in still water is 10 miles per hour, descends a river whose velocity is 4 miles per hour, and returns. She was away for 10 hours. How far did she go? 90. Three years ago A's age was -J of B's, and 9 years hence it will be f of it. What is the age of each ? 91. There is a number whose three digits are the same ; and if 4 times the sum of the digits is subtracted from the number, the remainder is 297. What is the number? 92. A woman being asked what she paid for her eggs, replied, " Six dozen cost as many cents as I can buy eggs for 32 cents." What was the price per dozen? 93. What fraction is that which will be doubled if the numerator is multiplied by 4 and 3 is added to the denom- inator ; but will be halved if 2 is added to the numerator and the denominator is multiplied by 4 ? 94. The stones which paved a square court-yard would just cover a rectangular surface whose length was 6 yards longer and whose breadth was 4 yards shorter than the side of the square. What was the area of the court? 95. A gentleman had not room in his stables for 8 of his horses, so he built an additional stable one-half the size of the other, when he l^d room for 8 horses more than he had. How many horses had he ? 96. A gentleman purchased two square lots of ground for $300. Each of them cost as many cents per square rod as there were rods in a side of the other, and the sum of 302 ELEMENTS OF ALGEBRA. the perimeters of both was 200 rods. What was the cost of each? 97. A gentleman who had a square lot of ground, reserved 10 square rods out of it, and sold the rest for $432, which was as many dollars per square rod as there were rods in the side of the original lot. What was the length of its side? 98. A and B hired a pasture, into which A put 4 horses, and B as many as cost him 18 shillings a week. Afterward B put in 2 additional horses, and found that he must pay 20 shillings per week. What was paid for the pasture per week? 99. The sum of two numbers is 40. If the greater is multiplied by 2, and the less by 3, the difference of the products will be 15. What are the numbers? 100. A general having lost a battle, found that he had only 3600 more than half his army left fit for action, 600 more than ^ of his men being disabled by wounds, and the rest, which were \ of the whole army, being killed or taken prisoners. How many men had he in the army? 101. Four places are situated in the order of the letters, A, B, C, D. The distance from A to D is 34 miles ; the distance from A to B is to the distance from C to D as 2 is to 3, and \ of the distance from A to B added to \ the dis- tance from C to D is three times the distance from B to C. What are the respective distances? — 21/2" 102. Given x 4- 1/3 = — = , to find the values of x. VS—x 103. Several persons incurred an expense of 112, which they were to share equally. If there had been 4 more in the company, the expense to each person would have been 50 cents less than it was. How many persons were there in the company? MISCELLANEOUS EXAMPLES. 303 104. It is between 11 and 12 o'clock, and the hour-hand and minute-hand make a straight line. What is the time? 105. A rectangular field, whose sides are to each other as 2 to 5, contains 4 acres. What is the length and breadth of the field ? 106. Divide 18 into two such parts that the squares of those parts may be to each other as 25 to 16. 107. What will be the payment which will discharge a debt of $2000 in four years, paying principal and interest in equal annual installments, interest at 6%? 108. A rectangular plat of ground has a walk 6 feet wide around the outside, which contains J as much area as the plat itself. If the sides are to each other as 8 to 4, what is the length and breadth of the plat? 109. Given 110. Given in/ : a;2 + 2/2 : . 3 : 10 ^ > , to find X lx^-y^:ix-yy.:61:lj and 2/. 111. There are four numbers in arithmetical progression such that the sum of the two least is 20, and the sum of the two greatest is 44. What are the numbers? 112. A farmer has tWo cubical granaries. The side of one is 8 yards longer than the side of the other, and the difierence in their solid contents is 117 cubic yards. What is the side of each ? 113. A merchant expended a sum of money in goods, which he sold for $56, and gained a per cent, equal to the number of dollars which the goods cost him. How much did they cost him? 114. The sum of three numbers in geometrical progression is 13, and the sum of the extremes multiplied by the mean is 30. What are the numbers? 304 ELEMENTS OF ALGEBRA. \x^ -\-xii =12] 115. Given < k to find x and y. [xy—2y^=^ Ij 116. There are two rectangular boxes, one containing 20 cubic feet more than the other. Their bases are squares, the sides of each being equal to the depth of the other. If the capacities of the boxes are in the ratio of 4 to 5, what is the depth of each box ? 117. What three numbers in geometrical progression are there whose sum is 14, and the sum of whose squares is 84? 118. What is the square root of a^a;* + h^x"^ -\- e"^ -{■ 2abx^ +2ac2;2 + 2bcx? 119. A merchant has three pieces of cloth whose lengths are in geometrical progression. The aggregate length of the three pieces is 70 yards, and the longest piece is 30 yards longer than the shortest. What is the length of each? 120. A father divided $2100 among his three sons, so that the shares were in geometrical progression, and the second had $300 more than the third. What was the share of each? 121. A vintner has two casks of wine, from each of which he draws 6 gallons, when he finds the quantities left are to each other as 4 to 7. He then puts into the less 3 gallons, and into the greater 4 gallons, when the quantities they contain are to each other as 7 to 12. How many gallons were there in each at first? 122. Some smugglers discovered a cave which would ex- actly hold their cargo, which consisted of 13 bales of cotton and 33 casks of wine. While they were unloading, a reve- nue cutter hove in sight, when they sailed away with 9 casks and 5 bales, leaving the cave two thirds full. How manj bales, or how many casks, would the cave hold? MISCELLANEOUS EXAMPLES. 805 123. A farmer sold a meadow at such a rate that the price per acre was to the number of acres as 2 to 3. If he had received $270 more for it, the price per acre would have been to the number of acres as 3 to 2. How many acres did he sell, and at what price per acre ? 4Q 124. Given \/x^ = 3x, to find x. Vx 125. The sum of two numbers is to their difference as 4 to 1, and the sum of their cubes is 152. What are the numbers ? 126. A and B set out from two towns which were 204 miles apart, and traveled in a direct line untU. they met. A traveled 8 miles per hour; and the number of hours before they met was greater by 3 than the number of miles B traveled per hour. How far did each travel? 127. A merchant bought a number of pieces of cloth for $225, which he sold at 116 a piece, and gained by the sale as much as one piece cost him. How many pieces were there? 128. There are three numbers in arithmetical progression whose sum is 15. If 1, 4, and 19 be added to them respect- ively, they will be in geometrical progression. What are the numbers? 129. A and B agreed to reap a field of grain for 90 shil- lings. A could reap it in 9 days, and they promised to complete it in 5 days. They were obliged, however, to call to their assistance C, an inferior workman, who worked the last two days, in consequence of which B received 3s. 9d. less than he otherwise would have received. In what time could B and C reap the field? 