UC-NRLF *B 535 310 Digitized by the Internet Archive in 2008 with funding from Microsoft Corporation http://www.archive.org/details/elementsofalgebrOOmilnrich ELEMENTS OF ALGEBRA A COURSE FOR GRAMMAR SCHOOLS AND BEGINNERS IN PUBLIC AND PRIVATE SCHOOLS BY WILLIAM J. MILNE, Ph.D., LL.D. Pbesidbiit of Nbw York State Normal College, Albany, N.Y. o»:o NEW YORK:. CINCINNATI:- CHICAGO AMERICAN BOOK COMPANY Copyright, 1894, by AMEKICAN BOOK COMPANY. [All rights reserved.] MILNE'S EL. ALGc f>rinte& b^ mniiiam ITvison «ew l!?orh, xa. S. a. PREFACE. Algebra has not always proved to be an interesting subject to the younger classes in our secondary or lower schools ; indeed, in very many instances it has been greatly disliked by the students in such institutions. Two causes, chiefly, have conspired to produce this unfortunate con- dition of affairs, — one the unattractive and uninteresting method of presenting the subject ; the other, the difficulty of the examples and the complexity of the problems pre- sented to the pupils for solution. It is believed that this text-book presents the elementary facts of the science in such a manner that a deep interest will be awakened in algebraic processes, and that the examples which the student is required to solve are quite within the scope of his ability to master. The author has in several instances departed from the order of classification commonly followed in text-books on algebra, because he has preferred to arouse an interest in the subject rather than to follow an order which is scientific, but which does not introduce the student to the attractive features of the study until he has mastered all the pro- cesses employed with most forms of algebraic quantities. And yet in no instance have erroneous mathematical ideas 4 PREFACE. been taught, nor has correct reasoning ever been sacrificed for the purpose of exciting such interest. The ideas of number which the pupil has gained in arithmetic have been associated with those involved in algebra in such a way that no difficulty may be experienced in passing from reasoning upon definite numbers to reason- ing upon general numbers. The treatment of equations is introduced at the begin- ning, and it is presented throughout the book, wherever it is possible to do so advantageously, because, since ele- mentary algebra treats of almost nothing except the equation and general numbers, the student should be led to a comprehension of the simpler forms of the equation as soon as possible. The method of presentation exemplified in the other books of the series has been followed here, because it is recognized as pedagogically correct and because it has met with general approbation. The work is designed to present the merest elements of the science, and yet it is believed that the method of pres- entation, the illustration and application of mathematical principles, and the knowledge gained from the solution of problems, will familiarize the student with the funda- mental principles of the science to such a degree that easy and rapid progress in the more abstract phases of the subject will be secured whenever he pursues the subject farther. WILLIAM J. MILNE. ►State Normal College, Albany, N.Y. CONTENTS. PAGE Algebraic Processes 7 Algebraic Expressions 19 Terms in Algebraic Expressions 23 Positive and Negative Quantities 25 Addition 20 Subtraction 32 Transposition in Equations 41 Equations and Problems * 43 Multiplication 47 Special Cases in Multiplication 54 Simultaneous P^quations 58 Division 63 General Review Exercises 70 Factoring 73 Equations solved by Factoring 80 Common Divisors or Factors 85 Common Multiples 88 Fractions 91 Reduction of Fraction.s 92 Clearing Equations of Fractions ... * 101 Addition and Subtraction of Fractions 107 Multiplication of Fractions ,110 Division of Fractions 114 5 6 CONTENTS. PAGE Equations 116 Review of E(iuations . . , . 120 Involution 131 Evolution 134 Quadratic Equations 150 Pure Quadratic Equations 150 Affected Quadratic Equations 152 Simultaneous Quadratic Equations 158 General Review 162 Questions for Review 177 Answers 184 ELEMEJN^TS OF ALaEBRA. o>vas f 3000, what was the cost ? Solution. Let X = the cost of the house ai id lot. Then lx= the gain. And • l^x = the sellmg price. . |x = 3000. Jx = 1000. X = 2000, the cost of the house and lot. 73. A man paid a debt of $7500 in 4 months, paying each month twice as much as the month before. How much did he pay the first month ? ALGEBRAIC PROCESSES. 17 74. A man traveled 324 miles; he went three times as far by steamboat as by stage, and eight times as far by railroad as by steamboat. How many miles did he travel by each conveyance ? 75. Harry solved 8 problems more than Edward, and Edward solved ^ as many as Harry. How many did each solve ? 76. A man paid $150 for a wagon, which was 50 per cent or one half more than the original cost. What was the original cost of the wagon ? 77. A. man bought a lot for a certain sum ; he built upon it a house costing twice as much as the lot ; he furnished the house at an expense equal to i the price of the lot. The man, who had only f 6000 in cash, found that he could pay for all his expenditures except furnishing the house. What was the amount of each item ? 78. In an orchard there are 210 trees, arranged in rows. There are 5 rows with a certain number of trees in a row, and 8 rows with twice that number of trees in a row. How many trees are there in the different rows ? 79. A man earned daily for 4 days three times as much as he paid for his board. After paying his board for 4 days, he had $ 8 left. How much did he receive per day, and how much did he pay for his board ? 80. A man on being asked how old he was said that y^^^ of his age added to twice his age made 84 years. How old was he ? 81. A man borrowed as much money as he had, and then spent i of the whole sum. If he had f 4 left, how much had he at first ? MiLNE's EL. OF ALG. — 2. 18 ELEMENTS OF ALGEBRA. 82. A man had twice as many 5-cent pieces as dimes, and his money amounted to 200 cents. How many pieces had he of each kind ? Solution. Let x — the number of dimes. Then "Ix — the number of 5-cent pieces. And 10 X = the number of cents in the dime pieces. 10 X = the number of cents in 5-cent pieces. .-. 20x==200. X — 10, the number of dimes. 2 X = 20, the number of 5-cent pieces. 83. A sum of money amounting to f 5 is composed of half and quarter dollars. The number of half dollars is double the number of quarter dollars. How many pieces are there of each kind ? 84. A man worked 10 days for a certain sum per day ; his wife also worked at the same rate, and his son at \ the rate of his father. They received in all $27.50. What were the daily wages of each ? 85. In a certain field the length is twice the breadth, and the distance around the field is 90 rods. What are the length and the breadth of the field ? 86. A tree 45 feet high was broken off so that the part left standing was twice the part broken off. What was the length of each part? 87. Divide $800 among three men so that the first and third shall together have three times as much as the second, and the first shall have double what the third has. 88. The income from a very successful business quadru- pled every year for three years. If the entire income for the three years was $42000, what was the income for each year? ALGEBRAIC EXPRESSIONS. 19 ALGEBRAIC EXPRESSIONS. 8. Quantities connected by algebraic signs are called Algebraic Expressions. Thus, a + b and 2 a — 3 he are algebraic expressions. 9. The signs employed in arithmetic are generally used for the same purposes in algebra. 10. What do the following expressions indicate ? 1. 2 4-5; 17-6; 16 -- 4 ; 15 x 3. 2 . a + 6 ; c — f/ ; c -^ d ; d x a. 3. ff + 6 -h c ; « — & -f c ; a — b — c; a X b X c 4. a — b — c-\-cl: a -\-b — d —e; a x b x c X d. M lilt ipUcat ion is also indicated by writing letters, or a letter and a figure, side by side without any sign between them, or with a dot be- tween them. Thus, a x b x c may be written abc or a-b- c. So, also, 2 X X X y X z may be written 2 xyz or 2 -x- y - z. What do the following expressions indicate ? 5. 4a6; 5a6; 3a 4-2&; 4ic -f 3.V. 6. 4 xyz ; 6 abc ; 6ax -\-3ay; 5cd — 3d. 7. 3xy — 4:yz -\-2z', 'i xyz — 2 axy — 3 by. 8. 5 xyz -\- 4:xy — 3y; 2 a&c — abd -j- 3 rf. 11. The product arising from using a quantity a certain number of times as a factor is called a Power of that quantity. Thus, 4 is a power of 2 ; 64 is a power of 4 and also of 8. Powers are indicated by a small figure or letter, called an Exponent, written a little above and at the right of the quantity, showing the number of times the quantity is to be used as a factor. Thus, a^ shows that a is to be used as a factor ^'i?e times, or that it is equal toaxaxaxrtxn:. 20 j:lements of algebra. Powers are named from the number of times a quantity is used as a factor. Thus, a' is read the seventh power of a or a seventh. The second power is also called the square, and the third power the cube of a quantity. 12. Eead the following expressions and state what they indicate. 1. a^', a^\ x^] ^y^'i ^'Vj ^y? ^V- 2. d^^h-, a-'+&'; o?-\-h^\ a^-h\ 3. d^h -\- alP- ; c^x — Jfy'^ ; aV + h^x^ ; d^x^y — Ifx^y^. 4. a- 4- ^2 _ ^2 . ^2^2 ^_ ^2^2 _|_ ^2^2 . ^^^3 ^ ^^^2^2 _ ^3^ 13. A figure or letter placed before a quantity to show how many times it is taken additively is called a Coefficient. Thus, in the expression 5 ic, 5 indicates that x is to be taken five times, or that it is equal to x -\- x -\- x -\- x -\- x. In the expression 4 6c, 4 may be regarded as the coefficient of ftc, or 4 h may be regarded as the coefficient of c. When no coefficient is \vritten, it is manifestly 1. 14. What do the following expressions indicate ? 1. bd'x'^. Interpretation. The expression indicates five times the product of a used twice as a factor, multiplied by x used three times as a factor, multiplied by z used five times as a factor. It is usually read^ve a square^ x cube, z fifth. 2. ^x'y^ 5. 2x2 + 2/'. 8. :^x'-\-2y-3z. 3. 4.^y\ 6. 3x-4.y\ 9. 2d'x' -'dxy -^2%^ 4. 6axy^. 7. 4:xy — 3f. 10. 5a^y+3xY-^^cz'. 15. When several quantities are inclosed in parentheses, (), they are to be subjected to the same process. ALGEBRAIC EXPRESSIONS. 21 Thus, 2 X (x + ^) indicates two times the sum of x and y ; 3 (a + 6 — c) indicates three times the remainder when c has been subtracted from the sum of a and h. 16. 1. Interpret and read the expression a x (6 + c) or Interpretation. The expression means that the sum of h and c is to be multiplied by a. It is usually read a times the quantity h plus c. Interpret and read the following : 2. x(y-\-z). 7. a-{-4:d(ax — cy). 3. 5(a-c). 8. 3x + 4.{2y-3z), 4. 3d(c-y). 9. 2(3x-5y)-h{6x-3y). 5. 4aj(c-t-2d). 10. 3a(x-{-c) — oy{z-{-d), 6. 3ac(2x — 3y^). 11. 5aa.'(a — 6) — ;^c^(c + d). Write the following in algebraic expressions : 1. The sum of five times a and three times the square of x. Solution. 5 « + 3 x^, 2. Three times b, diminished by 5 times a raised to the fourth power. 3. The product of a, b, and a — c. 4. Seven times the product of x times y, increased by three times the cube of z. 5. Three times x, diminished by five times the sum of a, b, and c. 6. Six times the square of m, increased by the product of m and tl 7. The product of a used five times as a factor, multiplied by the sum of b and c. 8. Twelve times the square of a, diminished by five times the cube of b. 22 ELEMENTS OF ALGEBRA. 9. Eight times the product of a and b, diminished by four times the fourth power of c, or c used four times as a factor. 10. Six times the product of the second power of a mul- tiplied by 71, increased by five times the product of a times the second power of 7i. 11. Four times the product of x by the second power of y, multiplied by the cube of z, diminished by a times the fourth power of c. 12. The fourth power of a plus the cube of b, plus three times the product of the square of a by the square of 6, diminished by the cube of d. 17. When a = 1, b = 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, 2a-[-36-c2-f(? = 2-|-6-9H-4 = 3. 1. 3 abed. 13. 12 be + e(c — a). 2. oabcd'. 14. 3crd — 5abc. 3. 2a'b'c\ 15. 9a-4.b-\-2cd. 4. 3 abode. 16. 3 abd — ^ e -\- cd. 5. 3a-\-:)b. 17. d'y-c- — d- ■\- e\ 6. 4a-fo6. 18. {a-\-b)-\-e\d — c). 7. 2e—3c-\-d. 19. b{d - a -^ c) - bd. 8. 4 ab-cd'. 20. abc + ade — d{e — c). 9. 10 d^b — 3 ab-. 21. b'C-d'- — ab -^ cd. 10. 3d — 2b'-'{-e. 22. b{d -^ a)-\-e{b -\- c). 11. 5cd — 5bc — oab. 23. c^-f- 4a&c 4- d(e — d). 12. Sa'b'^-\-c^-de. 24. {a-\-b)-\-3{c^d) ~{-2(?-d). ALGEBRAIC EXPRESSIONS. 23 TERMS IN ALGEBRAIC EXPRESSIONS. 18. An algebraic expressiou whose parts are not sepa- rated by the signs + or — is called a Term. Thus, 5 X, 3 by, 4 x^y'^z^^ a%hl^e are terms. In the expression 3 x + 2 ay + 3 cd — ^ there are four terms ; in the expression xy'^ — x^y'^ there are but two terms. 19. A term which has the sign -h before it is called a Positive Term. When the first term of an expression is positive, the sign + is usually omitted before it. Thus in the expression 2a — 36 + 4c — 5(^+3 6 the first, third, and fifth terms are positive. 20. A term which has the sign — before it is called a Negative Term. Thus in the expression 2a — 35 + 4c — 5 d—Z e, the second, fourth, and fifth terms are negative. 21. Terms which contain the same letters with the same exponents are called Similar Terms. The coefficients or signs need not be alike, however. Thus, 5x3 and — Ix^ are similar terms ; also 3(x + a)^ and b{x+ a)^. Expressions like hx^ and cx^ may be considered similar terms by regarding h and c as coefficients. 22. Terms which contain different letters, or the same letters with different exponents, are called Dissimilar Terms. Thus, Zxy and Zpq are dissimilar terms ; also 5 x^y and 5 xy^. 23. An algebraic expression consisting of but one term is called a Monomial. Thus, a&, Zaxyz, bx'^y'^z are monomials. 24. An algebraic expression consisting of more than one term is called a Polynomial. Thus, a 4- 6 + c and 3 x + 4 ?/ are polynomials. 24 ELEMENTS OF ALGEBRA. 25. A polynomial of two terms is called a Binomial. Thus, 8 X + 2 ?/, a — 6, and 4 a?) — 3 cd are binomials. 26. A polynomial of three terms is called a Trinomial. Thus, a + 6 + c and 2x — 2y -\- bz are trinomials. 27. Select from the following : 1st, the positive terms. 5th, the monomials. 2d, the negative terms. 6th, the binomials. 3d, the similar terms. 7th, the trinomials. 4th, the dissimilar terms. 8th, the polynomials. 1. 2air, 3aj2?/, 2a + 3&, 6a;V, oaa-^ 3a;-f-2&. 2. 3 x^ + 2 1/^, 5 a — 3 6 + c, — 4 ax, Q a — c -{- d, a^ -f- 61 3. a + 6 4- 2 c + 3 d, aaj — 6a; — 3 c -f 2 d, ax-\- bx. 4. aV + 2 ax, 3a -\-b, 2a — c -\- d, 3 aV — 4 ax. Write the following : 5. Four positive terms; three negative terms. 6. An expression containing three positive and two negative terms. 7. Three similar terms; four dissimilar terms. ' 8. An expression containing four positive similar terms, and one containing four positive dissimilar terms. 9. An expression containing five similar terms, three of which are positive and two negative. 10. An expression containing five dissimilar terms, four of which are negative and one positive. 11. Six positive monomials; three positive similar mono- mials ; four negative monomials ; three negative similar monomials. 12. Four polynomials ; four binomials ; four trinomials. ALGEBRAIC EXPRESSIONS. 25 POSITIVE AND NEGATIVE QUANTITIES. 28. In arithmetic the signs -|- and — are used to indicate operations to be performed, but in algebra they are used also as Signs of Opposition. > Thus if gains are considered j:)osi^i>e quantities, losses will be nega- tive; if distances north from a given point are considered positive^ distances sonth will be considered negative^ etc. 1. Place appropriately before each of the following quan- tities the sign -|- or the sign — : Mr. A gains $40 and loses $20. Mr. S earns $25 and spends $ 15. A ship sails 40 miles north from a given merid- ian, and then 20 miles south. A thermometer indicates on Monday 15° above zero, and on Wednesday 10° below zero. 2. A man deposits in a bank $50, and then draws out $20. Indicate the transaction by using the signs -f and — . 3. Mr. H buys 25 horses and sells 10 of them; he then buys 15 more and sells 8 of them. Indicate the transac- tions by proper signs. 4. A vessel sailed 40 miles east, and was then driven by adverse winds 30 miles west. Indicate the directions by proper signs. 5. How much worse off is a man who is $50 in debt than if he had nothing? Indicate his condition by the sign -f or — . A negative quantity is sometimes regarded as less than zero. 6. Two vessels left the same port, one sailing west 10 miles per hour and the other sailing east 8 miles per hour. How far were they apart at the end of two hours ? Indi- cate the distance sailed by each vessel and the direction. ADDITION. 29. 1. How many days are 5 days, 4 days, and 3 days ? 2. How many cVs are o ri, 4 d, 3 fZ, and 6d? 3. How many c's are 6 c, 3 c, 5 c, and 2c? 4. How many a&'s are 3 ab, 2 a6, 4 a6, and 5 a6 ? 5. When no sign is prefixed to a quantity, what sign is it assumed to have ? 6. When positive quantities are added, what is the sign of the sum ? 7. If Henry owes one boy 3 cents, another 5 cents, and another 6 cents, how much does he owe ? 8. If the sign — is placed before each sum that he owes, what sign should be placed before the entire sum ? 9. What sign will the sum of negative quantities have? 10. If a vessel sails -f- 5 mi., + 8 mi., -|- 9 mi., and is driven back —4 mi., — 2 mi., — 6 mi., how far is she from the sailing port ? 11. A boy's financial condition is represented as follows: Henry owes him 15 cents; David owes him 10 cents ; James owes him 8 cents. He owes William 9 cents, and Fred 10 cents. What is his financial condition ? How much is 15 cents, 10 cents, 8 cents, — 9 cents, — 10 cents ? 2Q ADDITION. 27 30. Principle. Only similar quantities can be united by addition into one term. Dissimilar quantities cannot be added, but in algebra an indicated operation is often regarded as an operation per- formed. It is important to remember that fact. Thus, a and h cannot be added, and yet a + 6 is called their sum, though the operation is only indicated. 31. To add similar monomials. 1. 2. 3. 4. 5. 6. 3aj 9a& — 3 mn - ^'y ax - be Ix 4.ab — 2 mn - -5x^y ^ax — 7 be X Sab — 9 mn - -Sx'^y 5 ax - 26c 5ic 6ab — mn - -*4.x-y 4 ax -126c 2x 2ab — 5mn - -Ixhi Sax - 6bc 7. Express in the simplest form 5a-|-3a — 2a — 7a + 12 a + 3 a. Solution. The sum of the positive quantities is 23 a, and the sum of the negative quantities — 9a. 23a — 9a = 14a. Hence the sum of the quantities, or the simplest form of the expression, is 14 a. Express in the simplest form : 8. 8a — 2a +a — 3o — a + 7a. 9 . 4 x-y^ -h 3 x'-y^ — x;-y- -\- 5 xhf — 10 xh/. 10. 7 mx -\- 4: 7nx — 5 mx — 2mx — () mx + mx. 11. oy —Sy-\-^y — 10 y -\-6y — y. 12. Sm-^ Sm — om — 2 m -\-6m — 4: m. 13. 7 6c-h3 6c — 4 6c — 5 6c-i-8 6c — 6c. 14. ^xy -\-2xy — b xy -\- 10 xy — 7xy — 4t xy -\- o xy. 15 . Q> x^z — 4: xh -\- S x-z + 8 x-z — 5 xh -f- 3 0^2; — 10 xrz. 28 ELEMENTS OF ALGEBRA. 16. 15 mn + 6 mn — 10 mn — 4 mn — 3 mn -f 4 m?i. 17. 4 a2^> _ 3 a% + 7 a-h - 14 a-& - 3 orb + 20 a^^?. — 18. 25 aa; — 17 a.f — 13 aa? -f- 19 ax -|- G ax — 20 ax. 19. 3(a6)- + 9(a&)--(a6)- + 7(a&)--9(a6)l — 20. 8(a-6)+4(a-^)-6(a-6)-2(a-&)+3(a~6). 21. 7 2/^2; — 4 2/-2; 4- ?/-2; — 6 ^2: + 2 2/-2;. — 22. 5(a;+2/)-2(x-hy)-3(.^4-2/)+8(x+7/)-2(x4-2/). 23. 4(a + 6)2 + 10(a + ^)'-7(a + ^)2-2(a + 6)2 + 5(a + 6)'. \ 24. 3cd — 2cd'\-bcd-\-lcd-3cd-^cd. 25. 9(x//)'^ - 3{xyy + 4(x?/)3 - 5(.x?/)-^ + 2{xyy - 6{xy)\ 32. To add when some terms are dissimilar. 1 . Add x + Sy — z, x — 2y, x -\- 4: y -}- 3 z. PROCESS. Explanation. Forconvenience in adding, similar x -\- 3y — z terms are written in the same column, and the sim- ^. _ 2 » plest form of the sum is obtained by beginning at J ,. either the right or left hand column and adding ^' "» y ~ r *^ q^qy^ column separately. The dissimilar terms in 3x 4- 5y 4- 2z ^^® result are connected with their proper signs. ^ Rule. Write similar terms in the same column. Add each column separately by finding the difference of the sums of the positive and the negative terms. Connect the results with their proper signs. Find the sum of each of the following : 2. 3. 4. 2a-46 10X + 32/+ z Sxy + 2y^- z' 6 a — 2 6 — 5 X — y — 2 xy -^ 6 y^ — 5 z' 2 a + 3 6 2x — 2y-{-z 7 xy — 4 y- -\- i z' — 5 a — 4 6 x-\-7y~lz 2y^ + 5z' ADDITIOX. 29 5 . Find the sum of 2 c -{- 5 d, 7c — d, c? — 4c, 2d — c. 6. Find the sum of 6 7n — 4:n, 2m-\-3n, 5?i — 7m, 2n — 3m. Express in the simplest form : 7. 2a4-2&-h3c + 46-4a4-6a-2c. 8. xh -\- o xz- — 7 xy -^ 6 xz- — 2 xh + 4 a?i/ + 4 xh — xz^. 9. 3w-^4:X — 7y-^2r — 2w—x-\-3y-\-4v — 3x^4.W'-6v. 10. a'b'-\-c'-\-cd-2 c--3 cc?4-o a^b'+cd-3 c--2 a'b'-c'. 11. Add ab + a-c-5,3ab-3 a^c -{-7,2 a^c- 2 ab - 3. 12. Add 5a-\-3b — 2c-{-d,2b-\-c-3d,7a-ob-\-c. 13. Add 6m+8n-f ir— ?/, 2m — 2 n-f-3 0/^+41/, — 7?i— 5a; + 2^. 14. Add 3x-{'7 y — 4:Z -{-6 w, 7 z — 4:X — 2y — oiv, x-{-y -\- Z -{- IV. 15 . Add 4:Xry — 3 xy^ — 2 oi^y^, 4 xy^ — 3 x-y — 2 x-y^, 4 a^2/^ — 2 x^y — 3 xy\ 16. Add 3 aj"* + 2 y'\ — 4a;"* + 5 2/", 5 a;"' — 4 ?/", 7 o.-"^ — 2 2/". 17 . Add 8 a-b'x' — 3ab-\- ed, 2 a-b'iv' + a6 — 4 ed, 2ab - a'b"x\ 18. Add 7 m^ — 6 mn -\- 5 ir, 4 ?;??i — 3 m^ — 11^ , 5 m^ — 4 n^ 19. Add Uaa^-Say^-{-6az% 20ay^ -24: ax^ -12 az^, 32 aa.-^ - 40 ay' + 15 a^^ 20. Add 10 a^b - 12 a'bc- 15 b'c'-{- 10, - 4: a'b -{- S a^bc _ 10 bV - 4, 2 a26 + 12 a'bc -f- 5 6V + 2. 21. Add 4a.'^-6aa^ + 5a2a;-5a^ 3 a;^ -f- 4 aa.-^ -|- 2 a^a; -h 6 a^, — 17 ar^ + 19 ax- — 15 a^a; + 8 a^. 22. Add (j(c-{-d) -3{c-{-d) -\.a(c-{-d) -26(c+d)-(c+d). 23. Add 7a — 36-fc + m, 36 — 7a — c + m. 30 ELEMENTS OF ALGEBRA. | i 24. Add Sax-^2{x + a)-\-3b, 9ax-{- 6{x -\- a) -9b, \ llx-f 6b — 7ax — H{X'j-a), \ 25. Add6{x-{-y)+3z — S,2{x-\-y)-2z-^4:,Sz-3{x+y). . I 33. A letter may sometimes represent some definite ! number. i Thus, a may represent 5 ; then 2 a will represent 10 ; 8 a, 15, etc. A letter may also represent any number, whatever its i value. j Thus, 5 times n may represent 5 times any number ; 8 times n or | 8 n may stand for 8 times any number. ; 34. Letters used to represent quantities having a definite : value, or letters which represent any number or quantity are : called Known Numbers or Quantities. | The Jirst letters of the alphabet, as a, b, c, etc., are used j to represent known numbers or quantities. ' ] Thus, a, b, c, (?, etc., are usually considered known quantities ; ] that is, they either stand for known numbers or for any number. 1 PKOBI.EMS. i 35. 1. A man bought a barrels of flour, b barrels of i sugar, and 8 barrels of molasses. How many barrels in all did he buy ? ] 2. Edith bought a ribbon for m cents, a pencil for d ; cents, and a book for 6 cents. How many cents did she pay ■ for all? "^ i 1 3. A farmer sold some sheep for c dollars, a cow for 7i 1 dollars, and a horse for as much as he received for the sheep : and cow. How much did he receive for all ? ; 4. George walked a miles, he then rode 3 miles on his i bicycle, and b miles on the cars. How far did he travel ? ; ADDITIOX. 31 5. A man began business with 2c dollars. The first year he gained ^ as much as he had; the second year -^as much as he had at the end of the first year; and the third year $ 400. How much did he gain in the three years ? 6. The letter h represents an odd number. What will represent the next even number ? What the next odd number ? 7. Laura is m years old ; Lizzie is twice as old as Laura ; and iNIabel's age is equal to i the ages of the other two. W^hat is the sum of their ages ? 8. A merchant took in c dollars one week, d dollars the next, and $ 75 the next. How many dollars did he receive in the three weeks ? 9. What is the sum of x -\- x -\- x -\- etc. taken seven times ? Oi X -\- X -\- X -{- etc. taken a times ? 10. A grocer sold h pounds of sugar, c pounds of coifee, d pounds of tea, and 2 pounds of chocolate. How many pounds of groceries did he sell ? 11. A man paid m dollars for a coat, n dollars for a waistcoat, p dollars for trousers, g dollars for a pair of boots, and r dollars for a hat. How much did his outfit cost him ? 12. A man paid a dollars for a farm ; he then expended upon improvements d dollars, and sold it for h dollars more than the entire cost. How much did he receive for it ? 13. My fare to San Francisco was a dollars, my sleeping- car charges c dollars, my meals cost me h dollars, and my other expenses $ 25. How much did I expend before I reached San Francisco ? SUBTRACTION. 36. 1. What is the difference between 8 days and 5 days ? 2. What is the difference between 11 cents and 4 cents ? 3. What is the difference between 12 d and 5d? 4. What is the difference between 15c and 8c? 5. What is the diff'erence between 12/ and 8/? 6. Subtract: Sx from 13a? ; 4,y from l;")?/ ; 6 c from 13c; Sd from 20 d; 10a;^2/ from 14 a?^?/; 8afy from 21 a^?/^; 4a7i/^ from 14 a??/-; 10a^6^ from 18a^6^; Sa^x-y-^ from 15 aV^/^; Aabcd from 13a6cd; 6a6ca; from 18a6ca7; lOaaj^^;^ from 20axh/z^, 7. What is the remainder when 4:aVy^ is subtracted from 10 a^x^y'^ ? What is the sum of 10 aVy and — 4 a^x^y^ ? 8. What is the remainder when 3a6c^ is subtracted from 12 ah(^ ? What is the sum of 12 ahc^ and - 3 ahc' ? 9. Instead of subtracting a positive quantity, what may be done to secure the same result ? 10. What is the result when 8 is subtracted from 15? What, when 8 — 5 is subtracted from 15 ? 11. Why is the result 5 more in the latter case than in the former ? 32 SUBTRACTION. 33 12. What is the result when IQa is subtracted from 18a ? What, when 10a — 8a is subtracted from 18a ? 13. Why is the result greater by 8a in the latter case than in the former ? 14. Instead of subtracting a negative quantity, what may be done to secure the same result ? 15. One vessel was 40 miles east and another 20 miles west from a given meridian. Indicate their relations by proper signs. How far apart were they ? 16. Mr. A's property is worth $ 25a and Mr. B is f 8a in debt. Indicate their financial conditions by proper signs. What was the difference in their financial condition ? 17. A thermometer indicated a temperature of 35° above zero on Jan. 5, and of 15° below zero on Jan 6. Indicate the temperature by proper signs. What was the difference in temperature ? 37. Principles. 1. The difference between similar quan- tities, only, can be expressed in one term. 2. Subtracting a positive quantity is the same as adding a numerically equal negative quantity. 3. Subtracting a negative quantity is the same as adding a numerically equal positive quantity. 38. To subtract when the terms are positive. 1. From 10 a subtract 4 a. PROCESS. Explanation. When four times any number is taken -j rw from ten times that number, the remainder is six times the number ; therefore, when 4 a is subtracted from 10 a, "* ^ the remainder is 6 a. Or, since subtracting a positive ^^— — number or quantity is the same as adding an equal nega- ^^ tive quantity (Prin. 2), 4 a may be subtracted from 10a by changing the sign of 4 a and adding the quantities. Therefore, to subtract 4 a from 10 a, we find the sum of 10 a and — 4 a, which is 6 a. milne's el. of alg. — 3. 34 ELEMENTS OF ALGEBRA. 2. From 13 m take 15 m. PROCESS. 13m 15 m - 2 m ExpLANATiox. After subtracting from 13 m as much as we can of 15 m, there will be 2 m yet to be subtracted, or the result will be —2 m. Or, since subtracting a posi- tive number or quantity is the same as adding an equal negative quantity (Prin. 2), 15 m may be subtracted from 13 m by finding the suiA of 13 m and — 15 m, which is — 2 m. Therefore, when 15 m is taken from 13 m, the result is — 2 m. 3. From 19 a? Take Ax 4. 7 ab Sab 5. 18 m2 13 m^ 6. 16 xy 20 xy 7. 8. 6 x^y-z 9 mn 8 x^yh 14 mn Subtract the following : 9. 8a;-|-32/ from 12a;-f72/. 10. 5a6-f 3 c from 10 a6 + 2 c. 11. 4^>-f 9cZfrom 86 + 5d 12. 5x-\-y from Sx-{- 3 y. 13 . 4:X-{-Sz from 5x-\-9z. 14. 3a + 2^ + 5c from 7a + 5Z> + 6c. 15. 7a + 2& from 9a -1-6. 16. 5x'-\-Sy'--\-6xyhom7x^-^3y^ + 2xy. 17. 4 a;?/ -f- 3 2; from 10 a;?/ 4- 5 2. 18. 5(a2 + 62)-h7c2-f-2d from 6{a^ -\-b^)-\' 4.c^ + 4d. 19. 3a(j9 + g) + 2 from 6a(p + g)+7. 20. 5a;-h72/ + 42;-h5 from 5a7-h42/ + 82;-h6. 21. 3x^-{-2y -^z-\-Av ivom Tx^-^Sy + dz^v. 22. 2a:-f-5(.v + 2;)-h8 from 7x -\-2(y + z)-\-10. 23. 3a2 + 562 + 4c2 + d2 ^^.^^ a^ j^Qb' + c^ -\- 5d\ SUBTRACTION. 35 39. To subtract when some terms are negative. 1. From 7 x—2y subtract Ax— 3y. Explanation. Since the subtrahend is composed PROCP.SS. ^£ ^^^ terms, each term must be subtracted separately. 7 X — 2y Subtracting 4 x from 7 x — 2 ?/, leaves ?jX — 2y,or the Ax — 3v ^^^^^^ ^^y ^^ obtained by adding — 4x to 1 x — 2y. But since the subtrahend was 3 y less than 4 x, to obtain the true remainder, 3 y must be added to Sx — 2y, 3x -\- y which gives ^x -\- y. Therefore the subtraction may be performed by changing the sign of each term of the subtrahend and adding the quantities. Rule. Write similar terms in the same column. Change the sign of each term of the subtrahend from -^ to —, or from — to -\-, or conceive it to he changed, and proceed as in addition. 2. 3. 4. 5. 6. YmmAa^x 9x^yz 6y-Az 7x^-\-5y' 3a-3b-{-c Take 3a^x -'7x'yz 6y-\-2z Ax^'-Sf- 3a + 3h-c 7. 8. 9. From Aah-3c + d 5a^- ^3xy-^x 10 ax - -13y + z^ Take -2a6-h5c -d 6x^ -\-5xy — X 5ax - - "^y- z" 10. From 12 ar^— 20 's^f + ^xy subtract 9 :f?y^ — 2xy-^A^. 11. From 4 a6 + 3 6^ - 6 cd subtract (jab-2h--\-2 cd. 12. From 9 x^y — 6 xy- — 2xy-\-5 subtract 3 oi^y -{-xy -\-^. 1 3 . From 7 ic"* -|- 2 x'^y'' — 5 y"" subtract 4 x"" — 2 x'^y'' — 9 y"". 14. From 8(m-f-?i2)-12(m2 + 7i) subtract 12{m ^ n-) -^(m'^n). 15. From A:x^^7f-3z'-\-i^ subtract -2a^ -\-y^-3r\ 16. From b{p -f- q) - 6(r + s) -}- 15 subtract 8(p + q) + 2(r + 60 + 25. 36 ELEMENTS OF ALGEBRA. : 17. From IS m'nx^ -\- 12 a'bc" ^ abd subtract ~ 2 Tii^n:^^ I -I- 5 abd. I 18. From a^ -2ab -\- c^ - 3b^ subtract 2a'-2ab + 3b'. \ 19. From 2x'' -{-2y^ ^ 4.xy subtract 2 a^ + 2 ?/- + 4 xy. J 20. From ^x^ -\- Sx^y- — Sy^ subtract the sum of x'^ — xry- ' -f 3?/^ and 2x^ + 2x-y' -9y\ ] 21. From 5 x^y^ -\- 10 x^y — G yz^ subtract lOa?^?/ — 4.ry' '. H- 5yz^ ] 22. From 30;"^ — 4i»''i/'"-f 4i/"* subtract 4 a;"^ + 2 x"?/"* — ?/^ ^ 23. From Abx^ + 3 a?/- + 4 — c?/ subtract cy — 5 — bx". j 24. From 10?7i^ — 4m?? — 3n^ — 18 subtract the sum of \ m^ — 3 mn + 4 and 5m- — 2 mn + 6 ti^. i 25 . From a^x — x^ -\- x^y — 8 subtract 3 a^x —5—xr-\-2 a%. ; 26. From 150^2^- — 15 subtract AxY -{- z- — 4y" — W. i 27. From 2a;4-2/ + :s + ?« subtract a? + 2?/ -j- 2;$; -|-2?/. 28. F>om 16 a^b^— 12 be -\-14:b'- subtract the sum of \ a'b'-3bc + A and 10 a'b^ -J^S be + 5 b\ ] 29. F^rom ax -\- by subtract ex — dy, PROCESS. Explanation.. Since a and c may be re- \ garded as the coefficients of x, and 6 and - d \ ax -f- oy ^l^g coefficients of y, the difference between i ex — dy the quantities may be found by writing tiie ; ', r — r— - difference between the coefficients of x and // ; (a e)X -f-(0 -\- a)y respectively for the coefficients of tlie remain- ; der. Since c cannot be subtracted from a, the \ subtraction is indicated by (a — c), and since — d cannot be sub- | tracted from 6, the subtraction is indicated by (6 + d), consequently ; the remainder may be written (a — c)x + (6 + d)y. 30. From ax-^2y subtract 2x — by, 31. 'FTom2cx — ay + 3z subtract cy-\-5x — az. SUBTRACTION. 37 32. From ex — 12ab}j + ^aV subtract 6x— 10 ahy-\- 3 b^. 33. From 2a(x — y)-\-4: abx subtract 3c(x — y)-^2 ax. 34. From ax -^by -{- cz subtract bx + cy + dz. 35 . From 3x-\-7 y -\- Sz subtract box ~ ay -\- az. 36. From oay -\-2cz -{- 6x subtract cy — az — dx. 37. From (a — b)x -\- {c -^ d)y subtract ax-\-dy, 38. From 7 a;^ -f- 5 ay^ -i-6z^ subtract 2 ax^ -f- 3 y- — abz^. 39 . From ( m -f- n ) x^ -\- ( m — n)y--\- z^ subtract mno^ — 4 1/^ -f az\ 40. From «(.r + y) -|- ^(^' — Z/) + <^^ subtract b(x -^ y) — a{x — y) — cx. 41. Express the difference between m and ?i. 42. Write the number one less than x\ the number one greater. . 43. A man sold a horse for $125 and gained a dollars. What did the horse cost? 44. A girl earns b cents a day and spends 25 cents a week. How much has she left at the end of the week ? 45. The sum of two numbers is 30, and x represents one of them. What represents the other? 46. The difference between two numbers is 5. How may the numbers be represented ? 47. A merchant bought a hat for b dollars and a coat for c dollars, and sold the two for d dollars. What repre- sents his gain ? 48. A man whose income is a dollars spends m dollars for rent, n dollars for living expenses, and 100 dollars for other expenses. What represents the amount he saves? 38 ELEMENTS OF ALGEBRA. 