130. Find two quantities such that their sum, their prod- uct, and the sum of their squares shall be equal to each other. 306 ELEMENTS OF ALGEBRA. 131. Find two quantities such that their product shall be equal to the difference of their squares, and the sum of their squares shall be equal to the difference of their cubes. 132. A sets out fj-om London to York, and B, at the same time, from York to London, both traveling uniformly. A reaches York 25 hours and B reaches London 36 hours after they have met on the road. In what time did they each perform the journey? 133. From two towns, which were 102 miles apart, two persons, A and B, set out to meet each other. A traveled 3 miles the first day, 5 miles the second day, 7 miles the third day, and so on. B traveled 4 miles the first day, 6 the next, 8 the next, and so on. In how many days did they meet? 134. Given x* — 2x^+x^ 30, to find x. Eesolve into factors by partially extracting the square root and factoring the remainder. 135. Given x^ — Gx"^ -\-llx = 6, to find x. Multiply both members of the equatioa by x, and reaolve into factors by extracting the square root partially and factoring the remainder. 136. Given {-==+2/^ VSMT^=45 ) ^^^^^^^, f a;*+«* = 17] 137. Given i [- , to find x and y. { x^y -|-a;^^ = 10 J 138. A railway train, after traveling 2 hours, is detained by an accident 1 hour. It then proceeds, for the rest of the distance, at f of its former rate, and arrives 7^ hours behind time. If the accident had occurred 50 miles further on, the train would have arrived 6\ hours behind time. What was the whole distance traveled by the train? TEST QUESTIONS. 36/. What is the difference between the arithmetical and the algebraic solution of a problem? Illustrate by the solution of a problem. What is an equation? What is a problem? What is a solution of a problem? What is a statement of a problem? Define quantity. What are used to express quantity? How is the word quantity used in algebra? What are known quantities? How are they represented? What are unknown quantities? How are they represented? Since the value of neither a nor x is known, what is the propriety in calling a a known quantity and x an unknown quantity? Define algebra. What is the sign of addition? What is it called? What is the sign of subtraction? What does it show? What are the signs of multiplication? Illustrate the use of each. What is the sign of division? In what other way may division be indicated? What is the sign of equality? What is formed when it is written between two equal expressions? What are the signs of aggregation? Illustrate their use. • What is the sign of involution? What is it called? Illustrate its use. When no exponent is written, what is the exponent? What is a power of a quantity? Illustrate the powers of numbers and literal quantities. How are powers named? What other name is given to the second power? To the third power? What is a root of a quantity? Illustrate the roots of numbers and literal quantities. How are roots named? What other name is given to the second root? To the third root? What is the sign of evolution? Illustrate its use. What is the index of a root? When no index is written at the opening of the radical sign, what root is indicated? What is the ambiguous sign? What is a coefficient? What are the various kinds of coefficients? Illustrate the use of each. When no coefficient is expressed, what is the coefficient? 308 ELEMENTS OF ALGEBRA. What is an algebraic expression? "What are the terms of an alge- braic expression? What is a positive term? When the first term of an expression has no sign written, what sign is it understood to have? What is a negative term? What are similar terms? Illustrate them. What are dissimilar terms? What is a monomial? Illus- trate. What is a polynomial? What is a binomial? What is a trinomial? Define addition. Define sum. State the principles of addition. Illustrate their application. Give the cases in addition. Illustrate each by the solution of an appropriate example. Give the rule for addition. How may dissimilar terms be added when they have a common factor? Define subtraction; minuend; subtrahend; difference, or remainder. What are the principles of subtraction? Illustrate their appli- cation. What are the cases in subtraction? What is the rule for subtraction? Show the truth of principles (1) and (2). How may dissimilar terms, which have a common factor, be subtracted? Give the principles relating to the use of the parenthesis. Illustrate their application. What are the members of an equation? Which is the first mem- ber? The second? Define transposition. What is an axiom? Give five axioms and illustrate their truth. What is the principle relating to the transposition of quantities? What is the rule for the solution of equations that require transposition? What is meant by verifying a result? How may a result be verified? If the same quantity with the same sign is found on opposite sides of an equation, what may be done? What is the efiTect upon an equation if the signs of all the terms are changed at the same tijne? Define multiplication; multiplicand; multiplier; product; factors of the product. What are the signs of multiplication? Illustrate their use. What are the principles relating to multiplication? Show the truth of principles (2) and (4). What are the cases in multiplica- tion? What is the rule for Case I? What is the rule for Case II? Solve an example and explain the solution. What is it to expand an expression? What is the square of the sum of two quantities? Illustrate. What is the square of the difference of two quantities? Illustrate. What is the product of the sum and difference of two quantities? TEST QUESTIONS. 309 Illustrate. What ia the product of two binomial quantities having a common term? Illustrate. Define division; dividend; divisor; quotient; remainder. Give the principles of division. Show the truth of principles (1) and (3). Deduce the law of signs in division from the law of signs in multipli- cation. What is Case I? Solve an example under Case I. What is the rule? When an equal factor is found in both dividend and divisor, what may be done with it? What is Case II? Solve an example under Case II, explain the solution, and deduce a rule. What are the principles relating to quantities having zero for an exponent, and to those having negative exponents? Develop the principles. Solve an example illustrating each principle. Define an exact divisor ; factors ; a. prime quantity ; a prime factor; factoring. What is Case I in factoring? Solve an example under this case. Give the rule. What is Case II in factoring? Solve an example under Case II. Give the rule. What is Case III? Solve an example. Give an explanation of the process. Give the rule. What is Case IV in factoring? Solve an example. Give the rule. What is Case V? What is a, quadratic trinomial? Solve an example under Case V. Give the rule. What is Case "VI in factor- ing? When is the difierence of the same powers of two quantities divisible by the difference of the quantities? Solve examples illus- trating the principle. What is the order and arrangement of the quantities in the quotient? What are the signs of the terms in the quotient? What is a demonstration? Demonstrate the principle just stated. When is the difference of the same powers of two quantities divisible by the sum of the quantities? State the principle and demonstrate it. What are the signs of the terms in the quotient? What is Case VII? When is the sum of the same powers