49. A lady paid a dollars for a dress, c dollars for a hat, and $25 for a cloak. How much had she left from a $50 bill? 50. A man paid 40 dollars for a cords of wood, and sold it at 3 dollars a corvl. How much did he gain? 51. A farmer sold some grain for b dollars, some fruit for d dollars, and some hay for e dollars. He received in part payment a horse worth / dollars. How much remained still to be paid? 40. The parenthesis, ( ), the vinculum, , the bracket, [ ], and the brace, \\, are called Signs of Aggregation. They show that the quantities indicated by them are to be subjected to the same process. Thus, (a + 6) X c, a + 6 X c, and {a -h h} xc show that the sum of a and h is to be multiplied by c. 41. The subtrahend is sometimes expressed with a sign of aggregation and written after the minuend with the sign — between them. Thus, when b -\- c ~ d is subtracted from a + h, the result is some- times indicated as follows : a -\- b —(b + c — d)» 1. What change must be made iu the signs of the terms of the subtrahend when it is subtracted from the minuend ? 2. When a quantity in parenthesis is preceded by the sign — , what change must be made in the signs of the terms when the subtraction is performed, or when the paren- thesis or other similar sign is removed ? The term parenthesis is commonly used to include all signs of aggregation. 42. Principles. 1. A j^arenthesis, preceded by the minus sign, may be removed from an expression by changing the signs of all the terms in ptarenthesis. SUBTRACTION. 39 2. A parenthesis, preceded by the minus sign, may he used to inclose an expression by changing the signs of all the terms to be inclosed in parenthesis. When quantities are inclosed in a parenthesis preceded by the plus sign, the parenthesis may be removed without any change of signs, and, consequently, any number of terms may be inclosed in a paren- thesis with t\\e plus sign without any change of signs. The student should remember that in expressions like — (x"^ — ?/ + 2) the sign of x^ is plus, and the expression is the same as if written -(+ x*^ - ?/ + z). Simplify the following: 1. 20-(3-f4-6 + o). Solution. 20 - 3 - 4 + 6 - 5 = 14. 2. 2o-(6 + 9-3-7+13). 3. o-(3 + 2-6)-(-2-4-M). 4. (7-5 + 3)-(8-9 + 2). 5. lo-(3 + 4-6)-(8-6H-7). 6. (4 + 10 -8) -(3 + 7 -4). 7. 19-(12-5-f 8)-(ll-5-3). 8. 30-(-6 + 8-l)-(18-10 + 4). 9. (14-5 + 2)-(18-20 + o). 10. lo-(3-f 8-7) + 16-(10 + 4-f-o-8). 11. (5 + 7-8)-6-(-5 + 3-6 + 2). 12. 16-[8-(5 + 6-4)-f-12]. Solution. 1(3 - [8 - (5 + 6 - 4) + 12] =: 16 _ [8 - 5 - 6 + 4 + 12] -10-8 + + 0-4-12 = 3. 40 ELEMENTS OF ALGEBRA. 13. 25 -[13- 4 + (3 -10 -2)]. 14. 10- 58 -(15 + 7 + 3) 4- 6J. 15. 17-(3 + 8)-[12-(3 + 8)-5]. 16. _ J- 16 + 13 -(6-1 4-4)4-5 -10|. 17. (3 + 7-4)-[14-(13 + 7) + 5]. Simplify the following : 18. a — {b — c-{-d — e). Solution, a -{b - c + d -e) =a-b-^c-d + e. 19 . 2x — {x — ox-\-3x — Sx). 20. 7m — (3n 4-2m — 6m 4 >?.). 21. 4 or 4- 7 ax — (5 ax + 3 ax — 2 oj- + 10 ax). 22. a'-\-b'-{-2ab-2a'-2b'). 23. {6xy-{-2z)-{4.z-{-3xy-2z-{-5). ~ 24. a — 6 — (tt + 6 — c — 3). 25 . a 4- 6 - (2 a - 3 6) - (5 a 4- 7 6) - ( - 13 a + 2 6) . 26. (a + b-{-c)-{-{-a-\-b-c)-(:a-b-\-c), 27. x-l-x-^2x-{x-\-2x)-2x^. 28. 3x — lx — 3z — {2y — z)'], 29. a- — a — (4a — 1/ — 3a2 — 1). 30. m + n — (m 4- ^0 — \ ^'^ — ^^ ~" (^^ + ^0 — ^^ • 31. {x'-]-2xy + y')-{2xy-x'-f). 32. 9a; — [8x — 6ic — 3a;]. 33. (a;4-10)-^,a;-3a;4-25-10|. 34. 8a-(6a-5) — (5a + ll-4a). 36. a-[26-(3c+26)-a]. SUBTRACTION. 41 TRANSPOSITION IN EQUATIONS. 43. 1. If a certain number, diminished by 3, equals 15, what is the number ? If a; — 3 = 15, what is the value of o^ ? 2. If a certain number, increased by 3, equals 15, what is the number ? If a; -f 3 = 15, what is the value of a;? 3. In the equation x — 3 = 15, what is done with the 3 in obtaining the value of a;? In the equation a; = 15 + 3, how does the sign of 3 compare with its sign in the original equation ? 4. In the equation a; -f- 3 = 15, what is done with the 3 in obtaining the value of a; ? In a; = 15 — 3, how does the sign of 3 compare with its sign in the original equation ? 5. In changing the 3's from one side or member of the equation to the other, what change was made in the sign ? 6. When a member or quantity is changed from one member of an equation to the other, what change must be made in its sign ? 7. If any number, as 5, is added to one member of the equation 2 -f- 3 = 5, what must be done to the other member to preserve the equality ? 8. If any number, as 3, is subtracted from one member of the equation 2 + 3 = 5, what must be done to the other member to preserve the equality ? 9. If one member of the equation 2 -f 3 = 5 is multiplied by any number, as 4, what must be done to the other mem- ber to preserve the equality ? 10. If one member of the equation 2 + 3 = 5 is divided by any number, as 5, what must be done to the other member to preserve the equality ? 42 ELEMENTS OF ALGEBRA. 11. If one member of the equation 5 + 3 = 8 is raised to any power, as the second power, what must be done to the other member to preserve the equality ? 12. What, then, may be done to the members of an equa- tion without destroying the equality ? 44. The parts on each side of the sign of equality are called the Members of an Equation. 45. The part of an equation on the left of the sign of equality is called the First Member. c 46. The part of an equation on the right of the sign of equality is called the Second Member. 47. The process of changing a quantity from one member of an equation to another is called Transposition. 48. A truth that does not need demonstration is called an Axiom. Axioms. 1. Things that are equal to the same thing are equal to each other. 2. If equals are added to equals, the sums are equal. ' 3. If equals are subtracted from equals^ the remainders are equal. 4. If equals are multiplied by equals, the products are equal. 5. If equals are divided by equals, the quotients are equal. 6. Equal powers of equal quantities are equal. 49. Principle. A quantity may be transposed from one member of an equation to another by changing its sign from + to — , or from — to -\-. SUBTRACTION. 43 EQUATIONS AND PROBLEMS. 50. 1. Given 2 x — 3 = x -\- 6, to find the value of x. PROCESS. Explanation. Since the known and the un- o r> . \ a known quantities are found in both members of ^x — o — x -f- o ^j^g equation, to find the vahie of x, the known + 3 = -{-3 quantities must be collected in one member and 7k r~^ the unknown in the other. ^ ' Since — 3 is found in the first member, it may X = x be caused to disappear by adding 3 to both mem- ] _ Q bers ( Ax. 2) , which gives the equation, 2x = x-\-9. ~~ Since x is found in the second member, it may be caused to disappear by subtracting x from ^^' both members (Ax. 3), which gives as a result- 2x — 3 = x -\-6 ing equation, x = 9. o^ _ J. fi-4-S ^^' since a quantity may be changed from one ~ member of an equation to the other by changing ^= " its sign (Prin.), — 3 may be transposed to the second member by changing it to + 3, and x VERIFICATION. ^^^ ^^ transposed to the first member by changing it to — x. Then, the resulting equa- 18 — 3 = 9 + 6 tion will be 2x-ic = 6 + 3. 15 = 15 By uniting the terms, x = 9. The result may be verified by substituting the value of x for x in the original equation. If both members are then identical^ the value of the unknown quantity is correct. Thus, if 9 is substituted for x in the original equation, the equation becomes 18 — 3 = 9 + 6, or 15 = 15. Therefore, 9 is the correct value of x. Rule. Transpose the terms so that the unknown quantities stand in the first member of the equation^ and the known quantities in the second. Unite similar terms, and divide each member of the equation by the coefficient of the unknoicn quantity. Verification. Substitute the value of the unknown quantity for the quantity in the original equation. If both . members are then identical in value, the value of the unknoivn quantity found is correct. 44 ELEMENTS OF ALGEBRA. Transpose and find the value of x. 2. 0^ + 4 = 10. 3. .X- — 3 = 4. y 4. ^a;+l = o. 5. 6x--G= 12. 6. 4a.' +3 = 15. 7. 8a;-2 = 14. 8. 3a;+5 = 26. 9. 9a;-5 = 3L 10. 5i« + 2 = 10 4-T. 11. 7aj-l = 30 + 4. 12. 20? -10 = 3 + 5. 13. 4x — 2a; = 3H-7. 14. 6aj-3 = 2a; + 9. 15. 5a;— 15 + 2 = 2— 3a?4-l. 16. 6a;- 2^ + 4 = 16-8. 17. 7a; — 3 = 2a; — 4 4-11. 18. 2a; + 4 = 9a;-10. 19. 6a; + 25 = 18 — a;. 20. 3a;-4 = 12-4. 21 . 5 a; — 5 = 67 — 3 a;. 22. 10 a; -20 = 24 -12 a;. 23. 3a;-14 = 10-.x. 24. 2 a; -16 = 20 -4 a;. 25. 15 a; -39 = 29 -2 a;. 26. 3a;-18 = 31-4a;. 27. 4a; -14 = 49 -3a;. 28. 5a; -20 = 25 -4a;. 29. 2a; -36 = 60 -6a;. 30. 3 a; — 20 -!-.« = 44 -4 a;. 31. 3a; + 3 = 5 + 8.^;- 7. 32. 9a;4-15 = 6 + 7a; + 3. 33. 5 a; -f- 2 a; = 9 a; + 5 — 15. 34. 8a; -10 = 10 + 2a; 4-4. 35.-"ir^6a;-6 = 6 + 3-3a;. 36. x-:^ = 18 - 4 .f - 3. 37. 3a; + 6a; =1-8 - a; + 2. 38. 3^— 6 = a; + 14 — 4. 39. 9a;+13=26 + 2.^;+l. 40. 4a; + 4- 3a; =16-2.7;. 41. 27 a; -14 = 190 -41a;. 42. ^x- 12= 4a; + 18. 43. 5a;- 15 = 3 a; +9. 44. 18a; + 9 = 15a; + 30. 45. 7.T-3+2a;=5a;-20 + l. 46. a; + 14-2a; = 6a;-21. 47. 9a; + 3-24 = 5.i;-25. 48. 10;r -4 + 3= 6a;+ 19. 49. 3a;-15-10 = 20-2.^•. 50. 5 a; — 5 — 20 + 6 a; = 41. 51. 2 a; — 25 = 35 — a; — 3 a;. 52. 3a;-19 = 20-10.T+13. 53. 5a;— 16 = 25— a;+40— 3a;. 54. 6a;-5-30=10-4a;-5a;. 55. 7a;-30 = 10 + 16-7a;. 56. 5a; — 50 = 25 — 5a; +25. 57. 10a; -22 = 17 -2a; -a;. SUBTRACTION. 45 PROBLEMS. 51. 1. Twice a certain number increased by 15 is equal to the number increased by 19. What is the number ? 2. What number is that whose double exceeds the num- ber by 12 ? 3. Ten times a certain number diminished by 13 is equal to the number plus o. What is the number ? 4. What number diminished by 8 equals 6? 5. Six times a certain number plus 7 equals five times the number plus 12. What is the number ? 6. Three boys together had 90 cents. The first had 10 cents more than the second, and the second had 1 cent more than the third. How much had each ? 7. A father gave a certain sum to his youngest son, and 4 cents more to the next older, and 10 cents to the oldest. If he gave to all 20 cents, how much did he give to each ? 8. The greater of two numbers exceeds the less by 14, and the sum of the numbers is 34. What are the numbers ? 9. A and B started in business, A furnishing $4000 more than B. Three times B's capital was then equal to A's. How much did each furnish ? 10. A and B had the same sum of money. A gave B $4, and then B had double the amount A had left^^— How much had each at first ? 11. A tourist rode 32 miles upon a bicycle. A certain number of miles was down hill, twice as far plus 8 miles was level, and the distance up hill was ^ as far as the dis- tance on a level. How many miles did he travel upon each kind of road. 46 ELEMENTS OF ALGEBRA. j 12. A man has two horses, of unequal value, together | worth $200. If he shouki put a saddle worth $30 on the ; poorer horse, the horse and saddle would together be equal ■ in value to the better horse ? What is the value of each ? | 13. Six hundred gallons of water are discharged into a ■ cistern by 3 pipes. The second discharges 100 gallons j more than the first, and the third discharges three times as j much as the first. How many gallons are discharged by each , pipe ? I 14. A drover being asked the number of his cattle said ^ that if he had three times as many as he then had and 25 more, he would have 1000. How many cattle had he ? f^r 15. A man bought a watch and chain for $60. The ! watch cost 12 times as much as the chain lacking $ 5. ■ What was the cost of each? .- j 16. A tenement house contained 90 persons, men, women, \ and children. If there were 4 more men than women, and 10 more children than men and women together, how many ; were there of each ? : 17. A steamer and its cargo are together worth $ 120,000. If the steamer lacks only $8400 of being worth twice as much as the cargo, what is the value of each ? 18. A clerk's expenses are $400 per year, and his brother's are $600 per year. If the brother has three times as large an annual salary and he has left at the end of the year a sum equal to twice his brother's salary, what is the salary of each ? 19. If a house and lot cost four times as much as the lot, and the house cost $ 2500 more than twice as much as the lot, what was the cost of each ? MULTIPLICATION. ^ 52. 1. What is the sum of ^m-{-5m -{-5m? Or, how much is 3 times 5m? 8 times bm"^ 2. What is the sum of Sxy -\- %xy -\-%xy -{-^xy? Or, how much is 4 times ^xy? 10 times ^xy? 3. How much is 6 times 1 he? Which quantity is the multiplier ? Which is the multiplicand ? What sign has the multiplier ? What sign has the multiplicand ? What sign has the product ? 4. When a positive quantity is multiplied by a positive quantity, what is the sign of the product ? ^ 5. If a vessel sails south 8 miles per hour, indicated by — 8 mi., how far will she sail in 5 hours ? What will be the sign of the product ? '•^-^ 6. How much is 4 times —5xy? 6 times —6ab? 7 times —Scd? What is the sign of the multiplier in each case ? What is the sign of the multiplicand ? What is the sign of the product ? — 4 a6 x 8 = ? — Z 1, "y I 7. When a negative quantity is multiplied by a positive quantity, what is the sign of the product ? — 8. How does the product of 6 times 7 compare with the product of 7 times 6 ? What effect upon the product has it to change the order of the factors, when the numbers or quantities are abstract ? ""'"■■' 47 48 ELEMENTS OF ALGEBRA. 9. How, then, will the product of — 3 a multiplied by -j- 4 compare with the product of -f- 4 multiplied by — 3a ? What is the product ? — 5a;2/x6— ? 6 x — o xy = ? ~7a6x4 = ? 4.x-7ab=?^^ ^^^ 10. When a positive quantity is multiplied by a negative quantity, what is the sign of the product ? 11. How much is 6 times — 3 a ?^ 2 times — 3 a ? (6 — 2) times — 3 a, or 6 times — 3 a, —2 times —3a? 12. Since. 2 times —3 a, or —6 a, must be subtracted from — 18 a to obtain the correct product, what will be the sign of — 6 a after it is subtracted ? -4~ 13. Since — 2 times — 3 a gives a product of -{-6 a, what may be inferred regarding the sign of the product when a negative quantity is multiplied by a negative quantity ? 14. What is an exponent ? What does it show ? What does a^ mean ? When w* is multiplied by a^, how many times is a used as a factor to obtain the product ? How many times, when a^ is multiplied by a^ ? 16. How, then, may the number of times a quantity is used as a factor in multiplication be determined from the exponents of the quantities in the expressions multiplied ? IHow may the exponent of a quantity in the product be determined ? fe I 16. 3a2x6=? 10a"^x5 = ? 20a2x3= ? 25a^2/x2 = ? I How is the coefficient of the product determined from the I coefficients of the factors, or from the multiplier and the I multiplicand ? 63. Multiplication is indicated in four ways: 1. By the sign x , read multiplied by or times. Thus, a X h shows that a is to be multiplied by b. MULTIPLIC A TION. 49 2. By the dot (•), read multiplied by or times. Thus, a ' b shows that a is to be multiplied by b. 3. By writing letters, or a number and a letter side by side. Thus, ab shows that a is to be multiplied by b ; and 5 a shows that a is to be multiplied by 6. 4. By a small figure or letter, called an Exponent, written a little above and at the right of a quantity, showing the number of times the quantity is to be used as a factor. Thus, a^ shows that a is to be used as a factor 5 times, or that it is equal to « x a x a x a x a or aaaaa. 54. Principles. 1. The sign of any term of the product is -f- tvhen its factors have like sig7is, and — ivheji they have UNLIKE signs. 2. The coefficient of a quantity in the product is equal to the product of the coefficients of its factors. 3. The exponent of a quantity in the proditct is equal to the sum of its exponents in the factors. ^^^ 55. To multiply when the multiplier is a monomial. 1. What is the product of 5 x-yz multiplied by 3 a6a; ? PROCESS. Explanation. The coefficieut of the product is ob- oxryz tained by multiplying 5^y 3 (Prin. 2). The literal 3abx quantities are multiplied by adding their exponents (Prin. 3). Hence, the product is I6abx^yz. 15 abx^yz 2. What is the product of 3 6— ^.c multiplied by JU5 V ? y PROCESS. Explanation. The product of 3 6 multiplied by * o, 5c2 is 15 6/j2^ But, since the entire multipli- ~ _ cand is 3 b^— c, the product of c multiplied by 5 c^ - ^^' must be subtracted from^l5 6c2. The product 15 h(^ -__ 5 (^ of c multiplied by bd^ is 5c^, which subtracted from 15 6c2 gives the entire product 15 6c2 — 5 c^. milne's el. of alg. — 4. 50 ELEMENTS OF ALGEBRA. Rule. Multiply each term of the midtipUcanQl^ by the multiplier, as folloivs : To the product of the numerical coefficients, annex each literal factor ivith an exponent equal to the sum of the expo- nents of that letter in both factors. Write the sign + before each term of the product ivhen its factors have like signs, and — ivhen they have unlike signs. - 3. 4. 6. 6. 7. 8. 9. 10. Multiply 10 10a -6a 22a; 18 -Ub 17c 24a; By 4-4 3 -5 6x^ 3 3c -9a; -/ 11. 12. 13. 14. 15. Multiply 5 oc^yz — 15 xy^ 16c^dz -42mV ~25a'b By 2 x^yz — 3 xy''' 16. 17. — 4:dz — 3 m7i 5 a6^ 18. 19. Multiply (a-\-b) 4(x- ■y) 3a; 4-2?/ -52; 2x-y-2y'z' By 4 -5 2a 3a; Multiply : 20. 3 a^a; — 5 x-y + 2 2/^ by — 4 xy. 21. a^ -^2 ab + b- by ab. 22. 6 m -t- 7 mn + 5 n^ by — 3 mn, 23. x^-2o^-\-5x^-}-x-3hy9x^ 24. 9a^-lSab-^4:b^-6hjl2a^b', 25 . x^ — Sxy — 3xz-\- yh^ by 2 ax. 26. 4 a6 -f 3 a-b - 5ab' - 2 a^hy 3 b\ 27. 3 m^ — 10 mn — 8 n^ by 4 7n?i. 28. x'^-^x^-^-x^ -\-x-{-l\)y —6x. 29. 6a2-18a&-hl5c2-20a6c + 14by3a252. MULTIPLICATION. 51 56. To multiply when the multiplier is a polynomial. 1. Multiply a-\-bhj a-{-b. Solution. a -\- b a -^ b a times a -\- b = a'^ -{- ab b times a -\- b — ab -\- b^ (a + ?>) times (a 4- b) = a*^ + 2 ab + b'^ Rule. Multiply each term of the midtiplicand by each term of the multiplier, and add the partial products. 2. 2ab-Zc 4ab -}- c -3c-2 3. X -y Sa%'- \2abe 2 abc - x^-S:x^y-\-Sx-^yi-xy^ - x^y + SxY-Sxy^-{-y^ 8 aVy^ - 10 abc - -3c2 x* -ix^y-^G x'hf - 4 xy^ + y^ Multiply : 4. x-\-y hy X -\-y. 17. 5m — 4:7i hj Am -{- oy. 5. m 4- ^i by m — n. 18. x-\-2yhyx-{-5 y. 6. a--^2cdj-\-b-hy a-\-b. 19. 1 -\- x -{- x^- by 1 — x. 7. 2 a — 5 6 by 2 a -h 5 6. 20. x -{- y -\-lhy x — y — 1. 8. aj-f4byx — 10. 21. 2a7 + 42/ by 3ic — 2i/. 9. 3y-^2zhy 2y-\-3z. 22. a + 6 - 2c by 2a - 6. 10. 3 a + 76 by 3 a — 76. 23. 4 a; + 7 by 3 a: — 2. 11. 2ic + l by 3a.- — 6. 24. 3 am + 6c by 8ac + cl 12. 2a — 36 by 3a + 56. 25. 3 a;?/ — 6 ?/ by 4 iui/ + 8 y. 13. 3??i + 47? by 2m + 3?^ 26. a; + ?/ — 2; by a; + ?/. 14. 5?/ — 32? by 4?/ — 4^;. 27. a — 6 — cbya — c. 15. 26 — 5c by 36+ 8c. 28. 2a -{- x — y by a — x. 16. 3a; -20 by 8a; + 4. 29. 2x -\- 3y — 6 by x -\- oy. 52 ELEMENTS OF ALGEBRA. 30. 3xz-\-2y^hy Sxz-4:y\ 31. 7n -f ?i + 1 by m — n + 1. 32. ox-{-4:hj5x— 9. 33. 3 XT — 4:y- by 3 or ^7 yi 34. cf -f- 3 d'h + 3 ah- + W by a -|- Z>. 35. 4ar — 12ii7?/-f-9/ by 2i» — 3?/. 36. m^ + 7/i'7i -j- mhr -f wi^i^^ 4- ^^^ by y>i — ?i. / PROBLEMS. 57. 1. If from three times a number 4 is subtracted and the remainder is multiplied by 6, the result is 12. AV'hat is the number ? 2. If from two times a number 4 is subtracted and the remainder is multiplied by 3, the result equals two times the sum of that number and 2. What is the number? 3. A father is four times as old as his son, and 5 years ago he was seven times as old as his son. What is the age of each ? 4. Samuel and John together have 40 cents. If John had 5 cents less, and Samuel 5 cents more, Samuel would have three times as much money as John. How many cents has each ? 6. A commenced business with three times as much capital as B. During the first year A lost \ of his money, and B gained $500. The amount of A's and B's money was then equal. How much had each at first ? 6. A is 50 years of age ; B is 10. When will A be three times as old as B ? 7. Six men hired a boat, but 2 of them being unable to pay their share, the other 4 were obliged to pay 1 dollar more each. For how much did they hire the boat ? MULTIPLICATION. 63 8. Three times the difference between a certain number and 10 equals two times the sum of the number and 10. What is the number ? 9. Express the product of the factors 2, x, y, z, x^, y, Az, 10. What will d quarts of milk cost at /cents per quart? 11. How far will a man travel in a hours if he goes b-\-6 miles per hour? 12. A farmer has a cows and three times as many sheep less 8. How many animals does he own ? 13. A man sold 20 acres of land at a dollars per acre. With a part of the money he bought 3 horses at d dollars each. How much money had he left ? 14. If a men can do some work in 12 days, how long will it take one man to do the same work ? 15. A starts in business with b dollars; B starts with c dollars. In one year A gains as much more, while B gains i as much more. How much has each at the end of the year ? 16. What will 10 bushels of potatoes cost at 2 m cents per bushel ? 17. A man earns f 2 per day and pays fa per week for his board. How much money will he have at the end of b -weeks ? 18. An engine pumps 150 gallons of w^ater into a tank each day ; 10 c gallons are drawn off. How much water will remain in the tank at the end of 4 days ? 19. The daily w^ages of a mechanic are a dollars. How much will the wages of 10 mechanics for c days be ? 64 ELEMENTS OF ALGEBRA. SPECIAL CASES IN MULTIPLICATION. 58. The square of the sum of two quantities. a -f b a -f b m + n -f m~ + mn mn n^ 71V 4- 2 mil + n- a^ -\- ab ab + b^ a- + 2 a6 + &^ 1. How is the first term of the second power, or square, j of the quantities obtained from the quantities ? How is ] the second term obtained ? The third term ? 2. What signs have the terms ? 59. Principle. The square of the sum of two quantities is equal to the square of the first quanilty, plus twice the pro- duct of the first and second, plus the square of the secoyid. Write out the products or powers of the following : 1. {x-\-y)(x-{-y). 13. Square 2 a; -f- 5. 2. (6+c)(6 + c). 14. Square 3m + 1. ^ 3. (m + -)(m + 2). 16. Square 2a + 5&. 4. {a -{-x){a -\-x). 16. Square a^ + 6-. 5. (if 4- 3) (a; + 3). 17. Square aj^ + 3. 6. (?/-|-l)(2/ + l). 18. Square 2 m' -f 3 nl 7. {2x -^y){2x-^y). 19. Square a6 -h 2 c. 8. {m-\-2n){m -\-2n). 20. Square 2 icy + 2;. 9. (3a + &)(3a4-^). 21. Square 2/^ + 4 2;l 10. (2i«-f 32/)(2a; + 32/). 22. Square 0^ + 8. j 11. (a + 4 6) (a + 4 6). 23. Square 5 a -f " &• 12. (2m+2M)(2m+2n). 24. Square 4 a^ -j- 3 &-. MULTIPLICATION^. 55 60. The square of the difference of the two quantities. X — y c - d X - y c — d or— xy c- — cd — 072/ 4- r - cd-\-d' x'^2xy-{- 7/ c}-2cd^d' 1. How is the first term of the second power obtained from the terms of the quantity squared ? How is the second term obtained ? The third term ? 2. What signs connect the terms of the power ? 61. Principle. The square of the difference oftioo quan- tities is equal to the square of the first quantity, minus twice the product of the first and second, plus the square of the second. Write out the products or powers of the following : 1. (a — x){a — x). 13. Square 2a- 3 6. 2. {h-c){h-c). 14. Square m — 2n. 3. {m — 7i)(m — >i). 15. Square 2 6 - 4 d 4. {x-2)(x-2). 16. Square a^ - 2 h\ 5. (y-z){y-z). 17. Square he — xy. 6. (a-;U)(a-3 6). 18. Square 2x^ — o y-. 7. (b-2c){b-2c). 19. Square 2a — c. 8. {2x-2y){2x-2y), 20. Square 3m^— 1. 9. {b-o){b-5). 21. Square 3 mn — 4. 10. (y-l){y-l). 22. Square y^ — 6. 11. {ah -2) {ah -2). 23. Square Ax^ — by\ 12. (a; -4) (a; -4). 24. Square ah — 2 (?. 56 ELEME^v^TS OF ALGEBRA. 62. The product of the sum and difference of two quantities. X - y X -f y x^ — xy a;y - y' x" -r c -{- d c — d c^ + cd - cd - d^ 1. How are the terms of the product of the sum and difference of two quantities obtained from the quantities ? 2. What sign connects the terms ? 63. Principle. The product of the smn and difference of tivo quantities is equal to the difference of their squares. Write the products of the following : 1. {a-{-b){a-b). 13. (b + 2c){b-2c). 2. (m 4- >i) (^>i — n). 14. (Sx-\-8y)(3x-8y), 3. (a-^x){a — x). 15. (a; + 10) (a.- -10). 4. (2a-\-b)(2a-b). 16. {bc-\-ef){bc-ef). 6. (2x-\-y)(2x-y), 17. (Sx'-{-2f)(3^--2f), 6. (a + 4) (a -4). 18. (5a + 3a;)(5a-3i^0- 7. (2m+37i)(2m-3n). 19. (a^ + b')(a'-b'). 8. (y-^l)(y-l). 20. (mn + 4)(?/iyi — 4). 9. (x + 5)(x-o). 21. (x + 6){x-6). 10. (2 + y)(2-2/). 22. (4:y-\-7z)(^y-7z). 11. (ab-\-3c)(ab-3c). 23. (3x-^^y){Sx-4y). 12. (2m4-2n)(2m~2n). 24. {2ab+5c)(2ab-'5c). MULTIPLICATION. 57 64. The product of two binomials. X +3 a; +3 x —3 X 4-5 " X ^ 5 X — 5 x^-i-3x xF4-3x x^-3x 5a; + 15 — 5a; — 15 — 5 x + 15 a;2-f 8a;-fl5 ar'_2a;-15 x'-8x-{-15 1. How is the first term of each product obtained from the factors? 2. How is the second term of the product in the lirst example obtained from the factors? The second term in the second example? The second term in the third example? 3. How is the third term of the product in each example obtained from the factors ? 4. How are the signs determined which connect the terms ? 65. Principle. The product of two binomial quantities having a common term is equal to the square of the common term, the algebraic sum of the other two multiplied by the common term, and the algebraic product of the unlike terms. Write the products of the following : 1. (a; + 3) (a; + 4). 8. (aj_4)(a;-f 8). 2. {x-l){x^5). 9. (a; + 9) (a; + 3). 3. {x-2){x-3). 10. (a; -12) (a; + 6) 4. (x + %){x-l). 11. {x-5){x-l). 6. (a; + 5)(a;-f 10). 12. (a; + 14)(;i;-4) 6. (^-13) (a; -4- 3). 13. (,;_!) (^4. 8). 7. (a;-F20)(a; + 5). 14. (a; -5) (a; -4). 58 ELEMENTS OF ALGEBRA. 16. (x-\-ll)(x-2). 20. (a; 4-9) (a? 4- 12). 16. (x-25)(x-4.). 21. (a; -10) (a; 4- 12). 17. (a; + 5) (a; 4- 15). 22. (a; — 2.^) (a; 4- 47/). 18. (a; — 6)(a.' — 3). 23. (a; — a) (a; — 7a). 19. (aj4-6)(a;-3). 24. (x -{- 6 y) {x -\- 10 y) . SIMULTANEOUS EQUATIONS. 66. 1. If the sum of two numbers is 8, what are the numbers ? How many answers may be given to the ques- tion ? 2. Let X and y stand for the two numbers; then, in the equation x -\- y = S, how many values has x? How many has y? How many values has each unknown quantity in such an equation ? 3. In the equation x -^ y = 20, what is the value of y, if x = S? Ifa.' = 6? Ifa; = 4? Ifaj = 12? If a; = 10? 4. If the equations x-\-y = 6 and x — y =2 are added together (Ax. 2), what is the resulting equation? What is the value of x in these equations ? Of y? ^ 67. Equations in which the same unknown quantity has the same value are called Simultaneous Equations. 68. The process of deducing from simultaneous equations other equations containing a less number of unknown quan- tities than is found in the given equations, is called Elimina- tion. 69. Elimination by addition or subtraction. 1. If the equations x -{-3y = 9 and x — 3y = 3 are added, what is the resulting equation ? What quantity is elimi- nated by the addition ? SIMULTANEOUS EQUATIONS. 59 2. How may the equations 2 x — 4 ?/ = 4 and ic + 4 ?/ = 8 be combined so as to eliminate y ? 3. How may the equations 3 a; + 4 ?/ = 18 and 3 iK + ?/ = 9 be combined so as to eliminate x ? 4. When may a quantity be eliminated by addition ? When by subtraction ? 5. \i x-\-'6y = o and 2 x + 3 ?/ = 7, how may the values of X be found ? 6. If3a; — 2/ = 5 and 2x-{- y = 6^ how may the value of X be found ? 70. Principle. Quantities may he eliminated by addition or subtraction when they have the same coefficients. 71. 1. Find the value of x and y in the equations it* -|- 3 y = 11 and 2a.-- 4 7/ = 2. ., .. .. Explanation. Since the quantities X -\-f:y y = II \i) \l^^^f^i not the same coefficients, we must 2 a; — 4 7/ = 2 (2) multiply the equations by such numbers ~ ~ ~ Q. as will make the coefficients alike. If 4 0/* -f- Jw^^ = 44 (o) ^g ^jgj^ ^Q eliminate ?/, we must multiply 6 a; — 12 :y = 6 (4) (i) by 4 and (2) by 3 (Ax. 4). We may TT~ ~ ^. now eliminate y from equations (3) and 10 a; = oU (O) ^4>j |3y addition. From the resulting X = 5 (6) equation, 10 x = 50, the value of x is ., obtained by dividing each member by 10, 5 + o // = 11 (0 ^ijg coefficient of x. 3 y = i) (8) By substituting the value of x in equa- ^ / _ 2 (9) tion (1), equation (7) is obtained, and the ^ value of y is found to be 2. EuLE. If necessary, multiply one or both equations by such a quantity as ivill cause one unknown quantity to have the same coefficient in each equation. When the signs of the equal coefficients are alike^ subtract one equation from another ; when the signs are unlike^ add the equations. 60 ELEMENTS OF ALGEBRA. ; i Find the values of the unknown quantities in the following equations : 2. 3. 5. } X- y= 2} ' \ox-j-:Uj=]6) c x-2y= 4|. ^^ ^9x-^-ij/= 31^ \2x- y = lli . ' i2a;-8y = - 2)" I a:-f-3y = 17| ^^ |2?/+ z = 26) \2x-2ij= 2) ' \27j-\-2z = 2S^ f ^4- l/= 4| ^^ <2^ + 3. = 23| (4a;- ?/= 1) ' \3y-2z= 2) (2a; + 2^ = 20) ^^' (3a'+ ;i = 18 I (2.T- //= 3 1 ^g^ ( a; 4- 2// = 30 I ^ ( .. + 2,v=:23| ^^ r3.r- ./= 30 1 l3.b'+ 2/=34i * 1 ;«-3// = -30) ( 4.1- + 2// = 14 I ( 2a; + 3// = 14 ^ 7. ^//=14| y= 43 21. (9a;- y= 4 3 (3a; + 2/y=16 3 10. j^^^-^y = lM 22. 1^^'^- ' = ^^1 ( a;+ yz= 7) (l4?/-3;2= 3) ^^ ^2x- 5y=. 6| ^^ (5.. + 2,= 6| (ox- \2y = 16 3 ( 4a; + 3 // = 2 i 12. } ^+ ^=^n 24. }6-3.= 12) l79;-2y= 9) (ox- 2/=13f l6a;-oy= 2J ' l3.r-2y = 16i SIMULTANEOUS EQUATIONS. 61 PROBLEMS. 72. 1. The sum of two numbers is 10, and their differ- ence is 2. What are the numbers ? 2. The sum of two numbers is 14, and the greater plus two times the less is 20. What are the numbers ? 3. A and B together have $300. Three times B's money added to five times A's gives $ 1100. How much money has each ? 4. The sum of the ages of a father and son is 50 years. The difference between the father's age and two times the son's is 20 years. What is the age of each ? 5. A boy has 25 marbles in two pockets. Twice the number in one pocket equals three times the number in the other. How many marbles has he in each pocket ? 6. A farmer paid $3400 for 100 acres of land. For part of it he paid $ 30 per acre, and for part of it $ 40 per acre. How many acres were bought at each price ? 7. A boy spent 35 cents for oranges and pears, buying in all 13 oranges and pears. He paid 3 cents apiece for the oranges and 2 cents apiece for the pears. How many of each kind did he buy ? 8. Two men start in business with $ 5000 capital. Twice the amount B furnishes taken from twice the amount A furnishes will leave the amount B furnishes. What cajjital does each furnish ? 9. There are two numbers such that five times the first minus three times the second equals 4, and two times the first plus the second equals 17. What are the numbers ? 10. A farmer sold 5 horses and 7 cows to one person for f 745. To another person at the same price per head 62 ELEMENTS OF ALGEBRA. he sold 3 horses and 10 cows for $650. What was the price per head of each ? 11. A man and wife working for 6 days received $ 15. Again, the man worked for 4 days and the wife 5 days, and they received $ 11. What were the daily wages of each ? 12. The sum of two numbers is 26. The iirst minus twice the second is 8. What are the numbers ? 13. A purse contained $ 30 in one and two dollar bills. If the whole number of bills was 18, how many bills were there of each kind ? 14. A merchant sold 2 yards of velvet and 4 yards of broadcloth for $ 16. Again he sold 3 yards of velvet and 5 yards of broadcloth for $ 21. What was the price of each per yard ? 15. If A gives B f 5 of his money, B will have twice as much money as A has left; but if B gives A f 5 of his money, A will have three times as much as B has left. How much money has each ? 16. A boy who desired to purchase some writing pads and some pencils found that 2 pads and 7 pencils would cost him 31 cents, and that 3 pads and 4 pencils would cost 27 cents. What was the price of each ? 17. The wages of 10 men and 8 boys per day were $28, and the wages of 7 men and 10 boys at the same rate were $ 24. What were the daily wages of each ? 18. A man received at one time $ 17 for sawing 8 cords of wood and splitting 10 cords, and at another time f 13.50 for sawing 5 cords of wood and splitting 12 cords at the same rates as on the former occasion. What did he receive per cord for the sawing and for the splitting? DIVISION. 73. 1. Since +5 multiplied by +4 is +20, if -f 20 is divided by + o, what is the sign of the quotient? 2. What, then, is the sign of the quotient when a posi- tive quantity is divided by a positive quantity ? 3. Since -\- o multiplied by -■ 4 is —20, if —20 is divided by -f 5, what is the sign of the quotient ? 4. What, then, is the sign of the quotient when a nega- tive quantity is divided by a positive quantity ? 5. Since —5 multiplied by -f-4 is —20, if —20 is divided by — 5, what is the sign of the quotient ? 6. What, then, is the sign of the quotient when a nega- tive quantity is divided by a negative quantity ? 7. Since —5 multiplied by —4 is -f 20, if -f- 20 is divided by — 5, what is the sign of the quotient ? 8. What, then, is the sign of the quotient when a posi- tive quantity is divided by a negative quantity ? 9. What is the sign of the quotient when the dividend and the divisor have like signs ? What, w^hen they have unlike signs ? 10. How many times is 6 x contained in 12 .r ? 8 y in 24y ? 