of two quantities divisible by the sum of the quantities? State the princi- ple and demonstrate it. When is the sum of the same powers of two quantities divisible by the difference of the quantities? State the principle and demonstrate it. Write out the quotient of {x^ + y^) -^- {x + y); {x^ — y^)^{x-y)\ {x^-y") -^ {x + y). _ What is a common divisor of two or more quantities? What is the greatest common divisor? What would be a more appropriate term to apply to literal quantities? Why? When are quantities prime to each, other? What is the principle relating to the great- 310 ELEMENTS OF ALGEBRA. est common divisor? What is Case I? Solve an example. Give the rule. What is Case II in grffatest common divisors? Give the principles included under Case II. Show the truth of these princi- ples by examples. Solve an example under this case and give an explanation of the process. Give the rule. What changes may be made upon the quantities whose greatest common divisor is sought without affecting tue greatest common divisor? What is a multiple of two or more quantities? What is a com- mon multiple? What is the least common multiple? What would be a more appropriate term to apply to literal quantities? What is the principle relating to least common multiple? Give the rule for finding the least common multiple. Solve an example. What is a fraction? What is the unit of a fraction? What is a fractional unit? How many quantities are required t6 express a fraction? Why? What is the denominator of a fraction? What is the numerator? What are the terms of a fraction? What are frac- tional forms? Define an entire quantity; a mixed quantity. What is the sign of a fraction? To what does it belong? Illustrate its use by an example. What is meant by reduction of fractions? What is Case I in re- duction? When is a fraction in its lowest terms? What principle applies to the reduction of fractions to higher or lower terms? Give the rules. What is Case II? Solve an example and give the rule. What must be done, in examples under Case II, when the sign of the fraction is — ? What is Case III? Solve and explain an example. Give the rule. What is Case IV? What is the principle? Solve an example. Give the rule. What is Case V? What are similar frac- tions? Dissimilar fractions? When have fractions their least com- mon denominator? State the principles relating to the common and least common denominators of fractions. Solve an example. Give the rule. What should be done with mixed quantities before finding their least common denominator? What is meant by clearing an equation of fractions? What is the principle? Upon what axiom is it based? Solve an example, explain it, and deduce the rule. What must be done, in clearing an equation of fractions, if a fraction has the minus sign before it? What effect upon a fraction has multiplying it by its denominator? Solve an equation containing fractions. What is the principle relating to addition of fractions? Give the TEST QUESTIONS. 311 rule. What is the principle relating to subtraction of fractions? Give the rule. "What is the principle relating to multiplication of fractions? What is Case I? Solve an example. Give the rule. What is Case II? Solve an example. Give an explanation of the process. Give the rule. What should be done to shorten the process when possible? What is Case I in division of fractions? Solve an example. Give the rule. What is Case II? Solve an example. Give the rule. How should entire and mixed quantities be treated before dividing? What should be done, when possible, to shorten the process? Solve an example. Give the rule. What are complex fractional forms? How are they simplified? Define an equation; members of an equation; first member; second member; clearing of fractions; transposing an axiom; a statement of a problem; a solution of a problem. Give the axioms. How is the degree of an equation determined? Write equations of the first, of the second, and of the third degrees. What is an equation of the first degree called? Of the second? Of the third? What is a numerical equation? A literal equation? Illustrate each by examples. When the same expression is found in several terms of an equation, how may the solution be shortened? Illustrate. What are the directions for solving a problem? How may fractions be avoided in the solution of problems? How may problems, in which the ratio of the numbers is given, be solved? Illustrate. What is a general problem? By assigning numerical values to the literal quantities, how many results can be obtained? What are simultaneous equations? What are derived equations? What are independent equations? What are indeterminate equa- tions? What are the principles relating to indeterminate equations and simultaneous equations, containing two unknown quantities? What is elimination? What is Case I in elimination? What is the principle? Solve an example. Give the rule. What is Case II in elimination? Solve an example. Give the rule. What is Case HI in elimination? Solve an example. Deduce the rule. When there are three or more unknown quantities, how many independent equa- tions must there be? Give the principle. Solve an example contain- ing three or more unknown quantities, and deduce the rule. Give some expedients that may be resorted to in the solution of equations containing several unknown quantities. Si2 ELEMENTS OF ALGEBRA. What is Principle 1 relating to zero and infinity? Prove it. "What is Principle 2? Prove it. What is Principle 3? Prove it. What is Principle 4? Prove it. What is Principle 5? Prove it. Express, by algebraic formulas, the five principles just given. Solve and inter- pret problems involving the principles of zero and infinity. Solve a general problem and derive a general rule from the results. What is involution? A power? An exponent? How are powers named? What are the principles relating to the signs of the powers of positive and of negative quantities? What is Case I in involu- tion? Give the rule. How is a fraction raised to any power? What is Case II? What is Case III? Give the principle relating to the square of a polynomial. What is Case IV? Give the principles re- lating to the binomial theorem. Solve an example illustrating the application of the principles. What Ls evolution? What is a root? How are roots named? What is the radical, or root, sign? What is the index of a root? What is the index when none is expressed? For what are fractional exponents used? What does the numerator of a fractional exponent indicate? What the denominator? Why? What are the principles relating to the signs of roots? What is Case I in evolution? Give the rule. What is Case II ? Give an explanation of the solution of an example and deduce the rule. How is the root of a fraction found? What is Case III in evolution? Solve an example under this case and deduce the general rule for the extraction of the square root. Give the principles relating to the figures required to express the square of a number and the orders in the square root of a num- ber. What is the principle relating to the square of a number com- posed of tens and units? Extract the square root of a number, ex- plain the process, and deduce the rule. What is Case IV? Solve an example under this case, explain the process, and deduce a rule from the solution. Show how the formula for obtaining the complete divisor in extracting any root of a quantity may be obtained. What are the principles relating to the number of figures required to ex- press the cube of any number and the orders in the cube root of a number? What is the principle relating to the cube of any number composed of tens and units? Solve an example in cube root, explain the process, and give the rule. How are decimals pointed oflF into periods? How may a rule for the extraction of any root be formed? TEST QUESTIONS. 