11. How, then, is the coefficient of the quotient found? 63 64 ELEMENTS OF ALGEBRA. 12. Since x^ x x^ = .^•^ if x^ is divided by a--, what is the quotient ? What, when x^' is divided by x^ ? 13. Since a^ x a^ — «^ wliat is the quotient if a^ is divided by a^ ? What, if a^ is divided by a"' ? 14. How, then, is the exponent of a quantity in the quotient found ? ^ 74. Division is indicated in two ways : 1. By the sign -i-, read divided by. Thus, a -^ b shows that a is to be divided by b. 2. By writing the dividend above the divisor with a line between them. Thus, - shows that a is to be divided by b. b 75. Principles. 1. TJie sign of any term of the quotient is -\- when the dividend and divisor have like signSj 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 in the quotient is equql to its exponent in the dividend diminished by its exponent in the divisor. 76. The principle relating to the signs in division may be illustrated as follows : -f-ax + 6 = + a6 --ax + 6 = — a6 -{-ax — b — — ab — a x — b = -\-ab. >. Hence < -\-ab-^-\-b = -\-a — ab-7--\-b = — a — ab-. — 6 = + a ^-\-ab-{ — b = — a DIVISION. 65 77. To divide when the divisor is a monomial. 1. Divide Ux-yz^ by — 7 xyz. — Ixyz PROCESS. Explanation. Since the dividend and divisor . . 2 „ \\2jwe unlike signs, the sign of the quotient is — ±i^'^ (Prin. 1.) — 2 XZ^ Then 14 divided by - 7 is - 2 ; x2 divided hy X is X (Prin. 3) ; y divided by 2/ is 1, which need not appear in the quotient ; z^ divided by z is z^. Therefore the quo- tient is — 2 xz"^. 2. Divide 4 ax^y^ — 12 a^x^y^ — 20 a-xyh by 2 axy, PROCESS. 2axy 4 gg^V — 12 aVy^ — 20 a^xy^z 2 x^y^ — 6 axy — 10 ayh Explanation. When there are several terms in the dividend, each term must be divided separately. Rule. Divide each term of the dividend by the divisor as follows : Divide the coefficient of the dividend by the coefficient of the divisor. To this quotient annex each literal factor of that term of the dividend 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 ichen the terms of both dividend and divisor have like signs, and — when they have unlike signs. 3. 4. 5. 6. 7. 8. ivide 10 a 16 a; -14a6 IBxf - 20 a'b' — 18 m-n y 6a 2x lab -3xy - 4a6 6 m Find the quotients in the following : 9. 30 a^bx" -h 15 ax. 11. 21 ax'y ^ -7 ay. 10. - 24 xYz^ -5- 8 ojV. 12. - 9 abc^ -J- - 3 abc. milne's el. op alg. — 5. 66 ELEMENTS OF ALGEBRA. 13. 30nV^6n2. 18. - 55 abc'd ^ 11 abc, 14. -12arV--12a;i/. 19. 27 a^z' -i- - 9 x'z. 15. -20oc^y*^-10y\ 20. - 120 m^n ---- 15 mn, 16. - 100 x'yz^ 25 xyz. 21. 325 x^z' -i- 5 xyh\ 17. 80 icV - 20 xy^, 22. - 65 a:^^ -^ - 13 icV. Divide : 23. a^ici/ — 2 axy^ by a^/. 24. 9 xy + 15 a;?/V by 3 xyK 25 . 14 a^&'^c + 49 a'bc by 7 abc, 26. — a:^:^ — 3 a-^; 4-icV by — xz. 27. 4 c^d - 14 cd-' by 2 cd. 28. —5x^y + 10xy — 15xy-hy5xy, 29. 16 m-?i- — 12 mhi — 8 mn^ by 4 7/i?i. 30. 15 ax^ — 25 6x-^?/ + 35 cxy^ by — 5 a?. 31. 9 x^yz — 36 xyh^ -\- 45 ax^/a;^ by 9 xyz, 32. 42a^-14a;2 + 28a; + 35by 7. 33 . - 45 a2& V _ 60 a6c- + 30 a& V by - 15 abc. ' 34. 116 m' + 80 m*^ - 112 m^- 92 m by 4 m. 35. 3 x^yz'- - 15 c^yh^ + 6 aj^"^ + 18 xYz by - 3 a^^/^^- 36. 3 a.'S - 6 o:^ + 9 o;^ - 12 x^ by 3 .^^. 37. 30ary + 60aj2/_45a.y_|.75a;by 15 a;. 38. 24 abx - 16 aby + 32 a^bx^ - 8 a6 by 8 ab. 39. —x^y — x^yz + a;^'*^; — x^x^z^ -\- xyh^ by — xy. 40. 50 a^2/;33 _|. 35 ^2^2^ _ -1^5 ^^.2^^3 _ 20 ft^^/V by 5 xyz, 41 . 2 ?i2ajy - 3 7ia^2/' - ^ mnx^f + 3 ri^ory by nx^y\ 42. a(a; + 2/)^ - «^(^ + 2/)^ + a^b\x + 3/)^ by a(aj + yf. DivisioiNr. 67 78. To divide when the divisor is a polynomial. 1. Divide aF — 3a-b + 3 air — Irhy a — b. PROCESS. a^ — n^b a^-2ab + b^ - 2 a=^6 -h 3 aW - 2 a^6 + 2 a&' ab^ - W ab'- - W Explanation. The divisor is written at the right of the dividend, and the quotient below the divisor. The first term of the divisor is contained in the first term of the dividend oP' times. Therefore d^ is the first term of the quotient, o?- times tlie divisor, a — 6, is a^ — a%, and this subtracted from the partial dividend leaves a remainder of — 2 a^hy to which the next term of the dividend is annexed for a new dividend. The first term of the divisor is contained in the first term of the new dividend —2 ah times, consequently — 2 a?> is the second term of the quotient. —2 ah times the divisor, a — 6, is — 2 a% + 2 aft^, and this subtracted from the second partial dividend leaves a remainder of ah'^^ to which the next term of the dividend is annexed for a new dividend. The first term of the divisor is contained in the first term of the new dividend 6^ times, hence 6'^ is the third term of the quotient. &2 times a — 6, is ah"^ — h^, which subtracted from the third partial dividend leaves no remainder. Hence the quotient is a"^ — 2 ah -\- h^. Rule. Write the divisor at the right of the dividend, arranging the terms of each according to the ascending or descending powers of one of the literal quantities. Divide the first term of the dividend by the first term of the divisor, and write 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 one or more terms for a new dividend. Divide the new dividend as before, and continue to divide until there is no remainder, or until the first term of the divisor is not contained in the first term of the dividend. 68 ELEMENTS OF ALGEBRA. If there is a remainder after the last division, write it over the divisor in the form of a fraction, and annex it with its proper sign to the part of the quotient previously obtained. 15x2- sxy - 12 y2 i Sx-\-2y 16x^+ 10 xy \ 6x-6y - ISxy- 12 2/2 - ISxy - 12?/2 Divide : 2c-^d 4c-3cZ + 8c2-26c(iJ+ 15f72 + 8 8c^-20cd -6cd-\- l^d^ -6cd + 15(^2 + 8 4. «2 -f a6 — ac — be \ a + b a^ + ab \ a — c 8 2c-bd — ac -bc - ac -be 5. x^- -y' x-y x^- - xh/ x^ + xy + ?/2 x'y - _ yS x^y- - xy'^ xy^- 2,3 xy'2- _y!_ 6. x^-{-2xy + y^hjx-}-y. 7. m^ — n^ hj m ^ n. 8. oi:^-^3x^y + 3xy'^-\-y^hyx + y, 9. x'^Sx-lShjx-^e. 10. ci^ + Sx-i-Why x-{-3, 11. d'-10ab-24.b^hy a-{-2b. DIVISION. 69 12. ic2_4^^4 by X — 2. 13. a;^— 49 by aj2_j_7. 14. a*^-16by a'^-4. 15. 10iK'- + 14a?-12by 2aj + 4. 16. 2i^-a;'- + 3aj-9by 2x-3. 17. 16a'-24.ab-{-9b^hj 4:a-3b. 18. a-^ + b'' by a + &. 19. x'-Shj x-2. 20. a^ -\- a^y + ay^-{-y* hy a -{-y. 21. aj3_5a;2-a? + 14bya?--3a;-7. 22. ar^-86aj-140by a;-10. 23. 6x^ + 4.f-^6xy'^-16x^yhj3ccF-y^-2xy. 24. 18a;-l-17a^by 1-07. 25. a'-\-4.aV-{-16x'hy a'-{-2ax-{-4:X^. 26. 27aj3 + 8/by 3iB-f-22/. 27. 4aj^-.r^4-4a;by 2 + 2aj2 + 3a;. 28. x^ — ^x^ — exy — y^hyx^-^-Sx-^-y. 29. 36 + a;^-13a^by e + aj^ + Sa;. 30. 1 — a;— 3ic''^ — a^ by l4-2a; + a;2. PROBI.EMS. 79. 1. Express the sum of x and y divided by 2. 2. If a; oranges are worth 10 cents, how much is one orange worth ? 3. How many apples at 2 cents apiece, can be bought for m cents ? 4. Express the quotient of b divided by a. 5. If 5 bananas cost c cents, what will 2 bananas cost ? 25 70 ELEMENTS OF ALGEBRA. 6. How many days will it take a man to earn $ 15, if he receives a dollars a day ? 7. A farmer has a cows, ^ as many horses, and three times as many sheep as horses. How many animals has he altogether ? 8. Henry and James together had d marbles ; Henry had c times as many as James. What represents the num- ber that each had ? 9.' What is the cost per barrel, if b barrels of flour cost the sum of m and n dollars ? 10. A, B, and C start in business. A furnishes 2a dollars, which is c times as much as B furnishes; and C furnishes b times as much as B. How much do B and C furnish ? 11. A grocer mixed together equal quantities of three- kinds of colfee, worth a cents, b cents, and c cents per pound. What is the cost of one pound of the mixture ? GENERAL REVIEW EXERCISES. 80. 1. Add 6{x-\-7j)^3z-S, 2{x + y)-2z-{-^, Sz -S(x + y), 2. Add a'+b'-\-4:ab, 5a^-5b--5ab, -3a'-{-2b'-4.ab. S, Add ^x + ^y + z, x-j-iy-\-^z, ^x-^y-^^z. 4. Add 2x'^—5xy''-{-6, Sxy''+4:y''—5x'^, ex^'—lO + Txy^ 5. Add f (m — n)4-|(m -f n), f (m — n) — f (m + n), ^(m — n) — f(m + n), 6. From 9a; + 9a;--3a^ subtract 2 a; - 2 a;^ - 12 aj^. 7. From fa — ^6 + ^c subtract ^a + ift — fc. 8. From Bx'^y — Tx'^y^ -{-Wx'^z'' subtract 2x^y^ — 3x"'y -f-5a;"*2;^ GENERAL REVIEW EXERCISES. 71 9. From S(x+yy-2z^+S subtract 2z^-x^-\-4:{x-\-yy. 10. From Sa^ — 2a^x — 7 subtract 7 + 3a^ — a^x-\-2m. 11. Multiply m^ — m^ + m^ — m + 1 by m + 1. 12. Multiply a + b + c-j-e by a -{- b -{- c -{- e. 13. Multiply x^'{-2x^y-\-4.xY + Sxf-\-16y^ by x^2y. 14. Find the product of a; — 10, x -\- 4, and aj + 6. 15. Find the product of a — b, a + &, and a^ — b\ 16. Find the product of 2a— 5, 2a — 5, 2a — 5, and 2a + 5. 17. Divide 4:a' - 5a^b^ -\-b^ by 2a2-3a6 + 62. 18. Divide a^ + 1 by a + 1. 19. Divide ni^ — c^ -\- 2 cz — z^ by m + c — z. 20. Divide l — x — Sx'^—x^ by l + 2a;4-a^. 21. Divide 6x'-96 by 3a; -6. Find the value of x in the following : 22. 10a;-3a;4-4 = .T + 10-2aj-h8 + a?. 23. 6a; - 13 - 9 a; + a; = 4a;— 12 + 3 a; — 6aj- 13. 24. 5(a; + l) + 6(a; + 2)=6(a; + 7). 25. 3(a; + l) + 4(a; + 2) = 6(a; + 3). 26. ia;-4 + fa;= 16 + ia;-10. 27. a;(a;4-5)-6 = a;(a;-l) + 12. Find the values of x and y in the following : J 3a; -72/ = 13 I cSx-15y = 19^ ^^' (43; + 22/ = 40 1 ^^' |2a;+ y = 19\ ^ 3a; -32/ = - 6 I r4a;+ 22/ = 24| ^^' \7y^3x= 22) ^^* (3a;- y= 8) 72 ELEMENTS OF ALGEBRA. 32. A man being asked how much money he had, replied that $ 25 more than three times what he had would equal f 775. How much money had he ? 33. A man drove 155 miles in three days. On the second day he drove 15 miles farther than on the first, a.nd on the third day he drove 20 miles farther than on the first. How far did he drive each day ? 34. A and B start from two towns 231 miles apart and travel toward each other. A goes 15 miles per day, and B goes 18 miles per day. In how many days will they meet ? 35. A man worked 20 days during a certain month. A part of the time he received $ 1 per day, and part of the time $ 1^ per day. If he received $ 25 for his wages, how many days did he work at f 1, how many at $ 1^ per day ? 36. David and his father earned $ 100 during a certain month. David earned $ 10 more than ^ as much as his father. How much did each earn ? 37. A man bought an overcoat, a suit of clothes, and a pair of boots. The overcoat and the suit of clothes cost $ 55 ; the overcoat and the boots cost $ 30 ; and the suit of clothes and the boots cost f 35. What was the cost of each? 38. A and B each own half of a flock of 120 sheep. In settling the partnership, A takes 45 sheep, B takes 75 sheep and gives A $ 45 to make the division equal. What is the value of one sheep ? 39. A merchant has one quality of sugar worth 5 cents and another worth 8 cents a pound. He wishes to make a mixture of 100 pounds worth 7 cents a pound. How many pounds of each quality must he take ? FACTORING. 81. The quantities which, when multiplied together, pro- duce a quantity, are called Factors of the quantity. Thus, «, b, and (x -f p) are the factors of ah(x + y). a, 6, and (x + y) are called the prime factors of ab{x + 2/), because they have no common factors besides themselves and 1. 82. A factor of a quantity is an Exact Divisor of it. 83. The process of separating a quantity into its factors is called Factoring. 84. To factor a polynomial when all the terms have a com- mon factor. 1. What are the factors of 4 a^m — 6 am^x -f 10 aVx^ ? PROCESS. 2 am 4 a^m — 6 am^x + 10 a^mV 2a — 3 mx -{- 5 amx^ Explanation. By examining the terms of a polynomial, we find that 2 am is the highest factor common to all the terms. Dividing by 2 am, we obtain the other factor. Hence, the two factors are 2 am and 2a — S mx + 5 amx^. The same result may be secured by separating the terms of the polynomial into their prime factors and then selecting the common factors. Rule. Divide the polynomial by the highest factor or divisor common to all the terms. The divisor and quotient will be the fOjCtors sought. 73 74 ELEMENTS OF ALGEBRA. Find the factors of the following polynomials : 2. lSx^-27xy. 12. 20 m^ - 50 m^ + 30 m*. 3. 15a^2/2 + 20xY 13. xy' -\- 9 xY -]- 27 a^f, 4. 12 mV — 48 mn^. 14. 5 am^ + 10 mn + 15 mn^. 5. 5a6c-5ac2 4-15a6-c. 15. 4.5 xYz + 60 x'yz^ 6. 6aj2 + 4a;?/-8a^. 16. 39 a^y - 65 a^i/ + ^1 ^2A^- 7. 9 x'y^- 6 x^y-\- 12 a^yz. 17. 32 a'^6« + 96 a^6« - 8 aW. 8. 6 a^b + 21 a^b- IS a%^ 18. 25 a?i + 75 am^ - 15 anl 9. Sa'b-abc-abd. 19. 15 aar^ - 35 ory - 55 fta^^ 10. 6a*6' + 36a'62-42a%l 20. 45?>-V + 60^^ - 30aj%. 11. 15a26^+20aV-25a^6l 21. 72aY-36ay-108a2y\ 85. To factor a polynomial when some of the terms have a common factor. 1 . Factor S xy -\- 3 xz -{- ay -{- az. Solution. S xy -\- S xz -{- ay -\- az = 'Sx(y + 2)-\- a(y + z) = (Sx + a)(y + z) ^ Factor the following : 2. ax -\- by -\- bx -{- ay. 10. 2 a* — a*^ + 4 a — 2. 3. a6 + 26+3a + 6. 11. ax — ab — bx -}- b\ 4. 9 + Sx + 3y-i-xy. 12. x^ -\- x'^ -{- ax -\- a. 5. ax-{-ay—bx — by, 13. ax'^ -{- ay^ — bx^ ^ by\ 6. xy -\- 6 — bxy — 6b. 14. 1 + a^ — a" — a^ 7. y^ — y^-\-y — l. 15. 1 — a? — .r^ 4- a^. 8. x^ -\- x^y — xy^ — "(f. 16. x-y — x-z — xy'^ -{- xyz. 9. a;^^^ + 6a;^ — 6y — 6^. 17. a -{- ay -{- ay'^ + a^/^. FACTORING. 75 86. To separate a trinomial into two equal factors. (m-^n)(m -{- n)= m^ + 2 mn + nl (m — n)(m — n)^7n^ — 2 mn + ri^. 1. What is the product of (7n -{- n) (m -\- n) ? What, then, are the factors of m^ + 2 mn + n^? How are the terms of the factors found from the trinomial ? 2. What is the product of (m — n)(m^n)? What, then, are the factors of m^ — 2 mn + n^ ? How are the terms of the factors found from the trinomial ? 3. What term of the trinomial determines the sign which connects the terms of the binomial factors ? 87. One of the two equal factors of a quantity is called its square root. Rule. Find the square roots of the terms that are squares and connect these roots by the sign of the other term. The result will be one of the equal factors. The other term must always be twice the product of the square roots of the terms that are squares. Find the equal factors of the following trinomials : 1. x'^-{-2xy-{-y^ 9. 4:X^ -{-Sxy -{- 4:y\ 2. x^-i-Ax + A. 10. x^-lOx-^ 25. 3. 4:a" — 4:ab-{-b\ 11. l-h2z-\-z\ 4. 9m^ + 6mn-\-n\ 12. a^ -\- 4: a^b^ -\- 4:b\ 5. ar-4.xy-]-4:y\ 13. x"" - G xz -]- 9 z\ 6. 2/' + 22/ + l. 14. aj2^20aj + 100. 7. a-b^—Sab-\-16. 15. m^ + 8 m?i + 16 n^. 8. 4n2-20n + 25. 16. 9x^-18xy + 9y\ 76 ELEMENTS OF ALGEBRA. 17. 9 4-6a2-fa^ 23. a^ + 18 a^ -f- 81. 18. Aa'x'-Aa'b^x-hb', 24. 100 aj^ _ 20 a^ + 1. 19 . aW — 8 amn + 16 nl 25 . m V + 4 mn + 4. 20. 25a;* + 80aj2+64. 26. x^"" + 2 x'^y'' -{- y^\ 21. a* + 4a2iK2 + 4aj^ 27. 16 4-16m2 + 4m*. 22. 46V-12&cd4-9d2^ 28. 1 - 8 a?"* + 16 a;^^ 88. To resolve a binomial into two binomial factors. (^a+b)(a-b) = a^-b'. 1. What is the product of (a -{-b)(a — b)? What, then, are the factors of a^ —b^? How do these two factors differ ? 2. What is the product of {x + 3) (a; - 3)? What, then, are the factors of a?^ — 9 ? EuLE. Find the square root of each term of the binomial and make the sum of these square roots one factor and their dif- ference the other. A binomial cannot be factored by the above rule unless the second term is negative and the indices of the powers are even numbers. Sometimes the factors of such a quantity may themselves be re- solved into factors. Thus, x^-y^ = (x2 + 2/2)(X-^-2/2) Factor the following : 1. x'-y^ 6. aj2-25. 2. a;^ — 4. 7. 121 a.-^- 1002/'. 3. a- -9 62. 8. 16a- -96-. 4. m'-l. 9. a;^-144. 5. a'b'--c\ 10. l-z\ FACTORING. 77 11. x'^ — y^ 21. m^ — ^2^. 12. a;- -16. 22. 225 or - 100 2/'. 13. 9a;2-36. 23. 162/^-256. 14. Ax' -25, 24. aV-Odl 15. a;^-81. 25. a;- -121. 16. mV~ 640^2 23 c'd' - 1. 17. 4a2-166l 27. 25m2- 225711 18. x^-625. 28. x^-1, 19. 9x^ — 81. 29. a;2m__^2« 20. 25a-62_49c^d^ 30. a:«- - 169. 89. To factor a quadratic trinomial. (x-\-3){x + 2)=:x^ + 5x-\-6, (x-\-3){x-2) = x^-\-x-6, (x-3){x-{-2) = a^-x-6. (^x-3)(x-2) = x''-5x-{-6. 1. In the above examples, what terms of the product are alike ? 2. How may the first term of each factor be found from the product ? How may the second term of each factor be found from the product ? 3. How is the coefficient of the second term in the products found from the last terms of the factors ? How, then, may the sign of the second term of each factor be found from the second term of the product ? 90. A trinomial in the form, x^ ± ax ± b, in which b is the product of two quantities, and a their algebraic sum, is called a Quadratic Trinomial. 78 ELEMENTS OF ALGEBRA. 1. Eesolve a;^ — 9aj + 18 into two factors. PROCESS. 2 Q I 1 Q Explanation. The first term of each factor is aj- — y aj + 15 evidently x. Since 18 is the product of the other r 18 X 1 two quantities, and — 9 is their sum, we must j^g _ <{ 9x2 select from the different pairs of factors of 18 those I fi V S ^^^ whose sum is — 9. Therefore, — 6 and — 3 ^ are the other terms. Hence, x — 6 and x — 3 are — ^d^ — ^—o the factors required. (a7-6)(a;-3) 2. Factor a;^ -f- 4 a; - 21. Solution. 21 = 3 x 7, or 21 X 1. The factors whose algebraic sum is + 4 are + 7 and — 3. . '. (x + 7) and {x - 3) are the factors. Factor the following : 3. a^+Ca+S. 17. aj^ - 10 a? - 200. 4. x2 + 5a; — 14. 18. a2-6aaj — 55a;^. 5. i^2_0^_7 19. aj2-14a; + 40. 6. m--10m+9. 20. x^+xy-20y\ 7. y'-^^h-\-Q^, 21. 7n2 + 10m + 24. 8. n^-n-2. 22. a'-A.ah-2lh\ 9. n'' + n-2. 23. a^- 14a; + 48. 10. a.^-7a;-18. 24. x'^lx-lS. 11. aj2 + 4a;-12. 25. 0^ + 0-420. 12. a' -5a -24:. 26. a^ + lla& + 2862. . 13. x^-{-5x-S6. 27. a:2n_9^n_^20. 14. y''-Syz-^2z\ 28. aj^ _ 3 a^^ - 154. 15. m^-6mn-16n^ 29. m' + 2m-255. 16. a' + dax + Sx^ 30. a^^ + 12 x'^ + 35. FACTORING. 79 91. To factor the sum or the difference of two cubes. (a3 4- b') _j- (a + 5) = a^ - a6 + b^ 92. Principle. The sum of the cubes of two quantities is divisible by the sum of those quantities, and the difference of the cubes of two quantities is divisible by the difference of those quantities. 1 . Factor a^ — oc^. Solution. d^ — x^ is divisible hy a — x (Prin.) (^3 _ x^) ^(a-x)= a^ + ax + x2 .-. a^-x^={a- ic)(a2 + ax + x^) 2. Factor 7^ -|- 5^. Solution. r^ 4- s^ is divisible by ?• + s (Prin.) (r3 + s^) -t- (r + s) = r2 - rs + s2 .-. r^ + s^={r + s)(r2 - rs 4- s^) 3. Factor a^ — 6 V. Solution. a^ — hH^ is divisible by a — 6c (^3 _ ;^3c3) -^ (« _ he) = a2 + abc + 62c2 .-. a^ - 6'^c^ = (a - 6c) (a^ + abc + b'^c^) Observe carefully, in the examples solved above, the quantities in the quotients and the signs, and you will be able to write out the factors. Factor the following : 4. x^-{-y\ 10. s^-{-t\ 16. x'-^fz'. 5. a^+fe^. 11. c^-d^ 17. a^ + ftV. 6. m^ — n^. 12. 2/^ + 1. 18. mV + c^d^. 7. a^-ft^ 13. a^-1. 19. aV -Wy\ 8. m^ + n^ 14. a^ — cW. 20. aj^ + l. 9. Q^ — f, 15. m^H-sl 21. 2/^ — 1. 80 ELEMENTS OF ALGEBRA. \ \ EQUATIONS SOLVED BY FACTORING. ^ j 93. 1. Find the value of x in tlie equation x--\-l = o, \ PROCESS. Explanation. Transposing the = 2 -, ^ known quantity to the second mem- ; ^ +-•- = *> ^gj.^ the first member contains the ■ ar = 5 — 1 second power, only, of the unknown i X- = 4, quantity. Separating eacli member j ^ ^^ r> o _,. o o into two equal factors, the equation' becomes ic-a; = 2.2 or — 2-— 2.1 .'. x= ± Z Since each member is composed of j two equal factors, a factor in each: must be equal. Hence, x = + 2 or — 2 ; or x = i 2. The sign, ± , called the Ambiguous Sign, is a combination \ of the signs of addition and subtraction. \ Thus, a it 6 indicates that b may be added to or subtracted from a, \ 2. Find the value of x in the equation a;^ + 5 = 30 -f 11. j Solution. x^ + 5 = 30 4- H • • x2 = 30 + 11 - 5 x2 = 36 X'X = 6'Q or — 6.-6 .'. x= ±Q Fin d the value of X in the following equations : 3. a:2_4 = 5. 12. a;2 + 21 = 25. 4. a;2-9 = 16. 13. aj2 + l = 82. 6. X' -25 = 24.. 14. a;2 + 12 = 48. 6. aj2 _ 1 = 3. 15. x'-15 = 10. 7. 0^2 + 4 = 20. 16. aj2 _ 10 = 90. 8. a;2 + 5=41. 17. a^ + 15 _ 8 = 8. 9. a;2 + 2 = ll. 18. a;2+3-6=5-24-10 10. aj2-8 = 8. 19. x'-4.a' = 21a\ 11. «2 + 7 = ll. 20. a^ + 3c2=7cl FACTORmG. 81 94. 21. Findthe value of a; in the equation aj^ -1-4 a; -f 4=16. PROCESS. Explanation. Since 2 . ^ . ^ ^ each member of the equa- 00 -h 4 a; -h 4 = lb ^^^^ ^^^ 1^^ resolved into (a; + 2)(a;-f2) = 4-4or — 4. — 4 tico equal factors, one fac- .*. a; -f 2 = 4 or —4 ^^^ of ^^^^ first member J OP must be equal to one fac- and a; = 2 or — b , . .. a ^ tor of the second member. Hence, x + 2 = 4 or — 4, and X is found to have two values, 2 or — 6. 22. Find the value of x in the equation a^ -f 6 a; + 9 = 49. Solution. x^ + 6 ic + 9 = 49 (x + 3)(x + 3)=7.7or-7.-7 .-. cc + 3 = 7 or - 7 and X = 4 or — 10 Observe that the first member cf^nnot be separated into two equal factors except when the trinomial is a perfect square (Art. 87). Find the values of the unknown quantities in the follow- ing equations : 23. /- 10 ?/-h 25 = 16. 32. a?- -f 24 a; -f 144 = 225. 24. a;2-8a; + 16 = 81. 33. a;^ -f- 2 a; -f- 1 = 36. 25. a;--16a;-f 64 = 9. 34. y'^ -Uy -\- A9 = 9. 26. z'-{-12z + 36 = 64.. 35. z' + 30 z -\- 225 = 625, 27. a;2 -f 14 a; + 49 = 100. 36. ar' -^ 24a;-f 144 = 169. 28. a:^ _ 20 a; -f 100 = 25. 37. a^ -f 40 a; -f 400 = 900. 29. 2/' -18 2/ -f 81 = 16. 38. a;^ - 26a; -j- 169 = 196. 30. x'-{-22x-^121=:lU. 39. y'-2y-\-l = 25. 31. a;2-f 6 a; -f- 9 = 4. 40. a;--f 50a; -f- 625 =1600. MILNE ^S EL. OF ALG. — 6. 82 ELEMENTS OF ALGEBRA. 95. 41. Find the value of x in the equation a^-f 4 a; = —4. PROCESS. Explanation. By transposing — 4 to the first member, that member becomes a X -f- 4 X' = — 4 perfect square which may be resolved into aj2_|_4/p_|_4_Q lyQQ equal factors, {x + 2) (x + 2). Since ( X A- 2^ (x A- 2\ =^ Q the product of these factors is 0, one of on ^^® factors must be 0; and since both .*. i» + ^ = li factors are the same, each factor is equal and x= —2 to 0. Hence, the value of x is - 2. 42. Find the value of x in the equation a;- — 8 a; = — 16. Solution. ic2 _ g ^j ^ 16 _ (x - 4) (x - 4) = ... x-4 =0 and x = 4 Find the values of x in the following equations : 43. a;2 -f.l0a;4-25 = 0. 52. aj- + 20 a; 4- 100 = 0. 44. a;2 4-12a; + 36 = 0. 53. aj- + 24 a; = - 144. 45. a;2 + 6a; 4-0 = 0. 54. a;- -f- 36 a; = — 324. 46. a;2-18aj + 81 =0. 55. a;^ - 50 a? = - 625. 47. aj2-f 16 a; -f- 64 = 0. 56. a^ + 22 a; = - 121. 48. aj2 -I- 26 .T + 169 = 0. 57. a;^ - 100 a; = - 2500. 49. a;--14aj + 49 = 0. 58. a;- + 44 a; = - 484. 50. a;2 + 30a; + 225 = 0. 59. a;^ + 60 a; = - 900. 51. x^ -28 aj + 196 = 0. 60. x' + 80 a; = - 1600. 96. 61. Find the values of x in the equation x^-\-^^ - 14 = 0. PROCESS. Explanation. Factoring the first member of the equation, the factors a^ + 5 a: — 14 = are (x + 7) (:« - 2) . Since the prod- (^x ^l)(x — 2) = uct of the factors is equal to 0, one of • a;4-7 = 0oraj-— 2 = ^^® factors is equal to 0. Solving, * * ' ^ ^ the values of x are found to be — 7 and a; = — 7, or a; = 2 ^p 2. FACTORIlSrG. 83 62. Find the values of x in the equation cc^ -j- a; — 72 = 0. Solution. ic^ + oj - 72 = .-. cc + 9 = 0, orx-8=0 and X = — 9, OY X = 8 Find the vakies of x in the following equations : 63. x'^—2x — 15 = 0. 72. a;- + 15 oj 4-50 = 0. G4:. x^-{-6x-{-5 = 0. 73. x^-x-2=z0. 65. a;- + 10 a; +9 = 0. 74. o?'^- 18 a; + 77 = 0. 66. i»2-8a^ + 16 = 0. 75. a?- + 2 a.' - 120 = 0. 67. a;2+2ar-48 = 0. 76. ar'- 22 a? - 75 = 0. 68. a?- + 13 a; + 40 = 0. 77. x" - 6ax -^8a^ = 0. 69. a:^- 5 a? -24 = 0. 78. a.-^ + 3a; - 54 = 0. 70. a;- + 7 a; + 12 = 0. 79. a.-^ - 4 aa; — 96 a- = 0. 71. a;2+9a;-22 = 0. 80. a^ + 11 6x + 246- = 0. 97. 81. Find the value of x in the equation aj^ + Ga; + 7 = 23. PROCESS. a^^ 6a;+ 7 = 23 x^+6-x+ 7+2 = 23 + 2 aP-\- 6.^;+ 9 = 25 (a; + 3) (a; + 3) = 5- 5 or -5.-5 .-. aj + 3=5or— 5 and a; = 2 or — 8 Explanation. Equations like this may be solved in the same manner as the equations immediately preceding, by transposing all the quantities to the first member and then factoring ; or they may be solved by making the first member a perfect square by adding to or subtracting from both members some number. The first member of 84 ELEMENTS OF ALGEBRA. i the equation is a trinomial. A trinomial is a perfect square when \ it is composed of two terms that are perfect squares and when the ; other term is twice the product of the square roots of the terms that ^ are squares, x'^ -\- 6x are two terms of the trinomial which is to be | made a square, but the third term is to be found. Since the second term, 6 x, is twice the product of the square roots of the terms that are ] squares, and the square root of one of the terms is x, if 6 x is divided ; by 2 Xy the square root of the other term that is a square will be found. I Dividing, the quotient is 3, and S^, or 9, is the third term of the tri- 1 nomial. Since the given term is 7, 2 must be added to both mem- i bers to make the first member a perfect square, giving x"^-f 6x-f9=:25. : Factoring, (x + 3) (x + 3) = 5 • 5 or — 5 • - 5. Whence, x = 2 or ; -8. i 82. Find the value of x in the equation aj^~-12a;-f-33=46. ] Solution. x2 - 12 x + 33 = 46 • ; x2 - 12 X + 33 + 3 = 46 + 3 \ x2 - 12 X -f 36 = 49 I (x-6)(x- 6)=7.7or -7. -7 . ; .-. X- 6= 7 or -7 I and X = 13 or — 1 \ Solve the following equations : 83. a;2 + 10a; + 20 = lL 92. a:- - 24 a; -f 122 = 3. ^ 84. i»2^8x + 12 = 32. 93. a;^ - 30 a; -j- 220 = 76. ] 85. a;2_18a;+80 = 15. 94. aj^ ^ 40 a; + 200 = 425. \ 86. aj2 + 4a;-f 2 = 7. 95. a;^ _8a; + 15 = 99. ] 87. a^-20a; + 85 = 10. 96. aj^ + 12 aj + 27 = 40. j 88. a^ + 14a;-f 45 = 60. 97. a;^- 38x + 360 = 8. ] 89. a;2 + 22 a; + 100 = 60. 98. y^ + 2y-l = 2. ] 90. aj2 + 4a;H-l = 33. 99. a;^ - 6aj- 3 = 13. 91. 2/^ + 16 2/ + 54 = 90. 100. a;- + 8a;- 2 = 18. COMMON DIVISORS OR FACTORS. 98. 1. Name a common divisor or factor of 5 ic and 10 xy. Of 4a6 and 16 a6. 2. Name all the common divisors or factors of 24c x^y^ and 12ic^i/. Which of these is the highest common divisor or factor ? Name all the common divisors or factors of 15 a^y^ and 25 ab^. Which is the highest common divisor or factor ? 3. What prime factors or divisors are common to 24:X^y'^ and 12 :x?y ? To 15 a'h'' and 25 a6* ? 4. How may the highest common divisor or factor be obtained from the prime factors of 24 x^y'^ and 12 x^y ? How from the prime factors of 15 a^6^ and 25 aW ? 99. An exact divisor or factor of two or more quantities is called a Common Divisor or Factor of both of these quan- tities. Thus, 3 a is a common divisor or factor of 9 a?" and 12 a. 100. The divisor or factor of the highest degree that is an exact divisor of two or more quantities is called the Highest Common Divisor or Factor. Thus, 5 a^x is the highest common divisor of 20 a^x and 15 a^x^. 101. Principle. The highest common divisor or factor of two or more quantities is equal to the product of all their common prime factors, 85 86 ELEMENTS OF ALGEBRA. 102. To find the highest common divisor or factor of quan- tities that can be factored readily. 1. What is the highest common divisor of 4 a ^6 and 12 a'b'c? PROCESS. 4cr6 =2. 2. a. a. 6 12a'b'c = 3'2'2'a^a'b'b'C H. C. D. = 2 . 2 . a . a . 6 = 4 a-6 Explanation. Since the highest 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, a, 6 ; and their product is 4 a'^b. Therefore the highest common divisor is 4 a^b. 2. What is the highest common divisor of m{a^ — b^) andm(a2-2a6 + &')? Solution. m{a?' — 6^) = m(a + b) (a — b) m{aP- -2ab-\- b'^) = m(a -b){a- b) H. C. D. = m X (a - 5) = m (a - 6) Find the highest common divisor or factor of the fol- lowing : 3. lA QtFyz^ 2ind 21 xyh. 4. 18 xyz^ and 45 xyz, 6. 5a^6-, 30 ab% and 15 a^bc, 6. 22mVz,Um%h\'dudl21mhi'z\ 7. 20 abx^, 40 abV, and 120 a^'bx^ 8. 9 a^bmV, 27 b^m% and 81 bmV. 9. a^~l anda;2_2a: + l. 10. oc^ -\- 2 xy -\- y^ and x^— y\ 11. m^ 4- ^^ and m^ — n^ COMMON DIVISORS OR FACTORS. 87 12. a — h, a? — h^, and a^ — 2ah-\- h\ 13. x^ — 1 Siiid x^ — X'-2. 14. x^-2xdind2xi/-'4:i/, 15. yz — z Sind y^ — 1. 16. 1 — a^ and 1 + a^. 17. 3t^-\-2x — 3^nda^-\-5x + 6. 18. a?2_2a;-15anda;2 + 2a?-3. 19. a?2_3^._4an^^2_^._12. 20. x^ -l,x'- 1, and i»^ - 2 aj2 -f 1. 21. 0^-707 + 6 and a^ + 3 ic- 4. 22. aoj + bx, a-m — ?>-m, and a^ -\-2ab-\- 61 23. a;'^ 4- 2 ic — 35 and x^ -\-x — A2. 24. x^ — 4 ici/ + ^2/^ and 05^ — 4 y"^. 25. aj2-8a? + 15, a.-2-4a;-5, and a;^ - 3 a; - 10. 26. ^ + x-20,x'-x- 12, and a;2 - 2 aj - 8. 27. aj2 4-2a;i/ — 82/^ and aj2 — 5a?2/4-6i/l 28. aj2 + 4a:2/ — 21i/2 and aj^4-6a^2/ — '^2/^- 29. o^^^xy — 1 y'^ mdx^ — 2xy-\-y'^. 30. a;2 — 4.x — 5 and a;2 + 2aj — 35. 31. 0.^ - 42/2, ar^ + 4.xy — 12y% and a:^ _ 4.xy + 4 2/^. 32. ax — 3a, a^^ _ j^j + 12, and ax^ + 5aa; — 24a. 33. am + 2ma;, a^ + 4aa; + 4a^, and a^ — 2ax — Sx\ 34. b^ - c^, 52 _|_ 55c 4- 4c2, and 6^ _ 9&c - 10c\ 35. a;2-2a; + l, x2-8x + T, anda;2-4a;+3. 36. a^ - 9, a^ - 9a - 36, and a^ - 7a - 30. 37. 4a — 82/, a2 — 5a2/ + 62/^ and am — 2m2/. 38. 2a; + 62/, 2{x^ + 6xy -{-9y^), and 2 aa; + 6 ay. COMMON MULTIPLES. "^ 103. 1. What quantities will exactly contain 3, 5, x, and y ? 2. What quantities will exactly contain Sx-y^ and 9x^y? 3. By what quantity must 9x^y be multiplied so that it may contain 3 x^y^ ? 4. By what quantities, then, must the highest quantity be multiplied to obtain a quantity which will contain each of several quantities ? 5. By what quantities must the highest quantity be multiplied to obtain the lowest quantity which will contain each of several quantities ? 104. A quantity that will exactly contain a quantity is cabled a Multiple of the quantity. Thus, a^x is a multiple of a, a'^, a^, x, ax, and a^x. 105. The lowest quantity that will exactly contain each of two or more quantities is called the Lowest Common Multiple of the quantities. Thus, 4 x'^y is the lowest common multiple of 4 x, ?/, and x^. 106. Principle. The lowest common multiple of tivo or more quantities is equal to the product of the highest quantity multiplied by all the factoids of the other quantities not con- tained in the highest quantity. 88 COMMON MULTIPLES. 89 107. 1. What is the lowest common multiple of dx^yz* and 4:axyz^? PROCESS. 5 oi^yz* = 5 ' oc^ ' y ' z* 4 axyz^ = 4: • a - x ' y - z^ L. C. M. = 5 . 4 . a . a^ . 2/ • ^' = 20aa^yz' Explanation. Since the lowest common multiple is equal to the product of the highest quantity, multiplied by the factors of the other quantities 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 factors. Thus, the factors of the lowest common multiple are seen to be 5, 4, a, x^y y, z^j and their product, 20«x"2?/^*, is their lowest common multiple. 2. What is the lowest corjimon multiple of oj- — 3 x — 40 and 0^24. 3a; _ 10? Solution. x'^ - 3 ic - 40 = (x - 8) (a: + 5) ic2 + 3x- 10=(x-2)(x + 5) L.C.M. zz:(x-8)(x+ 6)(x-2) = a:3-5x2-34x+80 Find the lowest common multiple of the following : 3. ^a?h''c and l^o?hc\ 4. lOaWx and 15a%3(^. 5. Ua^bY^, 7bVy, and 35abcx. 6. 27 am, 33 a^my^ and 81 a V?/^. 7. 15 oiPyh and 21 x^y^z\ 8. iK- — 4 and a^ — 4 x + 4. 9. x^ — y^ and x^ -{-2xy -^ y^, 10. ar{x — y) and a{oi^ — y^), 11. d'-b', a2-h6^ and a'-b^ 90 ELEMENTS OF ALGEBRA. 12. a;2-9aj-22 and aP --- 13x^22 . 13. a;2-h5aj + 6 and aj2^6a;+8. 14. x^ — 16, a^ + 4a;4-4, and a.-^ — 4. 15. C-- 5c4- 4 and c2-8c + 16. 16. m(a + 6), m^(a — &), and mx{a? — h'^), 17. a;(a'^ — ft*^), a;(a — 6), and x^ {a? -\- ah -\- y^) , 18. 2a + l, 4cr— 1, and 8a'^ + L 19. a^ — a- 20 and a^ 4- a— 12. 20. x^y — xy'^y X' — y'^, and x^ -f ic^. 21. x^ — x, x^ — 1, xy — y, and ab(x^ — l). 22. ic'* — a^, a^ — a^, and x — a. 23. a^ - 5ab + W and a^ - 2 ab + 61 24. X- — i»-30 and a;2+llaj + 30. 25. a;2 — 1 and .r^ + 1. 26. ic2 — 1 and x2 + 4a; -f- 3. 27. 6(a4-6), &2, am(a — 6), and a^ -{- 2 ab -\- b\ 28. 15(a-6-a62), 21(a-^-a62), and 35(a&2_^ 6'^). 29. x^-i-5x-\-6, x2-a;-12, and aj2-2a;-8. 30. 0^-80^ + 15, a.'2-4iK-5, and ^2-20^-3. 31. x^-\-4:xy — 21y^, x^ — 2xy — 3y^, and x^ — 6xy — 7 y^. 32. aj2-3a;-28, ar^ + x-12, and a^-lOx-^21, 33. a^ + 2a;-35, oj^ + a; - 42, and a^ — 11 a; + 30. 34. a^ — y\ x^ -]- xy -{- y^, and o^ — y^. 35. a;2_3aj_4^ a.^-aj — 12, and a;2 + 4a; + 3. 36. a;^-l, a;- -f- 1, a;^ + 1, and x^-^l. FKACTIONS. 108. 1. When an3^thing is divided into two equal parts, what are the parts called? What, when it is divided into three equal parts ? Into four equal parts ? Into m equal parts ? Into n equal parts ? 2. What does - represent ? -? %? -? ~? -? _„ ? f? 3. Express ^ of a ; ^ of 6 ; ^ of c ; | of a ; f of d. 109. One or more of the equal parts of anything is called a Fraction. 110. A quantity no part of which is in the form of a fraction is called an Entire Quantity. Thus, a, 3 X, 4 a + 3 6, etc., are entire quantities. 111. A quantity composed of an entire quantity and a fraction is called a Mixed Quantity. Thus, 3 X + -^7 3 X - 3 w H- ^ "^ are mixed quantities. 5 c 112. The sign written before the dividing line is called the Sign of the Fraction. This sign belongs to the fraction as a whole and not to either the numerator or the denominator. Thus, in — ^ "*" the sign of the fraction is — , but the signs of the quantities a, ?>, and 2 are + . The sign before the dividing line simply shows whether the fraction is to be added or subtracted. 