313 What is a radical quantity? How may the root be indicated? Illustrate. What is the coefficient of a radical? How is the degree of a radical determined? What are similar radicals? What is a rational quantity? What is a surd or irrational quantity? Illus- trate. What is the principle relating to the root of the factors of a quantity? What is Case I in reduction of radicals? When is a radi- cal in its simplest form? Solve an example under Case I, and give the rule. When is a fractional radical in its simplest form? Solve an example illustrating the reduction of a fractional radical to its simplest form, and give the rule. What is Case II? Give the rule. How may the coefficient of a radical be placed under the radical sign? What is Case III? Give the rule. What is the principle re- lating to addition and subtraction of radicals? Give the rule for ad- dition; for subtraction; for multiplication; for division. Give the rule for the involution of radicals. Solve an example under evolu- tion of radicals, and give the rule. What is meant by rationaliza- tion? What is Case I in rationalization? Solve an example and give the rule. What is Case II? Solve an example, explain the process, and give the rule. What is Case III? Give the rule. What is an imaginary quantity? Illustrate. Give th«- principle relating to the form of imaginary quantities. How are imaginary quantities added and subtracted? How are imaginary quantities multiplied? What is the principle relating to the sign of the product of two imaginary quantities? Show that it is correct. What is a radical equation? Give the suggestions to guide in the solution of radical equations. What is a quadratic equation? A pure quadratic equation? Illustrate. By what other name is a pure quadratic equation some- times known? What is a root of an equation? What is the prin- ciple relating to the roots of a pure quadratic? Solve an example to illustrate the truth of the principle. What is an affected quadratic equation? Illustrate it. By what other name is an affected quadratic equation sometimes known? What is the principle relating to the roots of an affected quadratic? To what general form may affected quadratics be reduced? What is the first rule for the completion of the square? Solve an example by this rule and give the reason for the steps. Give the rule for writing «he value, of the unknown quantity in an affected quadratic. If the 27 314 ELEMENTS OF ALGEBRA. Bign of the second power of the unknown quantity is negative, what must be done before finding the value of the unknown quantity? When may the Hindoo method of completing the square be em- ployed? Explain the process. Give the rule. When the coefficient of the unknown quantity is an even number, how may the square be completed? Explain the process. How may the square be completed when the coefficient of the highest power is a perfect square? Solve an example and explain the process. When is an equation in the quadratic form? What is the general form for quadratic equations? Solve an equation in the quadratic form having fractional exponents. Solve an equation in the quad- ratic form in which the terms are polynomials. What is meant by the absolute term ? Solve a general quadratic equation, and from the solution deduce the principles relating to the formation of quadratic equations. What is a homogeneous equation? Into what classes may simul- taneous quadratic equations, which can be solved by the rules for quadratics, be grouped? Solve an example illustrative of each class. Define ratio; geometrical ratio; arithmetical ratio. When should the first term of a ratio be regarded as the dividend? When may either term be regarded as the dividend? What are the terms of a ratio? Define the antecedent; the consequent. What is the sign of ratio? What is a couplet? What is a simple ratio? How are ratios compounded? What is a duplicate ratio; a triplicate ratio? Illus- trate each. Give the principle relating to the changes that may be made upon a ratio without changing the ratio of the terms. What is a proportion? What is the sign of proportion? Define the antecedents of a proportion; the consequents; the extremes; the means; a mean proportional. Upon what are the changes that may be made upon a proportion based? What is principle (1) in propor- tion? Demonstrate it. Illustrate the truth of the principle with numbers. What is principle (2)? Demonstrate it. Illustrate its truth with numbers. What is principle (3)? Demonstrate it. What is principle (4) ? Demonstrate and illustrate with numbers. What is principle (5)? Demonstrate and illustrate with numbers. What is principle (6)? Demonstrate and illustrate with numbers. What is principle (7)? Demonstrate and illustrate with numbers. What is principle (8)? Demonstrate and illustrate with numbers. What 10 TEST QUESTIONS. 315 principle (9)? Demonstrate and illustrate with numbers. "What is principle (10) ? Demonstrate and illustrate with numbers. "What is principle (11)? Demonstrate and illustrate with numbers. What is principle (12)? Demonstrate and illustrate with numbers. Solve a problem illustrating the application of the principles of proportion. Show how certain fractional equations may be solved by proportion. What is a series? What are the extremes of a series? What are the means? What is an ascending series? What is a descending series? What is an arithmetical progression? What is the conmion difference? What is Case I? Give the fundamental formula for finding the last term. Show how it is deduced. What is Case II? Give the fundamental formula for finding the sum. Show how it is deduced. How may the formulas for finding any element be ob- tained? Give the various ways of representing the unknown terms in an arithmetical progression. What is a geometrical progression? What is the ratio? What is Case I? Show how the fundamental formula for finding the last term is obtained. What is Case II? Show how the fundamental formula for finding the sum of a series may be deduced. How may the formulas for finding any element be obtained? How may the unknown terms in a geometrical series be represented sometimes? What is the logarithm of a number? What is a base of logar- ithms? What is meant by the common system of logarithms? What is meant by the characteristic of a logarithm? The mantissa? What are the principles relating to the characteristics of logarithms? Ex- plain the construction of the tables of logarithms. How may the log- arithm of a number be found? How may a number be found whose logarithm is given? How may numbers be multiplied by the use of logarithms? How may numbers be divided by the use of logarithms? How may numbers be raised to any power by logarithms? How may the roots of numbers be extracted by the use of logarithms? AlfTSWEES. Page 8. 2. Ck)a.t, $24; vest, $6. 3. Henry, $9; James, $27. 4. 8bu.; 16 bu. 5. B, $200; A, $600. 6. 1st, 50; 2d, 100; 3d, 300. 7. Charles, 70; William, 280. 8. 130. Page 9. 9. Cow, $50; horse, $200. 10. B, 105; A, 315. 11. Sister, 120; brother, 360. 12. Less, 90; greater, 450. 13. B,$350; A, $1400. 14. Wheat, 220 bu. ; corn, 1100 bu. 15. Bye, 150 bu.; corn, 300 bu.; wheat, 900 bu. 16. A, $80; B, $160; C, $320. 17. let, 13; 2d, 39; 3d, 117. 18. $7260. 