91 92 ELEMENTS OF ALGEBRA. REDUCTION OP FRACTIONS. ] 113. To reduce fractions to higher or lower terms. \ 1. How many fourths are there in 1 half? In 3 halves? ] In 5 halves ? In a halves ? In b halves ? In n halves ? \ 2. How many sixths are there in 1 third ? In 2 thirds ? i In 8 thirds ? In x thirds ? In y thirds ? In a + 6 thirds ? i 3. Since - is equal to — ; and - is equal to ^; what ■ may be done to the terms of a fraction without changing ; the value of the fraction ? , 4. How many thirds are there in f ? How many halves are \ there in | ? How tnany fifths are there in f^ ? How many ! twelfths are there iu — ? In — ? In — ? ■ '■ 24 36 48 5. Change — — , -— , -— , — - to fractions whose denom- ! 4 2 8 (X 16 j inator is 16 a. \ 6. Reduce to equivalent fractions whose numerator is 3 aj, J^ _3 ^ ^xy xyz ax by al) Axy^^ 4:ayz I 7. What else, besides multiplying them by the same 1 quantity, may be done to the terms of a fractiou, without j changing the value of the fraction ? j 114. A fraction is expressed in its Lowest Terms when its numerator and denominator have no common divisor. 115. Principle. Multiplying or dividing both terms of a fraction by the same quantity does riot change the value of the fraction. FRACTIONS. 93 116. To express a fraction in higher terms. 1 . Change ^ — to a fraction whose denominator is 15 6Vd. obc PROCESS. Explanation. Since the fraction is 2a to be changed to an equivalent fraction expressed in higher terms, both terms of 56c the fraction must be multiplied by the 15b^c^d -^ 5bc = 3 bed same quantity, so that the value of the 2a X 3 6cd 6 abed ?^"^T '''''^ T ^^ ^^'^^S^^(}'^''^-)- y- — — = ^ In order to produce the required denom- OOC X 6 oca lob'Ca inator, the given denominator must be multiplied by 3 bed ; consequently the numerator must also be multiplied by 3 bed, 2. Change — - to a fraction whose denominator is 30. 5 5 TYhYh 3. Change to a fraction whose denominator is 24. 4. Change to a fraction whose denominator is 28. 5. Change ^ ^,^ to a fraction whose denominator is 45 6Vd2 n 15b'c 6. Change — — to a fraction wliose denominator is 33 a;. ^^ 7. Change — to a fraction whose denominator is 48. ^^ 8. Change ^ to a fraction whose numerator is 4a;-v. 3a4-2 9. Change to a fraction whose numerator is 12abc. ^''-'■^y 10. Change to a fraction whose denominator is 12xy. ^ 94 ELEMENTS OF ALGEBRA. 1 7 . . i 11. Change to a fraction whose denominator is i w- — }r, ' 12. Change — -!— - to a fraction whose denominator is ' 9 o m — 71 \ nr — 71"^. • 13. Change ^^-- to a fraction whose denominator is 75. J 1^ i 1 . * ' 14. Change to a fraction whose numerator is m + n. i m -f 71 i i 2a . i 15. Change to a fraction whose numerator is j 2ax-4.a. ^ + 2 t 16. Change ~-^ to a fraction whose numerator is a;- — - 1/^. > j 117. To express a fraction in its lowest terms. ' 15 x^v^ ' 1. Eeduce -- — ^ to its lowest terms. ' 20i»2/2 1 PROCESS. Explanation. Since the fraction is to be changed i . p. 2 2 Q to an equivalent fraction expressed in its lowest l y = zJ^ terms, the terms of the fraction may be divided by \ 20 X'lf 4 any quantity that will exactly divide each of them ^ (Prin.). Dividing by the quantity, ^xy'^^ the ex- ] pression is reduced to its lowest terms, since the terms have now no : common factor. Or, the terms may be divided by their highest com- ' mon divisor. \ Eeduce the following to their lowest terms : ■ 3a6c 13m^7i 17 m^nx^ ^ 9a62* ' 39 mV' * Q^S mnV* • g A.xyh g 112a6^ ^ 125aW \ 12 ^yz^' ' 252abxy' ' 625 aVz*' \ ^ 10 abx ^ 35a^y'z\ ^^ 2x [ ' Sba'bcx^' ' 105 xYz'' ' 4:X^-6ax j FRACTIONS. 95 Sab -^ ax — a ar^ — 6 a; — 40 a- — ab^ ax^ —a ' oif — S x — 70 12. ii±A. 18. ^' + '^-^. 24. "^-y" . a'-b' a'-9a+U a-'- - f m — n - ?ft-+8m + 15 a^ — 1 ?)i- — rr 711' —Zm — lo .r" + 1 14. «-^ ■ 20. '^•-2"'-^4 . 26 J(^±lL. ft2-2a6+&-^ ar'-l2a;+36 27{a^-y') 15 --g'-y- 21 ^'+^^-20 27 ^'^-^y x'+2xy+y' ' scr + ix-S ' 8ar'-27/ 16. „ ^ + ^ ■ 22. ^?^^. 28. '^'-l aj^ + 2 o; +1 m'^ — 7i^ 2xy -\-2y 118. To reduce an entire or mixed quantity to a fraction. 1 . How many thirds are there in 2 ? In 5 ? In 7 ? In a ? 2. How many fifths are there in 4 ? In 7 ? In m ? 3. How many fourths are there in 3^ ? In 5f ? In a; + - ? 1 . Eeduce a; -f - to a fractional form. PROCESS. Explanation. Since 1 is equal to __ ^J( ^, aj is equal to ^ ; consequently y y ^ I a^xy ^ a^ xy^a '^ y~ y '^ y y y y y y EuLE. Multiply the entire part by the denominator of the fraction ; to this product odd the numerator when the sign of the fraction is plus, and subtract it when it is minus, and write the result over the denominator. 96 ELEMENTS OF ALGEBRA. If the sign of the fraction is - , the signs of all the terms in the numerator must be changed when it is subtracted. The student must note very carefully that the sign of the fraction affects the whole numerator and not simply the first term. Eeduce the following to fractional forms : 2. 6x-{-^-^^ 14. X ^— 2 7 3. 4cf-^. 15. 3a; + 5 6. a.' + 7 9 . 8a-i^±^. , a — h 11. x-\- Qf a — X ax 2 ac - c" a X 4. 80.+^. !«• '*- 4 20,-1 18- 3+ ^ 19. 3a + x'-l ab — a 7. 2a;-^±i. '' ^ 20. a + a; + ^^±^. 8. 7x + ^-^-^" . 6 ^^ , , 2ac — c^ 21. a + cH a — c 5 22. 20.-3---^. a; + 2 3 23. m — 2rj + 10. 3mH 14 m-\-2n c 24. m + ^ — a; + 25. a — aj^- 12 2x y^^^ o. , a2-|-aj2-5 1^. iiX • OK /7 O^ J ! . a-^-x 13. 56 + ^^-^^ 26. a(m + n)+ '^^^ m — n FRACTTOXS. 97 119. To reduce a fraction to an entire or mixed quantity. 1. How many units are there in | ? In | ? In -2/ ? 2. How many units are there in ^ ? In ? 4 1 ^ ^ 2 1. Reduce ^^"^ to a mixed quantity. X PROCESS. Explanation. Since a fraction may be re- garded as an expression of unexecuted division, ^^ ~T ^ = a -I- - ^y performing the division indicated, the fraction X X i^ changed into the form of a mixed quantity. Reduce the following to entire or mixed quantities : 2. 23^. 10. t±^. 9 x-1 26 a& ^^ 2a^-4&"^ ' 11 * ' a 4- & 4:5x^y^z 5 gy + ax + x 15 xy a ^ 36ac + 4c ^3 2ab + b\ 9 a + 6 6. ^' + ^ 14. ^^^' + ^ a * oj — 4 ^^ 12x^-5y ^g^ ar^ + 2/^ 6a; x — y 2a^x — ao[p -^ 2 a6 + a6^ — a' 8. ^^^^^^^^^ — ^=^- 16. a a6 a—a? a?^ — x — 1 milne's el. of alg. — 7. 98 ELEMENTS OF ALGEBRA. 18. 2a2-2&2 a — b 19. x'-2x-{-l x^l 20. 2x' + 5 x-3 21. a'-\-b' a — b 22. 23. 24. 25. X--2 2x^-6x-{-4: 2x -3 5x — 1 x^ -- a^ — X -\- 1 120. To reduce dissimilar to similar fractions. 1. Into what fractions having the same fractional unit may ^, ^, | be changed ? 2. Into what fractions having the same fractional unit may — and be changed ? ^ 3m 2m ^ 3. Express - — , - — , and - — in equivalent fractions 3m 2m 6m having their lowest common denominator. 121. Fractions which have the same fractional unit are called Similar Fractions. 122. Fractions which have not the same fractional unit are called Dissimilar Fractions. Similar fractions have, therefore, a common denominator. 123. When similar fractions are expressed in their lowest terms, they have their Lowest Common Denominator. 124. Principles. 1. A common denominator of two or more fractions is a common multiple of their denominators. FRACTIONS. 99 2. The loivest common denominator of two or more frac- tions is the loivest common multiple of their denominators. 1. Keduce — and — - to similar fractions having their 2x (TX lowest common denominator. PROCESS. Explanation. Since the lowest common o 9 denominator of several fractions is the - = — = lowest common multiple of their denomi- LX IX X a^ ^a-x nators (Prin. 2), the lowest common multi- 3 3x2 6 pie of the denominators 2 x and a'-x must be — ^ = — ^ ^ = o~2~ found, which is 2 a^x. The fractions are ax ax X ^ - a-x ^^^^ reduced to fractions having the de- nominator 2 a^x, by multiplying the numer- ator and denominator of each fraction by the quotient of 2 a'^x divided by its denominator. 2 «% -^ 2x = a^^ the multiplier of the terms of the first fraction. 2 a^x -=- a'x = 2, the multiplier of the terms of the second fraction. EuLE. Find the lowest common midtiple of the denomina- tors of the fractions for the loivest common denominator. Divide this denominator by the denominator of each frac- tion^ and multiply the terms of the fraction by the quotient. Eeduce to similar fractions, having their lowest common denominator : 2. - and — be o m ;y ab 3. — and — n X , ^xy 1 2 a 4. — ^ and c '6 by 5. ^ and ^. 4 Zy 6. 2 b . 3cd — and xh xz' 7. ax be ab y ax ay 8. be ac ab a, — ? — ? — ' a^ ^z yz 9. 2 2 2 xy ax ayz 100 ELEMENTS OF ALGEBRA. 10 -^, ^1., ^. 14 ^-±1. ^-y ^' + y' a?h he 5V • 4 2c ' 2a 11. ? — ;r> r— 15. ? > — ^ ojy ar xy-z x -{- y x — y x' — y^ 12. H H % i^. 16. ^-. ^- 13 3 26 39 a + <^ a-5 ^^a-{-ba — bab -^ 3 5 9 16. 1 > — • 17. ? > — • 3aj 2?/ 6 a;-f2a;-2a; a + & 1 1 18. 19. 20. 21. aS _. &•• a - 6 a- + a6 + ^>' 1 3 X x-i-l' 4 a; + 4 aj^ — 1* 2 1 3 a2_62' ^_^' a2 + 6=^* a; + 2 a;-2 of' — 3 a; + 2 ar' H- a? — 2 o» n. 22. 23. 24. 25. aj2_^2a;-3 a)'^-2a?--15 a; + l a; — 1 2 ^ a;(a; — 2) 4a; — 8 4aj 5 3a; 4 -13a; 14.2a;' l-2a;' l-4a.^* 1 a 3 a a + b a' - 52' a^ _ ft*' 26. ^-y . ^ + y. a;2 — a;?/ + y^ ar^ 4- r ^2/ + ^^ FRACTIONS. 101 CLEARING EQUATIONS OP FRACTIONS. 125. 1. Five is one half of what number? 2. Eight is one third of what number? One fifth of what number? 3. li ^x equals 5, what is the value of a;? 4. If ^ a; = 6, what is the value of x ? 5. What effect has it upon the equality of the mem- bers of an equation to multiply both by the same quantity ? 6. If the members of the equation lx = 6 are multiplied by 5, what is the resulting equation ? 7. Multiply the following equations by such quantities as will change them into equations without fractions : lx = S, or ^ = 8; ix=7, or | = 7; X t X f\ X , X ^ X , X f-^ - = 4: — = "; — — = b; — — = o; 8 '10 '2 4: '36 ' ? + *:5=10; ^ + ^ = 20; - + - = 8. 4^8 '510 '3 5 8. How may an equation containing fractions be changed into an equation without fractions? 126. Changing an equation containing fractions into another equation without fractions is called Clearing an Equation of Fractions. 127. Principle. An equation may be cleared of fractions by multiplying each member by some multiple of the denomina- tors of the fractions. 102 ELEMENTS OF ALGEBRA. 1. Find the value of x in the equation a; + - = 8- PROCESS. a^ + ^ = 8 • Clearing of fractions, 3x-^ x = 24: Uniting terms, 4 a? = 24 Therefore, x = 6 Explanation. Since the equation contains a fraction, it may be cleared of fractions by multiplying each member by the denominator of the fraction (Prin.). The denominator is 3 ; therefore each mem- ber is multiplied by 3, giving as a resulting equation 3 ic + ic = 24. Uniting similar terms, 4 cc = 24 ; therefore, x = 6. 2. Find the value of x in the equation , X , X , X tju PROCESS. 4 2 5 2 Clearing of fractions, 20x'\-5x-\-10x-{-4:X = 390 Therefore, 39 x = 390 and a; = 10 Explanation. Since the equation may be cleared of fractions by multiplying by some multiple of the denominators (Prin.), this equa- tion may be cleared of fractions by multiplying both members by 4, 2, 5, and 2 successively, or by their product, or by any multiple of 4, 2, 5, and 2. Since the multiplier will be the smallest v^hen we multiply by the lowest common multiple of the denominators, for convenience we multiply both members by 20, the lowest common multiple of 4, 2, 6, and 2. Uniting terms and dividing by the coefficient of x, the result is cc = 10, Rule. Multiply both members of the equation by the least. or lowest, common multiple of the denominators. FRACTIONS. 103 1. An equation may also be cleared of fractions by multiplying each member by the product of all the denominators. 2. Multiplying a fraction by its denominator removes the de- nominator. 3. 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 re- moved. 3. Find the value of x in the equation x — ^_x — l_ X — 5 Solution. — ^^-^ Clearing of fractions, 9x — 27 — 4ic4-4 = 6x — 30 Transposing, 9x — 4aj— 6x = 27 — 4 — 30 Uniting terms, —x = —l Dividing by — 1, x = l Note 3 under the rule is a very important one. Observe its appli- cation in the above solution. Find the value of x and verify the result in the following : 4. a; + ? = 10. 10. 0.^4-^ + 1 = 11. 4 Z 6 5. x + f = 20. . 11. f + f + ^=9. 3 3 4 6 6. 2a;+f=9. 12. 2a; + ^+| = 37. 4 L . 1 X X , X 69 8. 4. + |=51. 14. - + | + fo+5=^^- e. | + 3. = io. 15. 1 + 1+1 + ^=17. 104 ELEMENTS OF ALGEBRA. ^9^12 12 10 5 ^ 2 4 4 ; 17. 2a; + ?^4-|| = 27. 25. ?+?L±^ = x-3. i 4 15 3 8 I 18. 3a;-^ + :^=70. 26. ^-'l^ = x-d. i o iZ 3 11 * 19. xi---- = 3S. 27. — -2a^ = ^^ + ^-21 ^ 7 5 7 5^ 20. 0. + ^ + ?^-? = ^ 28. ^Ii?+i = i5_2a.. '^ 5 6 2 2 3 3 ; 21. ll-^ + ^ + ^ = s. 29. 2-^±^^2a;-^^±^. i 3 5^4^12 4 3 i 22. 2x + ^-^-^ + ^ = l^. 30. gi+3^2^+3 7a;-5, 1 8 3 6 2 2 10 ^ o i 23. a; + ^-^+^=59. 31. 7a;+16^x + 8^^ ' ! 2 4 11 21 21 3 i „„ a; +3 x — 8 _x — o ^ 4 5 ~ 2 ■ i „„ 6a;-14 , 2a;-l „^ . i 33. — g i — = 2a;-5. 23a! +13 jg-l^g 4 2" ; „- 4a; + 2 3a;-5 , ., ] 36. ^±i + ^±3 = ?i±i+16. • 3 4 5 , 37. ^ — 7 I x — 7 _x-{-l 'I 10 ' 5 6 38 ^ + 3 x — l_x-\-42 a- 4- 5 FRACTIONS. 10, 39. 2 3 6 42. aj , ic-2 5a:- 1 5-^3=" 6 40. X 2.T-4 ^, .r + T 3 7 -"^ 2 • 43. a; a: 2a; _ a; -52 8 5 5 4 41. l-2.r 4-oa: 13 3 ~ 4 42* 44. a;-.4 x-1 a;-26 4 3 5 45 ^-1 4.-^'-- X — 2 2 ^^- 2 + 4 3 3 PROBIfEMS. 128. 1. A certain number diminisliecl by 1- and also by ^ of itself leaves a remainder of 19. What is the number? Solution. Let x = the number. Then - and - = the parts of the number. 5 6 ^ And x----=19 6 6 Clearing of fractions, 30x — Ox — 5x = 570 Uniting, 19x = 570 x = 30 2. What number is that the sum of whose third and fourth parts equals its sixth part plus 5 ? 3. A man left i of his property to his wife, i to his children, and the remainder, which was f 1200 to a public library. What was the value of his property ? 4. Out of a cask of wine I part leaked away ; afterward 10 gallons were drawn out. The cask was then | full. How many gallons did it hold ? 5. A man leased some property for 40 years. ^ of the time the lease has run is equal to ^ of the time it has to run. How many years has it run ? 6. The sum of two numbers is 35, and ^ of the less is equal to \ of the greater. What are the numbers ? 106 ELEMENTS OF ALGEBRA. \ 7. A man paid f as much for a wagon as for a horse, : and the price of the horse pUis |- of the price of the wagon \ was 100 dollars. What was the price of each ? i 8. The sum of two numbers is 76, and | of the less is | equal to \ of the greater. What are the numbers ? 9. There is a certain number the sum of whose fifth ' and seventh parts exceeds the difference of its seventh and \ fourth parts by 99 ? What is the number ? ; 10. Divide the number 50 into two such parts that \ of ^ one plus f of the other shall equal 35. \ 11. A son's age is f of his father's ; but 15 years ago the '■ son's age was \ of the father's. What are the ages of each ? \ 12. A man paid f 3 a head for some sheep. After 20 of \ them had died, he sold \ of the rjemainder at cost for f 60. \ How many sheep did he buy ? ' \ 13. A man wished to distribute some money among a ; certain number of children. He found that, if he gave ; to each of them 8 cents, he would have 10 cents left, but, if a he gave to each 10 cents, he lacked 10 cents of having • enough. How many children were there, and how much \ money had the man ? \ 14. A man and his oldest son can earn. $30 a week ; the | man and his youngest son can earn |^ as much ; and the two j sons can earn $ 19 a week. How much can each alone earn i per week ? j ! 15. A and B start in business with the same capital. A j gains $1775, and B loses $225. B then has | as much j money as A. What was their original capital ? i 16. Divide $440 between A, B, and C so that B shall , have $ 5 more than A, and C shall have f as much as A ■ and B. i FRACTIONS. lt)7 ADDITION AND SUBTRACTION OF FRACTIONS. 129. 1. Find the value of i + i; of i - ^ ; of ^* + ^- 2. What kind of fractions can be added or subtracted without changing their form ? 3. What must be done to dissimilar fractions before they can be added or subtracted ? How are dissimilar fractions made similar ? 130. Principles. 1. Only similar fractions can he added or subtracted. 2. Dissimilar fractions must be reduced to similar fractions before they can be united by addition or subtraction into one term, 1. What is the sum of % '^^ and ^? 2 5 4 Solution. a 2a 3c_10a . 8a 15c_ 18a4-15c 2 6 4 20 20 20 20 2. Subtract — from - — 3x Sy Solution. 7_b_2a_ 21 bx _ 16 ay _ 21 hx - 16 ay Sy Sx~ 2ixy 2^xy 24 xy 3. Simplify - H —^ ^* ^ ^ X x-2 x-\-2 Solution. 2 x±\_x^_ 2x'-S a;3 + 3a;-H2x x^~6x:'- + 2x ^ 8a;-^-8 X x-2 x+2 x(x2-4) x(x2-4) x(x2-4) x(x2-4) 108 ELEMENTS OF ALGEBRA. i i Eemember to change the signs of all the terms in the] numerator when you subtract a fraction. j Add: \ 4. — and -^. 14. -A_ and ah ah m -f n m — n 5. 2a6 ^„^ 5«d. i5_ OH- 1 ^^^^ a^_-^l. Zxy 3xy x — 1 x+1 6. ?and^^ 16. t|M^a.id-^. 2/ 71 x'—xy x—y 7. ^andll. 