19. 1st yr., $3450; 2d yr., $6900. Page 10. 20. Ist, $75; 2d, $225. 21. A owns 2000; B, 6000; C, 2000. 22. 3. 23. 2 ducks. 24. Younger daughter, $1000; elder, $2000; son, $3000. 25. 15 slate-pencils. (316) 26. 1st part, 2; 2d, 16; 3d, 6; 4th, 12. 27. 15. 28. 4. 29. 13. 30. B, $3100; A, $12400. 31. Daughter, $1200; son, $3600; widow, $9600. 32. Barley, 4; oats, 12; wheat, 16. Page II. 33. Ist, 12; 2d, 36; 3d, 96. 34. Cherry, 20; peach, 60; apple, 480. 35. John, 6 cts.; James, 36 ota. 36. Fiction, 4500. 37. Sarah, 10 cts.; Mary, 50 ''ts. 38. 1st, 62; 2d, 124; 3d, 31. 39. 1st yr., $1000; 4th yr., faOOO. 40. A, $3000; B, $2000; C, f">00. Page 17. 1. 5. 2. 4. 3. 11. 4. 15. 5. 5. 6. 1. 7. 1. 8. 9. 9. 3. 10. If. 11. 4. 12. 70. 13. 240. 14. 36. 15. 6. 16. 9. 17. 8. 18. 2A. 19. 5. 20. 6. 21. 12. 22. 13. 23. 27tV 24. 33^. ANSWEBS. 317 Page 21. 3. 266. 10. —26x'y^. 4. Wax. 11. 19x'y^. 5. 25x'y. 12. 4a. 6. —23z^y^. 13. a'x. 7. — ISci'. 14. 6l/^ 8. SOoj;. 15. —i(xyy. 9. 26TOre. 16. (k + j/)*. Page 22. 3. 6a + 46 — 6c. 4. 7a; — 6iy + z. 5. 8i + 9z — 2a2. Page 23. 6. — 22/ + 6z. 7. 2a^ + 6z— 62/+4a;. 8. 4ac + 4ay. 9. 166 + 5crf— 13e. 10. —3x'y + 6iy + z. 11. — 3a + 8c + 8d. 12. —5y + &to + z. 13. 10a26i'— 7cV+<'^. 14. 7a5 + 9/^+25. 15. 121' -H 3a: + 6. 16. biiax^ + 5|ai' + 5ix'y — 2A5». 17. 2a62 + ia' + 3f a6c + l^o^c + b' + lib^c + cK 18. 8{a; + 3,). 19. 7(a — 5)=' + 9(a;— 2/)^'. Page 24. 21. (2a— 36 + 4c+3d)a;. 22. (2a— 46 + 30 + 4)a;2. 23. (6 + 5a) (a + 6). 24. (3a + 26 + 7) (0 + 3). 25. (5a + 5)T/^+y. 2. 2. 5. 4. 8. 9. 3. 4. 6. 4. 9. 2. 4. 2. 7. 4. 10. 4. 11. Harvey, 7; Henry, 21; James, 42. Page 25. 12. C, $20; B, $40; A, $80. 13. Samuel, 5; Henry, 15; William, 30. 14. B, $200; C, $400. 15. 10. 16. Fiction, 2000; reference, 20000; historical, 6000. 17. A, $20000; B, $2000; C,$6000; D, $8000. 18. A, 612; B, 306; C, 204. 19. Board, $36; wages, $60. Page 28. 3. 9a. 4. 5xy.. 5. —2x'y^. 6. — 3a^. 7. —12x^y'z. 8. — 3o«6'c. Page 29. 9. 4a; + 4y. 10. a—b. 11. — 2a^ + 2z. 12. 2xhj^—3z. 13. — 2ay'z— ay. 14. p^qs + Spq^s. 15. 2m^na; — mnx. 16. 4x^y + By'. 17. —23y'—3z. 18. —4p'q'—pq. 19. 14a; V^"— 23^"- 20. — %2* + 2y*«. 21. 6p^q^ — 2^. 22. — 9m/z' — 3a^a. 23. — 4a''a;y+2aa;». 24. r^'s^'z — 4rsz'. 318 ELEMENTS OF ALGEBRA. Page 30. 3. 6a^x. 4. 8x^y^. 5. 4a; +32^. 6. 3^— 6a. 7. 4ax+5hf. 8. + 66— 8e. 9. 2x + 7y + 2z. 10. 2iy-\-&z—Sx^+y. 11. — 6 + 2e. 12. x + 6y — 72. 13. 3o2 + 552 + 6e\ 14. 6a» — 6c' — 6(i». 15. nx* — 7y\ 16. ISpX + ej^ — 2?''. 17. — ca-jfAay. Page 31. 18. 8jw + 7aa. 19. — 6a;»2/» + 24an/» + 14iy. 20. 9aiV — lljl^*- 21. a!» + 5a;j/ + 52» + w. 22. l&c» + 52/S + 43» — 7»-'* 23. 3a;5^ + 7a; + 4. 24. 6bx« + Say^ — 2by + 9. 25. a;2j(4 + 5a^— 9a! + 5. 26. 2x^y^ — 9xy^ — 9z* — 9. 27. 4ar' — 66s» -^Zrs—p — 7. 28. 15a5_39a;y — II2/* — 42 3*. 29. — *" — ea?*^, + 4y» + 43;^"*. 30. 3a;''" — 4a;»''y» — 4^1 J^. 4a! 2". 31. T/^+Sa+fy^". 32. 6(0 + 6)2 — 40 + 60. 33. 5/^^— 9#^+^+7l/i+^ 34. Va+T' — 5f^r+d. Page 32. 36. {a—e)y+ {d + 2)x. 37. —(a + b)x + 4(c + d)y. 38. f c — 6)z + (a + 26 — c3i/. 39. (a — 6)a! + (a + 5)2,+ (C-1)2. 40. {5a — c)y+{a + 2e)z + (d — 6)a;. 41. (a- 26)a;' + (3a + 2c)j> + (3 + c)x'2/. Page 33. 1. —6. 2.y. 3. 2a + 6. 4. 2o + 6. 5. 6. 6. a; + 2sr. 7. 2a — y. 8. x + 4y. 9. 7x — 7y. 10. 6x — 2/ + 22. 11. a; — a". 12. —xy + 3x^y — x'. 13. 7a;2 +4y^+ z^ 14. 6052 + 4ac2. Page 34. 15. 3a — 25 + e — 2cJ. 16. —hx^+ 7a;»+ 6y. 17. — 5a;»2, + 43, + l. 18. 06 — 26c — 46d — 6c. 19. &xy + llz. Page 37. 2.4. 8. 2. 14. 4. 3. 8. 9. 5. 15. 5. 4. 12. 10. 3. 16. li. 5. 7. 11. 4. 17. 6. 6. 10. 12. 2. 18. 4. 7. 4. 13. 3. 19. 7. Page 38. 20. 8. 25. 8. 31. 18 21. 3. 27. 46. 32. 10. 22. 11. 28. 18. 33. 12. 23. 2. 29. 43. 34. 11. 24. 17. 30. 15. 35. 13. ANSWERS. 319 Page 39. 37. John, 20; James, 30; Henry, 35. 38. In lat, 110; 2d, 130; 3d, 155. 39. 48. 40. $1500; $1650; $1800; $1950. 41. $2000; $2250; $2500; $2750. Page 44. 3. — 24. 7. — 20a;. 4. — 12. 8. 6a;6. 5. 21a. 9. 6a:». 6. —12a;. 10. 8x^y'. 11. 6x^yK 12. —l^x'm'y". 13. —40x'y^z\ 14. 16x*y^z'. 15. — 24a56'a;*. 16. 15a'b*x^y'. 17. —12c^d^y. 18. —15a*x^y'z. 19. 24x*y*z'. 20. —12aH^y*z^. 21. — 15a6a;V- 22. 2(a; + y). 23. — 12(a + 6). 24. 15(y + z)5. 25. 4(a — 6)«. 26. 6{e + dy. Page 45. 27. — 10(a; + y + z)'. 28. 12a;2". 29. — 20a3». 30. — 15a3a;''' + i. 31. 8a'7f+". 32. — 15a3"a;*". 33. 20a?»+'y»+". 34. 3x^y — 6y^. 35. 2x^yz — 4z^. 36. 12x''y — 6x^y'. 37. — 6a;*2/ — 4a;V- 38. — 16a;*j/222 — Sx'^z*. 39. 95;yz — 6aa/202. 40. 4a;*i/ + 2xy' + 3aa/3. 41. 6x^yz-\-2xyz — 6x'z'. 42. 18a;'3/* + 12icy* — ISxyH^. 43. l2a'bcd — 9a'cH — 9a'cd^. 44. — 25a2c2a;+30a2ra2— 20a26ra. 45. — 2Qa'b'cH + 12a26c2d2 + 12a62c2ci2. 46. — 6a'a;'^ + 4a'6ca;^ — 8a^a;'^. Page 48. 4. a:^ — y^. 5. 3a2 + lOac + 3c'. 6. 12a2 — 18o6 + 6b'. 7. 6y' +yz — l2z'. 8. 4a:2 + 6x2/ + 23/2. 9. 9a;2 — 24a«/ + 161/2. 10. 15o2— 29ac— 14c2. 11. a'x' + 2abxy + b'y'. 12. 4a2c2— 962c2. 13. 662d2 + 62cd— 1262c2. 14. 6a;*2/* + a;22/2s2 — 123*. 15. Qx^y'z^ + 4aa/« + Zx'z'' + 22/«. 16. Sa^i* + 6a62c2 + 8a6»c2 + 662c*. 17. 25a;*3/* — 15aa;22/« — 10a3;*2/ + 6o2a;2. 18. a» + 3a26 + Zah' + bK 19. x" + 6x' + 12a; + 8. 20. a^ — 2ay' + yK 21. 6a' — 3a26 — 9a62 + 66a. 22. a» — o". Page 47. 23. x^—y". 24. 6a;2 — 5a^ + 2aa — 63/^ + 23yz — 2az\ 25. 60* + lla'6 + o2e2 — 10a262 + 31a6c2— 15c*. 26. 21a;* — 34a;»y + 34a;23/2-t- 2a;^/' — 15y*. 320 ELEMENTS OF ALQEBBA. 27. 1 — 5j; + 11j;2 — 12a;8 + 6a;*. 24. A, 10; B, 10. 28. a* -1- a'x" + a;*. 25. Henry, 9; John, 12. 29. x^ + Ziy — Sxs +2y' — 5yz + 26. Ist, 20; 2d, 35. 22^. Page 50. 30. a^"- — Az™ 31. a;2» + 2j?'ir + S/""- 32. af'+» + a?^" + x^y" + jr+". 33. a;2">+2" + 22?"+"^+" + At2m+2n 27. C, $5; B, $10; A, $20. 28. Smaller, 10; larger, 40. 29. B,$1400; A, $2800. 30. Amount wanted, 46 lbs.; y 34. o^"- -2n — A2m— 2n in 1st firkin, 40 lbs.; 35. a;2 + 211/ + V^- in 2d firkin, 60 lbs. 36. 4j;2- -4j^ + 3,2. Page 51. 37. 91" - - 162/^ 1. c" + 2ed + d'. 38. i6x' — 362/2. 2. m^ + 2mn + n". 39. 9o2j; 2 + 6cim/ + ^"^ + 4y2!' 3. r" + 2rs + s^ 40. 4x^ - -Sai^jr — 43;z+8a^2. 4. x' + 4x + 4. 41. 9a« - -6aftc + 6a25c — 462c2. 5. o2 + 6a + 9. 42. a^+ab + a'b^+bK 6. 9a^ + 6ca; + x^. 43. a^ - -6='— 26c — c2. 7. 4a;2 + 16rc2/+16j/». 44. a' + Sa^b + 306" + js. 8. 9a' + 12ab + 4b'. 45. a* - -2a262 + 6^ 9. X* + 2x'y' + yK 46. a;*- -2a;2+l. 10. 16a;2 + 24a;2/ + 9y2. 47. a*- -2(i262 + 6*. 11. 9p' + 12p3 + 492. 48. l + 2a — 2a8 — a*. 12. 41* + 2l0xV + 252/*. 49. a;« - 4x^y 4a;«i/2 + 6a;V — Page 52. 50. a'6- -2o'26* + 2a*6i2 — 6'6. 13. a' — 2ac + c'. 51. a'66'6 — 2a'26'«+ 2a*6'6 — 6'6. 14. 3/2 — 2yz + z'. 15. r' — 2rs + s'. Page 48. 16. 62— 26c + c2. 17. a:2_2j;+l. 2. 6. 7. 2. 18. x'—4xy + 4y''. 3. 4. 8. 12. 19. o2 — 2ad-hd2. 4. 2. 9. —1. 20. 4r2— 12rs + 9s2. 5. 17. 10. 8. 21. 432— 48? + 9^ 6. 15. 11. 2i. 22. 9m2 — 24m™ + 16n''-. Page 49. 23. 4l)2 — 4»!0 + M)2. 24. 41* — 8x'y' + 4y*. 12. 38. 16. 25. 20. 1. 25. c2— d2. 13. 4. 17. IJ. 21. 17. 26. r'—s'. 14. 10. 18. 6. 23. 4. 27. m2— m2. 15. 14. 19. 3. 28. e2— a2. ANSWERS. 321 29. 30. '— 1. X' 4 — a;2. 31. 41'' — 16. 32. 4x* — y^. -y*. 33. X* 34. x'—y'. 35. 9»2— 4lo^ 36. 25a;22/2— 9. PaKe S3. 37. x^ + 7x+ 12. 38. j!^ — 2x— 15. 39. x^—x — li. 40. a;2— 101 + 24. _41. a« + (3 + 6)a + 36. 42. a^ + (m + »)o + nm. 43. 4a;2 — 2a; — 20. 44. 9a;2— 6a; — 35. 45. 43/i2 — 14y + 12. 46. 16a2 + (5+c)4a + 6c. 47. 25a2 + (26 — 2c)5a — 46c. 48. 9aV— 9aa; — 28. 49. 4a*a;i'— 8a»i — 12. 50. 4x*yo + 22x^y' + 28. Vage 57. 4 2. 7. 4a;i/. 6.-4 8. —3z\ 6. — Say. 9. 4m/z. Pagre S8. 10. — 5xyz. 17. 7». 11. 2o*6*. 18. 2y\ 12. 2/. 19. — 2ffli;»2^> 13. 2a;2. 20. 4s. 14 — 2r!!. 21. —18a;. 15. — 3«). 22. —24. 16. — 3ra». 23. — 9»"jr. 24 — 5arV* z. 25. -7^'" z'a; 9.(\ —5n^. m". 27. ai + y. 28. a(a; + y). 29. — 3(a: + z). 30. 31. 32. 34 35. 36. 37. 38. {c + d)K 20. 18(x + z)K "•{x — y). 5a 2 — 2x2. 39. a6 — c. ax — 2y. 40. 3xy + z. 31/ — 3a;. 41. — 3aa/ + oa;. 2a; + y. 42. — 6a;2/2 + 4 06-252. 43. o — 36 + ca, 44. X — y -{- xy^. 45. X- 46. 47. 48. 49. 50. 51. 52. 53. 54 6. 7. 8. 9. 10. 11. 12. 13. 14 15. ■2y+'-. X 2- 3a; + 3«. 3m2 m + 2 — c—3d + l + 3a;- 4df_ cd' Sa'y Page 59. 22/2 v+3vy — • vy -3 + 2{x + y). — a{b + c) — b{b + c)'. 3 — 2{a — c)2. -(a; + z)2 + 2(x + z)». Page 61. a — 6. a; + 2. 3 + a;. x^+yK 08 +y'. x^ + 23y + y^. r + «. a;»+3a;22/ + 3a!!/2+2/», c' + 2cd + d^. x^ + 5x + 7. 322 ELEMENTS OF ALGEBRA. Page ea. 16. a + x. 17. a+6 — c. 18. a^ — 2ay + y'. 19. x^ — ax — 6. 20. 5a^ + 2ab — 3bK 21. 3a;2 — 5y2 + 3a2. 22. 2a2— 3a + l. 23. 2a + 36. 24. 66'' + 12a6 + 27a^ — 1. 25. 5a3 + 4a2 + 3a + 2. 26. x^ — xy -\- y' — xz — yz-\- z^. 27. 6a;2— 72; + 8. 28. 8a;' + 12aa;2 — ISa^i — 27o'. 29. a' — 2ai + ix^. 30. x^-i-xz + zK 31. i' + a;^^ + ^^ + y'- 32. I* — x'y + a;2y2 — ^ys _|_ 2,4_ 33. x^—x^ + x*—x^+x'—x+l. 34. a:' + Sx^y + 9xy^ + 27yK 35. 27a3 — 18a26 + 12aA2 — 85*. 36. K»-i — x'^^y + af^^y^, etc. Page 63. 1. — aofi^a;", or — 1. 2. 2a»a;, or 2a;. 3. — 2a2a;», or — 2a2. 4. — 3a;y°2, or — 3xz. 5. 6x'{y + z)o, or 6a;^ 1 6. , ora;'' 1 8. a^ar'^jr"' 9. 12j;V"- 10. — 4a-26-8ca. ar^y-a x'y' 11. 12. afy^ Page 64. 3. c-t 3. 6. d — 3a. 4. o — 4. 5. 2a + 3. 7. a + 6. 8. 2o + 35. Page 65. 9. a2 + c. 12. 5o + 26. 10. 3a + 26. 13. 2e2— d. 1L7 + 56. 14. 6 — 3c. 15. 2m2 — 3m + 1. 16. 9 — 6o + a^ 17. 2m2 + 3mre + n^. 18. 3a2 — 6 + 62. 19. $110. 20. lst,|1000; 2d, $2000; 3d, $4000; 4th, $3500. 21. 5 beggars; 19 cents. 22. 6th, 10 years; 5th, 14 years; 4th, 18 years; 3d, 22 years; 2d, 26 years; 1st, 30 years. 23. 7 gallons. 24. $1 Page 66. 1. 20oj; + 20 + 8V'x+ 10a;2. 2. 2om + 5a; + 3/^+ 2 — iS. 3. Wa,'—b^ — 6{x + y)—6. 4. SVx— Vy — 32 + 22 + y. 5. (,a'^—h^)x^—ay-\-{e—Z)y^~ 3a«. 6. — 4a;2»— 6a;22/3 -j. g^Sj^ — 6s^2 + 4o2 + 4af + 6z. 7. a;« + 2x^y + x^y' + 2x^y + 4x^y^ + 2x^y^ — x*y^ — 2x'y^ — xy''. 8. a;2" + 4a;''*»y" + 2a?y + 9. 3 4- 2ar'^'"' — a?*?/" — 3ar-»y2» — 2ar2" + jf'" 4 3^'- + 2ar-'y-" — a^*". ANSWERS. 323 10. 9j^ + 6ar2«/»+'» + Sards'" — 11. i» + 5a;*j( + IQx'y^ + lOi^yS + 5xy* + 2/S. 12. o* — 8o2 + 16. 13. 81o* — 6480* + 1296. 14. x^ + 4xy+ 4y'. 15. 4a;2 + Wxy + 252/». 16. 9i* — 12a;23,a + 4y^. 17. Z*" + 4j;2>>y2'l -)- ^m. 18. I-*" — 4ar-^"^^" + 4^*". 19. 4x2 _ j,2_ 20. 9x^ — i9y^. 21. 16k* — 4^*. 22. aH^' — y'^^. 23. o^'ar-"" — a'y-^\ 24. 25. 26. 27. 28. 29. 30. 31. 32. 35. 36. 37. 38. 39. 40. 41. 42. Page 67. 2561' — 25921* + 6561. x + y. x' — Sx^y + Zxy^ — y'. 2a2» — 4a"6» + 262« 33. 34. a' 7*5 6-i». xi + 4a;'y + 6a;2y2 + 4!n/' +2/*. r-* + 4ar V' + 6ar V' + 4ar^y~' + y^- 2o — 35. 3 + 2o 4- c. 2a + 3c + d. Page 69. 2. 2, 2, 2, a, o, 6. 3. 2, 5, a;, ^t, y, y, y. 4. 3, 5, a, a, o, i/, y, z. 5. 2, 2, 5, a, X, x, x, y. 6. 2, 3, 7, a, X, y, y, y. 7. 2, 2, 3, 3, X, y, y, z, z, z. 8. 2, 2, 7, o, a, e, c, x. Page 70. 2. a2(56 + 6c). 3. 4a;2(22/2 + 822). 4. 6a^2(l + 2m/). 5. 9a;2/22(-c2 _|. 222). 6. a^xyz{xy + 2). 7. c(a2 + 62 + cd2). 8. a^(4a; + «^+3y2). 9. a'z(ay + a; + 3:^2/^2). 10. %2(j3.+ 62_j_j^2)_ 11. ml^{ay^^ + 2"^' + ayz). Page 71. 2. (a + 6) (a +6). 3. (x + y){x + y). 4. (6 — c)(6 — c). 5. {r + s){r + s). 6. (a;+l)(a;+l). 7. (a: + 2) (a; + 2). 8. (2,-l)(2/-l). 9. (22,-1) (22,-1). 10. (3a; + l)(3a;+l). 11. (3m + 3m) (3m + 3m). 12. (3 + a;) (3 + x). 13. {l — x'){l — x^). 14. (4m — l)(4m — 1). 15. (4 + 2a) (4 + 2o). 16. (6 + a2)(6 + a2). 17. (7 — a;»)(7 — a;'). 18. (9r — o)(9a; — o). 19. (2o''+36'')(2a'' + 36»). 324 ELEMENTS OF ALGEBRA. Page 73. 2. (a + 6)(o — 6). 3. (c + d)(c — d). 4. (m + n) (to — n). 6. (2*;+ 2i/) (21 — 22/). 6. {^x + y){Zx-y). 7. (»; + 32/)(a;-3j^). 8. (4j;+42/)(4a; — 4i/). 9. {^+h){¥-h)- 30. (i2/ + 22/z)(a^ — 2i/2). 11. (m* +7i'')(m + 7i)(m — n). 12. (a* + 6<)(a2 + 6'')(a + 6) (a-b). 13. (m" + ra'»)(m» — m"). 14. (3a« + 2J2») (So" — 262"). 15. (aS + 6*) (a* + 6^) (a^ + 6) (a»-6). Page 74. 2. (a; + 2)(a: + l). 3. {x + ^j{x + Z). 4. (j;-7)(l+3). 5. (1-9) (a; + 2). 6. (j; + 4)(a: + 2). 7. (a; +8) (a; + 4). 8. (1 — 13) (a; + 3). 9. (a; — 16) (a; + 4). 10. (2a; — 3) (2a; — 2). 11. (3a; — 6) (3a; — 3). 12. (2a; + 6a) (2a; + 2a). 13. (3a + 66) (3a + 46). Page 75. 1. a;' + x^y + x^y'' + x*y^ + x^y^ + x^y^ -{-xy^ + y''. 2. a;' + x''y + a;«3/2 + x^y^ + "^^y* "I" *'y° + *^^° + 2^'+ y ° • 3. a;« + a;2 + X + 1. 4. a;' + 2a;2 + 4a; + 8. 5. x' + x*y' + a;23/< + y^. Pngre 77. 1. a;' — a;^2/ + ^"'l/* — ^V + 2. a;' — a;'?/ + a;'?/' — a;«2/' + ajSj^i — x^y^ + x'y^—x'y'' + a;y8 — J/'. 3. a;' — a;2 + a; — 1. 4. a;' — 2a;2 + 4a; — 8. 5. a;" + a;*2/2 + a; V + 2/°- Page 78. 1. a;' — x^y + a;*y* — x'y' + x'y^ — xy^ +y'. 2. a;' — a;'y + ^^V^ — ^n^y' + a;*!/* — x'y^ -{- x^y' — xy'' + y', 3. a;*- a;«+a;i'— a; + l. Page 79. 1. x + y+ -^ x — y 2. x' + xy x — y 3. x'+x' V + xy' + 2'' + , 2y* c — y 4. a;* + x'y + x^y '+xy' + y' + 2y^ x — y Page §3. 3. 6m2«a;2. 14. a — 5. 4. 4r'sV. 15. a;— 2. 5. 7xV^'- 16. ix — y. 6. Bx^yz^. 17. a; + 3. 7. cay. 18. a; + 5. 8. a'xy'. 19. a; + 6. 9. 262c. 20. a; + 3. 10. 2iy\ 21. a; + 7. 11. 2r'sH. 22. a; + 6. 12. 13. ba'xy^. Sx'y'z'. 23. x + y- ANSWERS. 325 Page 86. 3. a;— 7. 13. Sj;^ — 1. 4.1 + 4. 14.1 + 3. 5. 1 — 3. 15. x—7. 6.1 + 6. 16. a; + 3. 7. 3i— 2. 17. o2_53, 8. K— 4y. 18. 1 + 2. 9. x—y. 19. a + 6. 10. x2— 2z + l. 20. 3x2+ 2s; + 1. 11. 2a;2+4a;+2. 21. x' + 4. 12. 3a; + 9. 22. a^ + 4. Page 89. 3. 40o262f!.. 4. 100a;'y»8^ 5. 70a^b^e^x'y. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 36j-'«'3*. a;' — x^y — xy' + y'. a;' + x^y — ly' — y'. a;* — 2iV+2/*. X* + x'y — xy' — y*. a^y^{x^ — z*). I* — 1. 17a!y'(x^—y^). x(x^ — l). a;(x* + a;' — x — 1). 8(1 — a;"). a;»+9a;2+26a;+24. a3_4o2_i7a + 60. a;' — lla;2 — 4a; + 44. X* — 6i» — 6x2 _|. 70a; _ 75. 28. j;6 — ^i^x*y — y^. a5 + 2a* — 16a — 32. j;5 j^4_ x»— 3x2— 4x+12. I* — 2ax' + 0^x2 — lOx' + 20ax2— 10o2x+ 25x2— 50ai+ 25o2. j;5^2x* — 161 — 32. Page 04. 3. 4. 5. 9. 10. 11. 12. 15. 16. 17, 18. 19. 20. 21. 12a _ 28' 30x2 86 lOa + 206 15 4ax2 6x + 4xy 2a2 + ax — 2oS — bx 3ax — ay + 35x — by a2 + 2ai + 62 15x + 35 30 6x I8X + 9' 9x 18x— 24 X T' 2g2 Zxym^ 13.-^. 14. 5xy 7m2_ 4z' y 5x 7yz 2aa2 3y' a + b a — 6 a — 6 o„ a;— 1 zz. x + 1 23. X — 1 22/ 24. x' + ax X — 25. x' X2+3/2 OR x + 3 27. x2 — 4x x + 4 10a; + 4y 5 X— 4 Page 96. 20x— 3y 4 326 ELEMENTS OF ALGEBRA. 8x—ee 4. 5. 4x + iy + 3 8a + 3g + 4 4 16a;+3y— 4 8 ISz— 2y — 3 10a— Si— 4 10. 11, 24a— 3y— 7 4 3cd + 4a + 6 12. 13. 14. 15. 16, 17. 18. 19. iacd-{-Sc — d cd Sao;' -|- 6a — x ax 5ri;+20 + 2c— d 5 g'- 2ac+c' u .. x^ — 9x + 6 x—2 2a' iO — X a' + 2ac— 2c' a — c 21'— 2y' 20.:^Sr '' 21. 22. 23. 2. 3. a' — 4aa;-|-4a!' — 2a& 2a6 1> °^ I o — 6 — o m' — 2m7i — 2ji' a-l a x + - Page 97. 4. 26 — X — 4 4 — X 8. a;'+a; + l 9. 10. 6^ o-f 6 5. a + a;. 6. o — a;. 7. x^—x + l. 2 11. 12. 13. 14. 2x. a'+a6 + 6' + -1 a — 6 1+- ox 2a— 26. x + y + l+y^^ ,x+y a'+a6 + 5'. 15. x + y x + y Page 99. 10. 11. 12. 13. a*'cs. Page 169. 16. 163:83,1234. 21. jr'^2r^2S". 17. 27ar«2r'i'. 22. ^f^g-^. 18. — 64a»2«y'. 23. a-*'2<»j(;*». 19. —320-'°^-'" 24. a-V'""^"*'- 20. 16a«3:'3r"'- 25. ±a;*y-»•^->•»' ANSWERS. 335 26. ± a'°"6'°"(r-5»»d-!+yK 7. 4 + 46" + 6*. 8. o* — 2a'b + 62. 9. a2+2a6 + 2ac+62+26c + ca. 10. z* + 8i8y + 24xiy' + 32a^»+ I63,*. 11. n*— 4n»m+ en^m^ — 4nm» + m*. 12. o» + 3a26 + 3^ + 5'. 13. z2» + 2a?y + 2/2« 14. 8a' + 36a'b + 54a62 + 276'. 15. 271' — 54x''y + 36xy' — 83/'. 16. n* + 2m2«2 + m*. 17. o*+5a*5 + 100362+ 10o263+ 5o6* + bK 18. x^+6x^y + 16x*y^+20xV+ Ibx^y* + 6aa/5 + y^. 19. 16a* + 64a36 + 960*62 + 64a6' + 166*. 20. o'+7o8a; + 21oSic2+35a*a;»+ SSo'z* + 2laH^ + 7aa;« + 1'. 21. 243i5 4- 810j;*2 + lOSOi'z* + 720a;aa3 + 240a3* + 32s5. 22. l&y* + 322/532 + 243/22* + 8^^ 4" 2*. 23. a2 + 2a6 + 2ac + 2ad+62 + 25c + 26d + c2+2cd + d2. Page 171. 2. a2+62+c2+2a6 + 2ac+26c. 3. a;2+3/2+22— 2i3/ + 2a:2— 2i/2. 4. o2 + c2 + (£2 + e2 + 2a. 336 ELEMENTS OF ALGEBRA. 10. a;5 + fe* + 10a;3 + lOi" + bx+\. 11. a;« — 61' + 15a;'' — 20a;S + Xbx^ — 61 + 1. 12. 1 + 5o + lOa" + 10a' + 5a* + aK 13. 1 — 7a + 21a2— 35a«+35a*— 21o5 + 7a« — a'. 14. a;*+4a;Sac+6a;2a2c2+4a:a»c'+ a*c*. 15. a;5 + 5i*6c + Ite'fi^c' + 10a;»6»c' + 5a;6*c* + ft'c'. 17. 16x* — QGx'y + 216a;2y2 — 216i2/» -'r 81?/*. 18. 27a» + 54o2c + 3600^ + 8c'. 10 g* a°6 I a'ft' I 2a6' . 6* ^'^^ 16^ 6 "^ 6 "*" 27 "^81" 20.16 + ^+i^+^+^*- ^ 3 3 27 81 01 . 3aa;2 , 3a»a; a? 21. k' 1 • y y^ y' 22. 16a*a:* + 96a>bx'y + 216a^b'x^y' + 216o6'a2'' + 815V. 23. 8a6a;3 + S6a*bH^y^ + 54a'b*xy^ + ^rb'y'. Page 179. 2. d=4xyz. d= 606=6. 3. 2xy^z^. Zbz^y. 4. ±4x^2/!!. ±3a'a;'2. 5. ±12. ±16. ±18. 6. 4. 8. 16. 7. ±6. 12. 8. ±(a + 5). ±(0 + 1). 9. ±(a2+o6). ±(m» + m2). Page ISO. 3. — 2a62(;. ", 4. 2o2c2a;. 13. a^xyV. 5. Saa/^z. 14. o^a;*?/'. 6. 2a62c2. 7. — 2o' 4a 1, * 15. — - t'6c*. 5«2 52,2 8. — oc2a;V- ,^ 2a; • aa;«y. 3j,2 10. oiyM. gj. 3 1 17. ^• 11. xy^z^w^. 6a2 Page 183. 3. a; + 2. 7. 3a — 26. 4. 6 + a;. 8. a + 6 — c. 5. 2a; + 1. 9. 2a;2— 3a;+l. 6. a + ib. 10. 2a2+a— 2. 11. x'—2x^ + Sx. 12. 4a2— 3aa; + 5a;2. 13. a — 5 — c. 14. 3x^ — 2a; + 6. 15. 2j;'+3a;2— a; — 1. 16. 7a;2 — 2a; — f. Page 186. 3. 53. 13. 3546. 4. 63. 14. 5555. 5. 66. 15. 472. 6. 96. 16. 3375. 7. 266. 17. .874. 8. 344. 18. .5555. 9. 821. 19. .306. 10. 886. 20. .315. 11. 969. 21. .8411 +. 12. 2424. Page 187. 22. ff|. 25. .86602+. 23. m- 26. .94868 +. 24. .70710+. ANSWEB& 337 3. j; + 4. 5. 6. 11. 12. 13. 14. 16. Page 189. 7. o + ~ a 3a + 1. 8. a — 4. 2x— 3. 9. 2a + 1. 3a; + 4. 10. Sa;'— 2a;+l. 2m> — 3ro + l. 1— a + a^ m — 1 • m y+2z. Page 190. 1— 2a. 17. a; +1. 18. a;+l. 3. 4. 5. 6. 7. 8. 9. 10. 11. 2. 3. 4. 5. 6. 12. 14. 15. 16. 17. Page 194. 42. 12. 1.259 +. 64. 13. 2.0800 +. 56. 14. .6463+. 89. .8617 +. 57. 15. .8735 +. 63. 16. .843 +. 177. 17. 19.51+. 126. 18. 12. 536. Page 197. QaVK hxy^\/Zx. VOa'hVF. VidxVZay. 7. ^Vxyl 8. &yzV'x.' 9. l&eyzV'^ 10. aVT^^. 11. x^x — y^. Page 198. a'^a — a:": IZ. a{a-\-h)Va. a(a + 6)1/02 — 5". {x'—y^Wxl x{i + y + y^)^. 4a(l + y)a!*. 29 19. \VK ''■ h, 20. 7/21. 2i--^Tn5: 22. y^T/iBEfi. 24.^pr7^3. 25. g^-ZlSoa. 26. Yv/lO. Page 199. 27. -v'a 28. :4-5^/»i^^. 29. l^a;(a; + 2,). 31. -|(2%)l 32. 5a;, 2)* Page 300. 2. V9a^. 3. i/lfeV. 4. fsSfi¥y^. 5. t/o2 + 2o6 + 62. 6. #'o» — Za^b +^3a5a — 6'. 7. i/S^ 8. i/9P^. 9. i/4a;* + 4a;2ya. 10. i/2a;' + 4a;2y + 2a!y". 11. V9a* — 9a"62. 12. v/a'— 026 — 062+6'. 13. t/o(!1; — y). 14. i/i(a; + jr)2. 338 ELEMENTS OF ALGEBRA. 2. 3. 4. 5. I 6. 7. 8. 9. 10. Page 201. ^'iooa|/9: f/oV, v'^. ^'^»^ i/6V> ^o^- 8 27' 27' 2T ^~«- I^'^"^ 11. 12. ^{a + by, ^(a"+6r Page 303. 2. 151/2: 4. 9^3: 6. 12#'6: 3. 11/3; 5. 56 1/2. 7. 3i\j>/6: Page 203. 8. 12/6: 13. m,V2. 9.1^/3: 14. —f 3: 10. HVI. 15. i/W. 11. A4x^y'\/2!a/. 16. lli/a 12. 3i/3. 17. 341/2: 18. 9al/5a; 19. (2a + 25 — 5c) 1/%: 20. 2yl/2: 21. 6ry2j^ 37 22. — T/3a. Page 205. 3. 12/10. 6. 24#^j;: 4. 36/2: 7. 241^2^. 5. ISx/S. 8. 30a^/^- 9. 403y#^ 15. x'yt/Ttitfy!^* 10. 120a;!/#^22^ 16. ayj^a^x^. 11. 12/70. 17, 2/t''27a5a;5y. 12. 30/21. 18. 12yV^^. 13. 200. 14. 3ab. 19. -/1-05. Page 206. 21. 1. 23. 27 — 10/2: 22. 46. 24. 4a;2— y. 25. /6"— 3/5"— /lT+5/3: 26. x — y, 27. x — 2V'^+y. 28. i2 +.2^ + 2/2. 29. — 6. 30. ai + 2ah^ + b^. 31. a — b. 32. o— aM — aM + 6. 33. a* + 6a' + 16. 34. 12 + 24/5"+ 30/2"+ 60/io: Page 207. ■ 3 6/3: 7. VK 5. 2/2a. 6. 2x/3: 8. 8f 4 9. |/io. 10. 3 ,,_ a^ i/ a'x'y^ Page SOS. "•r 12. 1 a; — y » - 13. fl?^. 14. — (2a3a;)i a 15. /3^. 16. 2/(1-2,) (1 + 3,) ». ^JVSTT^'IJS. 33c 18. ai + yi. 21. 2 + 31/2: Page 314. 19. J — yi 22. 4+3/5: 2. ^^^ 5. « . 20. 2. 23. 4 + 3/6: 5 /15 ^ 2/7" „ 2a Page 3. 9a;. 4. 4x^. 5. lea'b^gb. 6. 216a;v^i: 209. 7. 24aV&: 8. 324aa;. 9. + 6. 0. 0. — ::zl 7 /aa; 44/^ ' 7. 1". a 3/35 Pagc 315. g 2/2 + 2/3" Page 210. —1 10. (2a + 3«)2. g 2j:/S^— 2a:/r 11. 23+4l/i5. a — ^^6 12. 52+16/3: 13. 49 + 12i/5: jQ 2a;/F+2/^ 14. x + 2z^yi+yt x^ — y 15. J + 2a%^ + j(i jj 2a6/F+2a6/P" x — y Page 211. 2. 42^^^ 6. iVM. j„ 31/ a; — 1 + 3/a; + l — 2 3. 6acjf ii^ 7. 4i/2x'y. 4. 3ab-\/xyz. 8.f{x + y)K 5. o^Aa ^02. Page 216. Page 313. 2. 3o/ — 1. 5. —aV — 1. 3. 106/ — 1. 6. — 2ma;/— 3 3. T/4aa;. 7. ^a'b'yK 8. /3s^z. 4. IToa;/ — 1. 4. #'(3a22,)» 5. VS^. 9. ]^(3a^2,)^. Page 317. 6. i?'(32/^)^ 2.-2/5. 5. — BoVr Page 313. 2. 1/5"+ 1/2: 7. 2/5^3/^ 3. — 12/6: 6. 2. 3. VW+Ve. 8. /iH-/3S: 4.-36. 7. — 2l/— 1. 4. a;— t/s: 9. a; + /^ 2. 3v'2: 3. |. 4. ^• 5. a; + 3l/6: 10. x^ + /^ 3 6. l/a"+ /i. 5.1 — /—I. 6. 1 — /— L 340 ELEMENTS OF ALGEBRA. 2. o'+7a«6 + 21a55a+35a*6'+ 35a'6* + 21a265 + 7a6« + 6». 3. 16a* — 96a«6 + 2X&a^h^ — 216ai' + 816*. a^i , a52 6' 8 4^6 27 Page 218. ■ 1024 "*" 768 "*" 288 "^ Sggy' 5a:y* j/^_ 216 "*" 324 "^ 243* 8z3 36z'y 54a^i' 27y» ■ 125 "^ 175 "*" 245 "^ 343 7. a''+7ta'-'6+ "^'^^^ a"-''6''+ TO(n — 1) (m — 2) ttn-464_|- 2X3 m(M— 1)(to— 2Xn— 3) 2X3X4 n(ro— 1)(to— 2)(m— 3)(n— 4) 2X3X4X6 8. oo-^ + (n — 2)a»-86 + (n — 2)fn — 3) 2 (w— 2)(n— 3)(ro— 4) 2X3 (m — 2)(m — 3)(ro — 4)(?t — 5) 2X3X4 o»-66*. 'o»-*62 + 9. af + nf-i^+^^^^^-— iV-y+ ■af-*y*+ r(r — l)(r— 2) 2X3 '■('— !)('— 2)(r-3)^ _,_^, 2X3X4 r(r— 1)(>— 2)(r— 3)(r— 4) 2X3X4X5 10. l + 4a;2+9!/2+z2+4a;+6y+ 23 + 12ot/ + 4xz + 6ip. 11. 3a:^ — 42/*+2. 12. a;2— a;— 1. 13. m — l/mn + m. 14. a;2 — as/l/I'+y''. 15. 2j; + 2Vwc. 16. 2. 17. 2j!i/V^ 18. 4 + 4a2i/2x+y + 2a*a; + aY 19. 2»;«I. 20. 21. 22. 23. 8 a — 6 24 + 17/F 1 ' 2a' + 2aT/n'— z'- 2m2 — 2l/m* — 1 Page asi. 4. 4. 8. 27. 12. 9. 16. 5}. 5. 32. 9. 12. 13. 25. 17. 64. 6. 56. 10. 5. 14. 121. 18. 27. 7. 6. 11. 6. 15. 25. ANSWERS. 341 Pagre 233. 19. i 22. a — 1 23. ii- a 24. 4. 25. 4i. 26. 10. 27 ^ 28. 20. 2. 21. 100. 29. a(6 — 1)' 46 52+1 30. f 31. 4. 32. 4a2 (1+a) 2 33. 4a62 (1 + a)^ Page 325. 12. ±6. 13. d=8. 14. ii. 15. ±3/2. 16. ±2. 17. ±a/2l 18. ±2. 19. ±il/3: 3. ±7. 4. ±2. 5. ±6. 6. ±6. 7. ±3. 8. ±4. 9. ±5. 10. ±10. 11. ±5. Pag:e 226. 20. ±/oM^ 21. ±iVW. ~\a — 2 23. ± 1/(0 — 2)2 — 1. 1. ±f. 5. ±30; 2. ±4. 3. ±8. 4. ±8. 22. ±..p^ \a — 5 ±50. 6. Son's, 8; father's, 32. 7. $150. Page 237. 8. 12 ft.; 18 ft. 9. 3 and 9. 10. 7 and 8. 11. 6 and 7. 12. 8 and 12. 13. 12 and 20. 14. 8 yd. 15. Breadth, 36 rods; length, 40 rods. 16. 8 and 10. Vage 331. 4. 5, or — 9. 14. 30, or — 2. 5. 3, or — 9. 15. 32, or — 2. 6. 2, or — 10. 16. 2, or — 3J. 7. 1, or — 11. 17. 3, or — 4i. 8. 1, or — 21. 18. 3, or — 6i. 9. 1, or — 19. 19. 2, or — 5. 10. 1, or — 25. 20. 4, or — J. 11. 15, or — 3. 21. 20, or 1. 12. 11, or — 3. 22. 2, or — 5i. 13. 17, or — 3. 23. 4, or — 1. 3. 4. 5. 6. 7. 12. 13. 14. 15. 16. 20. 21. 22. 23. 24. 25. 26. Page 1, or — 2|. 2, or — 5J. 4, or — 51. 5, 01 — ^. l6,or— 4f. — la±iVa- 8, or — 2. 14, or — 1. 1, or — 18. X2, or — 1. 6.229 +, or 3.525 +, or 3, or —2h 13,or-4J. 7, or — If 2, or —5. 334. 8. 6, or — 4J. 9. 2, or — 2J. 10. 8, or — 7}. 11. 6, or — 4f + 36. 17. 5, or 4. 18. 2, or — 4J. 19. 2J, or — 3. - 2.729 +. - 2.325 +. 27. 14, or— 10. 28. 3, or — h 29. 7, or — If Page 236. 3. ±2, or ±l/^^. 4. 2, or f-^- 5. 2, or #'— 4. 6. ± 2, or ± V— 2. 7. ± il/6^ or ± V^^. 8. ± V% or ± V— 6. 342 ELEMENTS OF ALGEBRA. 9. 16, or 256. 11. 1, or — 64. 10. 4, or fm. 12. 625, or 256, 13. ^;P^ix;;^== Page S3 7. 15. ± 2, or ± V^^. 16. ±1, or ± V— 10. 17. 3, or —6. 18. ± 5, or ± 2. 19. 4, —3, or J± J i/^43. Page S3S. 20. 4, or —1. 21. 4, or 20. 22. 4, 2, or — |±Jl/l7: 23. 3, 2, or —3 ± VZ. 24. 3,— l,orl±^T/=10. 25. 9, —2, or |±Jl/l73. 26. 3, —4}, or — |±il/^55. 27. ± 4, or d= 1 1/15. 28. 4, or 69. Page 339. 2. 7 and 3. 3. 7 and 20. 4. 28 rods and 40 rods. 5. 20 sheep. 6. 40 in a row; 50 rows. 10 and 12. 11 persons. 6 days. 8ct.perdoz. $20. 20 persons. 13. 3 inches. Page 840. 14. $30. 15. A, $1.14039 + per rod; B, $.89039 + per rod. A dug 43.84 rods; B dug 56.16 rods. 16. 6 rods. 17. 12 yards and 24 yards. 18. 19, 2, 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 9 gallons. One, 9 mi.; 20. Silver, 2; other, 10 mi. copper, 25. Page 242. x^ — 1x = — 12. x^ + Zx = 10. x^ — IOj; = — 21. a;2-|-10a; = —24. a;2 -|- a; = 6. a;2 + 9a; = — 20. z^—4x = 12. a;2 + 10a; = —21. a;2 -|- (6 — o)a; = ab. x^ -\- {e — 6)a; = he. x^—3xVW= —10. a;2 — 4a; = 3. 6. 7. 8. 9. 10. 11. 12. 13. 14. Page 345. a;:=6; y = 2. x = 2, or 3; ^ = 3, or 2. Page 346. a; = 5, or IJ; y = 3, or 10. a; = 6, or — IJ ; y = 1, or — 4. a; == 3, or — 5 ; S^ = 2, or 6. a; = 5, or 6 ; 3/ = 4, or 3J. x^5, or 10; y = 10, or 5. a; = ± 6, or ±4 /^^ , 2^ =±4, or ±3l/^^, a; = 5,. or — 2 ; y = 2, or — 5. ANSWERS. 343 15. x = y 16. x = y-- 17. X-- y = 18. X-. y- 19. X-- y- 4, or 3; 3, or 4. 4, or —3; 3, or —4. 3, or 1 ; 1, or 3. 3, or 1 ; 20. : 1, or 3. a; = ±6; 3/= ±5. = 3, or— 1; 21. a:=±2; = 1, or— 3. y=±3. 22. x=± 7, or ± 51/2"; j;=±3,or ±2/2^ 23. x = %, or 17f; i, = 6, or — 13J. 24. 1 = 18, or 12}; y = Z, or —2}. 25. x = 2, or —46; 2/ = 3, or 15. 26. 1 = 4, or 2; y = 2, or 4. 27. a; = 6, or 4 ; y:=4, or 6. 28. j;=±2, or ±}l/2J 3^ = ± 3, or 4; 1 1/2. 29. 1 = 64, or 8; y = 8, or 64. 30. 1=4, or— 2; j/=2, or— 4 31. 1 = 2, or 1; 3^ = 1, or 2. Page 347. 32. x = Z, or —2; 3/^2, or — 3. 33. x = % or 3; « = 3, or 2. 34. 1 = 9, or 4; y = 4, or 9. 35. 1 = 6, or —102; y = 5, or 59. 36. a; = 3, or 1 ; ^ = 1, or 3. 37. x = 3, or9; 38. a; = 4, or3; y = 9, or 3. ^ = 3, or 4. 1. 2 and 6. 4. 9 and 11. 2. 10 and 2. 5. 36 a,nd 64. 3. 4 and 9. 6. 4 and 2. Page 248. 7. 8 and 6. 8. 8 and 6. 9. Y ± i i/26 — a^ and a = }v'25 — a^. 10. 6 and 4. 12. A'srate,36; 11. 48. B's rate, 24. 13. Linen, 16 yards; cotton, 48 yards. 14. 24 rods long; 18 rods wide. 15. 49 yards; $3 per yard. 16. Fore wheel, 12 feet; hind wheel, 15 feet. Page 240. 17. J(3 ± l/F) and i(l ± Vl). 18. A's, $192; B's, $224. 19. Length, 31 rods; breadth, 19 rods. 344 ELEMENTS OF ALGEBRA. 20. 8 men; 12 women. Page 272. Men, $3; women, $2. 2. 144. 4. 124. 21. Gold, 5; silver, 4. 3. 108. 5. 52i. 22. 18 and 3. 6. 80a. 23. 20 miles. 7. 9a + 96 + 36c. 8. n^x. Page 363. 9. —12. 11. 78. 2. 4i. 7. 4. 10. 330. 12. $3360. 3. IJ. 8. $12 and $28. Page 275. 4. 2f. 9. 12 and 18. 2. 30. 8. 2, 5, 8. 5. ± 6. 10. 12 and 20. 6. IJ. 3. 10. 9. 3, 5, 7. 4. 8. 10. 1, 2, 3, 4. 11. 200 bushels of wheat; 5. 3, 5, 7. 11. 3, 5, 7, 9. 300 bushels of oats. 6. 1, 3, 5. 12. 2, 4, 6, 8. Page 264. 14. 5 and 3. 15. 5 and 2. 16. 8 and 6. 17. 8 and 4. 18. 4 and 2. 19. 4 and 3. 3. 6. , 96'' 4. Page 266. 9. 4. 5. |. 6. 2a5 10. 11. 62 + 1 7. dziVW. 8. 4. 12. ± a(Vb—l)\ V6 a(6 — c) 2i/6^ 2a a + 1 Page 270. 2. 55. 3. 47. 4. 6. 5. llj. 6. 3. 7. — 68, 8. 30o. 9. 351. 10. 0. 11. 2™ — 1. 12. $1.72. 13. 214J feet. 7. 3, 6, 9. Page 276. 13. 2,5,8,11,14. 15. 1,4,7,10. 14. 1, 3, 5, 7, 9. 16. 234 Page 277. 2. 160. 4. 2187. 3. 512. 5. 512. Page 278. 6. 128a'. J__ 7. 768a8a:8. ' 729* 8. 2''-i. 11. $2187. 9. 3 X 4«-i. 12. $32000. 2. 2047. 3. 9841. Page 279. 4. 16380. 5. 265719. Page 280. 6. 2046a. 7. 59048a;2. 8. 2(2" — 1). 9. 3|||. 10. lOiMI. 11. 1364 12. 1022. 13. 4 14 12. 15. 3. ANSWERS. 345 16. '' . 18. $510. 2. 361. 6. 32767 +. x' — l 19. $1048575. 3. 1225. 7. 15625. I* 4. 2025. 8. 2744. 17. -^• 5. 841. Page 383. Page 294. 2. 7. 8. 1, 3, 9, 27. 2. 14 6. 16. K 1. 9. 1, 2, 4, 8. 3. 16. 7. 24. 4. f. 10. 2, 4, 8. 4 64 8. 42. 6. 5. 11. 4, 6, 9, 13J. 5. 36. 6. 1, 3, 9. 12. 2, 6, 18. H ,„ 4 7. 1, 2, 4 13. $629.38 +. 1. lOx^y — 2. 2. {6 — 5c)x*y-' + 2r^i + Pagre 290. 3x^ + 2a + ey + 2xK 2. 2.50243. 9. 3.68404. 3. 10cy~^ + 12m + 16aa; — 36. 3. 2.4,5484. 10. 3.58500. 4. 2.68664. 11. 3.44483. 4 iaR/2 — az + lOa— (12— 26)1. 5. 2.52504. 12. 3.50093. 6. 1.52634. 13. 3.27301. Page 295. 7. o;42813. 14. 0.37014. 5. o5 — 55. 8. 1.58433. 15. 0.22636. 6. 6a?»— 4isr-2_9ii»-ij,3^6^. 7. 9jri — 4yi. Page 291. m 2. 241.31. 7. 1528.6. 8. 4c" — 92r"". 3. 153.55. 8. 731.72. 9. I*" + ac^nyam + j,4>». 4. 1.7040. 9. .001765. 10. i" — x'y + ra/» — I/'. 5. .19339. 10. 965.06. 11. 18 — a;^?/ + a;*!/" — « V + 6. .09652. a:y — 32(5+2/6. Page 292. 12. a?^i + i»-22/ + a?'~y + 2. 8.51. 6. 418.2. a!»-V+a?^V + a^V- 3. 87.5. 7. 5.824. 13. a:^ + f. 4. 756. 8. .000598. 14 (2x + y){2x + y). 5. 74.87 +. 9. .0000225. 15. (i'»+3/^)(i + 3/)(a; — 3/). Page 293. 16. {x-7)(x + 5). 17. (1 — 9) (1 + 3). 2. .25. 3. .763. 7. 3130. 8. 21600. 18. {x*+y^){x^+y')(x+y)(x--y). 4. 30.2. 9. 41.6. 19. 1 — 1/. 22. a;!» — 5a; — 8. 5. 3650. 10. 4420. 20. 2x^ + 3. 23. 12aH^y'. 6. .15. 11. .428. 21.1 — 3. 2i.y(x^ — y'). 346 ELEMENTS OF ALGEBRA. 25. 27. 28. 29. X — 5 21 + 3' 26. x + i Pag:e 896. a — b- a + b- x^—1 1 x + 2' 30. 0. 31. 0. 33. 34. 35. 8a'b' a* — 6*' (x' + y')' x^ + y^ x — y — z 32. I* — 1 36. 37. x — 1. Page 297. 38. i/^— /^ 39. a;''+ 7262/+ 21251/2 +35a;V + SSj;^?/* + 2l3;2j/5 + 1xy» + yi . 40. 32a5 + 240a*6 + 7200^62 + lOSOa^fts + 810a6* + 24365. 41. a'" + lOa^fi + 45aS62 _j_ 120a'63+210a66*+252a565_j_ 210a^6«+ 120a36» + 45a26s + 10a6« + 61". 42. a;3 + 3a;2jr + Zxy'' + j/S. 43. [x-^yyVxy. 44. {x — y)V^. 45. 10 #'1: 46. a; — ?/. 47. a + 2/^+ 6. 48. 7. 49. 2m a — 6 60. a' — b^. 51. i^- oc 52. 6. 53. 12. 6m — 54. 55. 5. Page 398. 56. d= 2, or ± V—e. 57. ± Vt: 60. 4. 58. 81. 61. f. 59. 25. 62. 2 = 3, y = 4, z = 5. 63. 2 = 11^,2/ = -7i, 2 = 74:1. 64. 2 = 2, 2/ = 4, s = 3, w:i^3, i; = l. 65. 2 = — , y^ae, z^ — c a 66. 2 = Page 299. :2, 2/=±4. 67. 2 = 2, or 3 ; 2/ = 3, or 2. 68. 2 = 4, or 5 ; 2/ = 5, or 4. 69. 2 = 9, or 25; y = 25, or 9. 70. 2 = 4, or — 2 ; 2/= 2, or — 4. 71. 2 = d= 5, y = ± 4. 72. 2 = ±3, 2/ = ±4. 73. 2 = 2, or 3; 2/ = 3, or 2. 74. 2 = 2, or 16 ; 2/ = 2, or J. 75. 2 = 4, or 2 ; 2/ = 2, or 4. Page 300. 76. 2 = 5, or 4 ; 2/ = 4, or 5. 77. 2 = 4, or 1 ; y = 8. ANSWEBS. 347 78. x = 8, or 6; 2/ = 6, or 8. 79. 3, or — 2 ; 2, or —3. 80. 21 and 28. 81. 20 miuutes past 5. 82. 20. 83. 9 and 12. 84. 9 and 3. 85. 17, 14, 27, 8, 33. 86. 2 and 3. Page 301. 87. 276. 88. 50 apples, 150 pears. 89. 42 miles. 90. A's age, 21; B's age, 39. 91. 333. 92. 8 cents, 93. *. 94. 144 sq. yd. 95. 40 horses. 96. $180 and $120. Page 303. 97. 8 rods. 98. 30 shillings. 99. 27 and 13. 100. 24000men. 101. 12,4, and 18 miles. 102. 1 — 1/2: 103. 8 persons. Page 303. 104. 27-^ minutes past 11. lOo. 40 rods and 16 rods. ine. 10 and 8. 107. $577.18 +. 108. Length, 118.48+ feet; Breadth, 88.86+ feet. 109. a: = 18, or 6; y = 6, or 18. 110. a; = 20, or —16; y = 16, or — 20. 111. 7, 13, 19, 25. 112. 2 yards and 5 yards. 113. $40. 114. 1, 3, 9. Page 304. 115. x = ±S, y = ±l. 116. 5 feet and 4 feet. 117. 2, 4, 8. 118. fa + aa;2 + c. 119. 10, 20, 40. 120. $1600, $400, $100. 121. 38 gallons and 62 gallons. 122. 24 bales, or 72 casks. Page 305. 123. 18 acres ; $12 per acre. 124. 4. 125. 5 and 3. 126. A, 96; B, 108. 127. 15 pieces. 128. 2, 5, 8. 129. B, 15 days ; C, 18 days. 130. J(3 ± i/=3) and J(3 =F t/^=^). Page 306. 131. ±11/5" and i(5 ± VW). 132. A, 55 hours; B, 66 hours. 133. 6 days. 134. 3, or —2. 135. 1, 2, 3. 136. a; = ± 3, or ± 4 J/ = ± 4, or ±3. 137. a; = ±2, or ±1 jr = ± 1, or ±2. 138. 300 miles. i £ IPUCT GEBR^