17. ^^^ and ^-. 3a 6 ax xr —y x — y 8. and 18. and — '—-' a-{-x a — x a-fJ a — b 9. ^and^. 19. ^i^and^JLh^. 10. 2^ and ^^. 20. -^ and l^Zll. 3 c 12 c 2a;-l 4a;2_l 2 »^ , 2 a; 1 2 »; 11. ;^^ and ^^. 21. —5— and -=ii^. 12. ^and^. 22. - « and ^ a + 3 a + 5 a;(a — a;) a(a — .cj 13. l^^zll and 2£zil/. 23. AjL. and f'^^ ■ 2, X — y x — y a^x a^ — xr Subtract : 24. 3-* from ^. 26. '"-^ from ^''. y y ax y 25. I from ^. 27. 21 from ^. o o xy yz FRACTIONS. 109 28. -^— - from -• 31. — ^ from x-1 x-2 x-^2 x-\-3 29. ^^ from -^. 32. -^-^ from A. X- — 4: x — 2 it- — 2 a; 2x 30, 4^ from . ^ + ^ , . 33. —^ from 34. Find the sum of one half of c dollars and one third of c plus d dollars. 35. A man having b dollars paid out one half of his money, and then one fifth of the remainder. How much did he pay out ? How much had he left ? 36. A carpet cost a dollars, a table b dollars, and a desk one half as much as the carpet and the table. What was the cost of all ? Simplify : 37 ?^ , 4_xj/_3^ a-b a-{-b a b * 15 "^ 5 7 * ' 3 "^ 4 2^5 38. ^x-l-x-\-\x. X. y , x' 39. -f-^ab — \ab-{-\ab. 45. _ + -^4--.^ y x-\-y x-4-xy . xt/ xy 1 40. ^ + ^_^. ^' x^y x-y'^x^y 1 , 1 , 2a .. 3a; , a; , 3a; 47. -7—— + Z + :j 41. —-!--+— -. l-f-a 1-a 1-a- 5 4 10 2a__46 a 48. a-b ^ b-c c-a ' 7 9 3 ab be ac . 5 ax 2 ax 5 ax .^ 2x 3 5 ' 6 ~9" 12 * a;2-l a; + l a;-l* 110 ELEMENTS OF ALGEBRA. 50. 4^^.--^. + -^- 53. .-^ + - a;2— x— 2 x—2 x-\-l a; 4-1 x — 1 51. -H -H 64. 1 x-\-l X -\-2 X — 3 x — a x-^-a x ^^ 1 1 ^^ a , 3a 2 ax 5/3. —^ -— -• 55. h x^ — X — 2 x^ -{-X — 2 a — X a -{- x a^ -{-x^ 56. A man walked a miles the first day ; the second day he walked half as far as the first day ; and the third day half as far as the second. How far did he walk in the 3 days ? 57. What is the cost of a coat, hat, and pair of shoes, if the coat costs a dollars, the hat c dollars, and the shoes -| as much as the hat and coat ? 58. A train went e miles the first hour, /miles the second, and then went back ^ of the whole distance. How far was the train then from the starting point ? MULTIPLICATION OF FRACTIONS. 131. 1. How much is three times ^? Four times f ? Five times — ? 9 2. Express 2 x f in its lowest terms; 5 x y^-; 8 x g^j 3x— . 3. How may the products in 1 and 2 be obtained from the given fractions ? 4. In what two ways, then, may a fraction be multiplied by an integer ? 5. How much is 1 of 4 or 4 -3? i of i^ or i? -- 2? FRACTIONS. Ill 6. In what two ways, then, may a fraction be divided by an integer ? 132. Principles. 1. Multiplying the numerator or divid- ing the denominator of a fraction by any quantity, multiplies the fraction by that quantity. 2. Dividing the numerator or multiplying the denominator of a fraction by any quantity, divides the fraction by that quantity, 1. What is the product of - multiplied by -? PROCESS. Explanation. To multiply - by ^ is to find c X c ex times J part of -• J part of - is — (Prin. 2), and c ^. ^ times -^ is — (Prin. 1). That is, the product of the Sy Hy numerators is the numerator of the product, and the product of the denominators is the denominator of the product. 1. Reduce all entire and mixed quantities to the fractional form before multiplying. 2. Entire quantities may be expressed in the form of a fraction by writing 1 as a denominator. Thus, a may be written -• 3. When possible, factor the terms of the fractions and cancel equal factors from numerator and denominator. 2. Find the product of — — - x — X ^ a^_4 4a; x^^2x-3 Solution. x — ^^ x x2-4 4x x'^-2x-3 x-^ (x+l)(x-l) x±2_ X ^ ■ f-^ '- X (x-f2)(x-2) 4x (x-3)(x+l) Canceling equal factors from dividend and divisor, the result is x-\ 4xCx-2)' 112 ELEMENTS OF ALGEBRA. Multiply : 3. by abn. 9. by abcx 18 m + n 4. ;^;^ — "2 by 6 a. 10. ^^ ^bya;+2/. 5. n-by-- 11. 6. - — by 12. ax xyz „ 7 am 1 3 mn 7. — -— by 13. 9 bn 2 ax 8. ^±lby^. 14. _, , a6c aa; 5a;— 15 ar + 5x4-4 a^ — ab , a^ — 4 aft + 3 &^ a^-2ab + b'' ' a' -J)' ,- o;^ — ic — 2i ic^ — 3a; — 10 16. by . aj2^4a.- + 3 ^ a^-4 3m -12 , m4-2 17- ~TT-^ — T-^^y ?7i^ + 3 m + 2 m'^ — 5 m 4- 4 18. / + ^ by ^-^ 5aj-15 *'a^-4aj-21 20 , ar^ + 3a;-40 19. by — ' • aj2_64 J' 5 a; -25 20. ^l#by "^' 2^ a;^ + 2a;-63 v a;^ + 9a;4-18 a;2_4^_21 ^ 6a; + 54 FRACTIONS. 113 Simplify the following : 22. ^x4x^. DC ah ac 23. ^-=l!x-^X^^. X x+y x—y 24. (2--y 4 25. -+- 1 , ly x^y X yj \xf — y 26. (—^A^\(t^Z^\(tj^\ \^-\-\^x-^1h)\ x-2 J\x-5j 'x , X 2y\f ^xy 5 aj- — 3 xy^ 28. i^ X ^^^ X ^^^'^y^ X ^"-^ 9 0^2; ^ — y^ 8 ic?/^ a?2/2; \mr -\-m -\- \j\mr m 30.^X^X^^^20-15. a + 3 a— o a- — a— 2 31. If a cords of wood cost h dollars, how much will c cords cost at the same rate ? 32. If & pounds of sugar are worth c pounds of butter, how many pounds of butter are 6 pounds of sugar worth ? 33. A father works a weeks at n dollars per week, and his son works 4 weeks at ^ of the wages that the father receives. How much do both receive ? 34. What is the value of d acres of land at f of {d — e) dollars per acre ? 35. A walked c miles in b hours. How far did he walk in 1 hour ? In 5 hours ? In a hours ? In c + d hours ? milne's el. of alg. — 8. 114 ELEMENTS OF ALGEBRA. DIVISION OP FRACTIONS. 133. 1. How many times is ^ contained in 1 ? i in 1 ? 1 ml? yViiil^ -inl? ;i- in 1 ? -J_ in 1 ? 2. How does the number of times a fraction, having 1 for its numerator, is contained in 1 compare with its denominator ? 3. How many times is f contained in 1 ? | in 1 ? ^ in 1 ? ? in 1 ? c a 1. What is the value of — -^ - ? 2c d PROCESS Explanation. The fraction - is con- a ^^ b ^a b d bd tained in 1 d times; and ^ is contained in. 2c d 2 c a 2ac ^l p^j.^ ^f ^ ^^^^^^ ^^ d ^^^^^ ^^^^ -.a ^ h h since - is contained in 1 - times, it will be contained in — , — d a 2c 2c times - or -^ times. That is, the quotient of one fraction divided a 2ac by another is equal to the product of the dividend by the divisor inverted. ,1. Change entire and mixed quantities to the fractional form. 2. An entire quantity may be expressed as a fraction by writing 1 for its denominator. 3. When possible, factor the terms of the fractions and use cancellation. 2. Divide -^-Tli—by '"^^ x^-\-x-20 ' x'-25 Solution. x^ - 1 ^ .x+j_ ^ (x + l)(x - 1) ^ (x+5)(x-5) x2 + a;-20 5c2-25 (x + 5)(ac-4) a: + 1 Canceling equal factors, the quotient is (a;-l)(y. -5) ^^ x^-6x + b X — 4 X — 4: FRACTIONS. 115 Divide : 3- - by -• 12. -Hb by — ^ 2/ n ^ — y^ ^-\-y ^ by ^. 13. *i xy c 2y 10. 11. ^' 6a'b'c 9. 13 a; + 42 • a;^_2a:-15 '^ x'-{-Ax-^3 23^ a^-lOa + 24^ /a-6^ a-.4 a^+oa- 14 - \a-2 a^^6a-7 V2^3; "^ V 6(0^ + 2/) 25. 4:x'-4.y' by '^^-i^- 116 ELEMENTS OF ALGEBRA. 26. 4=4 by ^SZV.. 28. What is the cost of h tons of coal if 7 tons cost a dollars ? 29. How many times a dollars are 2 times c dollars ? 30. A man sells 4 horses at m dollars each, and after using n dollars of the money, divides the remainder equally among c children. What represents the share of each child ? 31. A man exchanges d horses for/ cows. At that rate what is the value of a horses ? 32. A son's age is — of his father's. The father's age m is represented by f of the difference between a and h. What represents the age of the son? EQUATIONS. 134. Elimination by Comparison. 1. cc -f 3 2/ = 5 and x — y=l are simultaneous equations. Transpose the term containing y in each to the second member. What are the resulting equations ? 2. Since the second members of these derived equations are each equal to x, how do they compare with each other? (Ax. 1.) 3. If these second members are formed into an equation, how many unknown quantities does it contain ? 4. How, then, may an unknown quantity be eliminated from two simultaneous equations by comparison ? EQUATIONS. 117 1. Given ^ !" -' ' ^^ >• to find the values of a; and v by comparison. 2x-y=S (1) x + Sy=zl9 (2) 2x = 'S + y .(3) .=i±i (4) x=lQ-Zy' (5) 3 + 2'=19 32, 2 " (6) 3 + 2/ = 38 -62/ (7) 72/ = 35 (8) 2/ = 5 (9) a; = 19 - 15 (10) a; = 4 (11) Solution. Transposing in (1), Dividing by the coefficient of x, Transposing in (2), Equating (4) and (5) (Ax. 1), Clearing of fractions, Transposing, etc., Dividing by the coefficient of ?/, Substituting the value of y in (6), EuLE. Fmd an expression for the value of the same unknown quantity in each equation. Make an equation of these values and solve it. Solve the following by elimination by comparison : ^ |4aj + 2/ = 10| ^ (2x-2y = -Q^ \ x + y= 7) ' \3x-2y= 6) • • ( a; + 3.y = 32i * \2x-y = 22) (X -\-2y = ll>^ (3x- y = U^ 4. ■< >• 8. s >■ - (6a;- 2/ = 40) \2x-^2y = 2%\ ^ ( x + lOy^l^^ ^ (3a:+ y = 26| (2a;- 2/= 7> * 1 x-\-2y = 21) 118 ELEMENTS OF ALGEBRA. 10. 11. 12. 13. 14. 16. 16. { \2x-\-3y = 2o\ (2x- y= 7 I (3a;-f 2i/ = 28 3 (Zx — 2y — 2^ \2x- y = b) (4aj-3?/= 1| \2x+ y = 23\ (2x^y = 27^ (3aj-2/= 3) (3a;-f 2i/ = 30| \bx-{- 2/ = 29) r 6 8 I ' X y X y 17. 18. 19. 20. { ax-\-hy =^m^ = n ) ax-\-cy •• 2 3" 5 {^2x-\- ay = b \ ax -\-2y = c I 3 2 9 2 21. <^ o ? + 5a; = 27 135. Elimination by Substitution. 1. In the equation x-\- y = 12, how may the value of x be found if y equals 5 ? 2. a; 4- 22/ = 8 and 2 a? 4-3?/ = 13 are simultaneous equa- tions. Find the value of x in the first equation by trans- posing y to the second member. 3. If this value of x should be substituted for x in the second equation, how many unknown quantities would the resulting equation contain ? 4. How, then, may an unknown quantity be eliminated from two simultaneous equations by substitution ? EQUATIONS. 119 1. Given < ~ , " . ^ to lind the values of x and v by substitution. Solution. Sx — 2y =: 1 (1) X + 4 y = 19 (2) From (1), x = ^ "^^^ (3) o Substituting (3) in (2), J-±-?l + 4 ?/ = 19 (4) o Clearing of fractions, 1 + 2 ?/ + 12 ?/ = 57 (5) Transposing, etc., 14?/ = 56 (6) 2/ = 4 (7) Substituting in (3), x = ^—^ (8) o X = 3 (9) Solve the following by elimination by substitution : ^ ( x-3y= 3j ^ r7a;-4^=81| (20^ + 4?/ = 16 i * (2x- 2/ = 25i r3a;-2iy=3| i x-\-2y = 20^ (207 + 32/ = 28 1 ^* i x-'2y= 4:1 5 (4^ + 42/=76i ^^ r3a.- y = 5, \Sx-{- 2/=39> * (32/- x = 9> (3x-{- 2/ = 14| ^^ f2aj- y= 18 | ( aj + 32/ = 26) ' I a;-22/=-6J 120 12. 13. 14. 15. 16. ELEMENTS OF ALGEBRA. \2x- y=20} I 6 a; + 5 ?/ = 180 3 ^ 5 a; + 3 ?/ = 46 1 \Sx + 2y = 29i ( 2x- y= 10 1 (16a;-52/ = 20ol x + l=ie] [ + y = 20 17. 18. 19. 20. (4:X-I5y = 24:^ [3x-2y = 19\ 3x-2y -+7 = 3 a b X y a b { 2jo 3 ' 2 y. 7 : = o X , y a . b + - = m X y > G . d + - = n X y REVIEW OP EQUATIONS. 136. 1. Definition of an Equation. 2. Definition of Members of an Equation. 3. Definition of Transposition. Eule. 4. Definition of an Axiom. Give the Axioms. 5. Definition of Simultaneous Equations. 6. Definition of Elimination. 7. Definition of Clearing of Fractions. Rnle. 137. The highest number of factors of unknown quanti- ties contained in any term determines the Degree of an Equation. Thus, when only one unknown quantity of the first power is found in any term, the equation is said to be of the first degree ; when the second power of an unknown quantity is found in any term or when the number of unknown factors in any term is two, the equation is said to be of the second degree, etc. EQUATIONS. 121 138. An equation of the first degree is also called a Simple Equation. Thus, x-\-y=li^ax-\-2 a^y = 10 are simple equations. 139. An equation of the second degree is also called a Quadratic Equation. Thus, x2 = c, X -\- xy = 4:, abcxy = 6 are quadratic equations. 140. Solve the following equations : 1. x-6-{-5x=4:-{-2x-{-2. 2. 4:{x-l)=:5{2x-S). 3. x{x-4:)-{-6 = (x-^)(x + 30). 4. (2x-^6)(x-\-l) = {2x-2){x-{'9), 5. ax — bx = cr — b^. 6. ax — ab = x — ex -f b'\ 7. a ■{■ b -{- 2 X = c^x. 8. a^x — a — b = b^x -{- 2 ab. 13. ^-x = ^-^-9. 3 11 14. ^(20.^-6) =|(a^4-4)-7. 15 11 ^ - 80 _ Sx-5 ^ ^ 6 15 ~ * 16. ^^3^ 3(a: + 3) ^ 3(.. + 3) _l, 7 2 2 122 ELEMENTS OF ALGEBRA. 17. 5 9 2 18. X , c a d 19. ax — h,^ x-{-ac \ Uj C C 20. 1 1 2a; x-1 x^l x""-! 21. 4 , 12 _ 48 x-2 x-\-2 i»2_4 22. a;_12ic-4_^ 7 0^-7 x-- 12 '^ x-1 23. X x^ — 6x 2 3 ^x-1 3 24. aj + 7 i» + o_^ 5a; + 3 2 3 ~ 4 25. ^^^ + ^^-^^'-^ = 10(x-l). oa 7 a; + 16 a; x-\-^ 21 3 4a; -11 Suggestion. The equation may be expressed as follows : 7x16 x_x + 8 Simplifying, 21 21 3 4x-ll 16^ x + S 21 4x-ll 20a; 36 5a; + 20 ^ 4a; 86 25 25 9a;-16 5 25* 28. 1 ^=_:^. a;-|-2 x-\-Q EQUATIONS. ' 128 29. 30. 31. 5a; , 10 ^ x + 6^ x-1-^- 15 1 44 x + 3 3(a; + 3) 15 3a; -2 6a; +13 _ 21- -3x 2x-5 10 32. 7 6a;-1 ^ 3(16-2a^) X — 1 X -\-l a^ — 1 32 10 0^ + 17 _ 12a; 4- 2 ^ 5a?-4 18 , 13X--16 9 See suggestion, Example 26. _. 18^ + 19 , lliu-f 21_9a.' + 15 35. 28 6ir + 14 14 6a? + l 2a;-4 ^ 2a;-l 15 7a.'-16~ 5 36 ^ ^' — -^ , ^' 4- 5 ^ g; 4- 13 1 16 4 8 16' 4 0^ + 3 ^ 8 37 + 19 _ 7 a? -29 9 ~ 18 5a;- 12* 38. -±-^ ' ^' x-^2 x-^ '3 a;- + 5a; + 6 39. 2^1^ + 1 1-^^ 40. l_3a; 6 l-2a; 4 10 a; - 4 x-5 a;^ - 9 a; + 20 b a 42. -^-t-A = a2 + 6^ bx ax 124 46. 47. 43. 44. ELEMENTS OF ALGEBRA. x'^— a _ a^x _2x _a bx b h X Sax --2 b ax — a ax 2 45 3b 2b b_(l±^^,^^b_x^ ex ( x-^5y= 7 X4tx-^3y -'I =11) \3x + 3y='d6\ x-y=8 9 48. i 49. < ^-•^ = 3 ^ + •^-=8 5 2 52. x-\-hy=Vl' l+3y= 7 53. Xx — y= b) 54. 55. ^ ax-{-by= c ^ \bx — ay = d) a , h - + - = m X y a b V — :=: ?i X y 50. i 3 8 X y _ ^ 56. — \- 7iy = m -{- n m mx , ii^v 9 , 2 n m 51. ^4-^ = 2 2 2 7 + ^ = 2^ 57. a b I 1^6 a J EQUATIONS. 125 PROBLEMS. 141. 1. The sum of two numbers is 100, and the less num- ber is 10 more than i of the greater. What are the numbers ? 2. A, B, and C build 216 rods of fence. A builds 7 rods a day, B builds 6 rods, and C builds 5^ rods. If A and B work the same number of days and C works twice as many days as the others, how many days does each work ? 3. A man has $ 6000 in cash. He expends part of the sum for a house, and invests the remainder where it brings him an income of f 250 per year. If in 4 years the amount invested together with the income of that time equals the amount paid for the house, how much is paid for the house ? How much is invested ? 4. A boy bought a certain number of apples at the rate of 3 for 2 cents. He sold them at the rate of 6 for 5 cents, and gained 12 cents. How many apples did he buy ? 5. A lady bought two pieces of cloth; the longer piece lacked 4 yards of being three times the length of the shorter. She paid f 2 per yard for the longer piece, and f 2^ for the shorter, and the shorter piece cost just ^ as much as the longer. How many yards were there in each' piece ? 6. The distance around a rectangular field is equal to 10 rods more than five times the breadth ; and the length is 1| times the breadth. What are the length and the breadth of the field ? 7. A man spends J of his annual income for his board, i- for traveling expenses, -^^ for clothes, ^ for other ex- penses, and saves f 380. What is his annual income ? 8. Of a company of soldiers f are on duty, ^ of the remainder are absent on leave, -^^ of the whole are sick, and 126 ELEMENTS OF ALGEBRA. the remaining 50 are off duty. How many soldiers are there in the company ? 9. A farm of 450 acres was divided between A^ B, and C so that A had f as many acres as B, and C had i as many acres as A and B together. How many acres had each ? 10. A boy bought a certain number of apples at the rate of 2 for 1 cent. He sold half of the number for 1 cent apiece, and the other half at the rate of 2 for 1 cent. He gained 10 cents by the transaction. How many apples did he buy ? ' ' Suggestion. Let 2x equal the number of apples. 11. The length of a certain field is twice the breadth. If the length were | as much, and the breadth li- as much, the entire distance around the field would be 64 rods. What are the length and the breadth of the field ? 12. Find a number whose half, third, and fourth added together equal the number plus 2. 13. A man in business gained $ 100, and then lost ^ of all that he had. He afterward gained $ 150, when he found that his money was just equal to his original capital. What was the original capital ? 14. Two numbers are to each other as 3 to 4. If 10 is subtracted from each, the smaller one will be | of the larger. What are the numbers ? Suggestion. Let 3 x and 4 x represent the numbers. 15. A woman sold some eggs at 2 cents apiece ; but when she came to deliver them, four of the eggs being broken, she received only 192 cents. How many eggs had she ? 16. Two numbers are to each other as 5 to 7. One half of the first plus the second equals one half of the second plus 12. What are the numbers ? EQUATIONS. 127 17. A said to B, "Give me $100, and I shall have as much money as you have left." B said to A, "Give me $ 100, and I shall have three times as much money as you have left." How much money had each ? 18. A man gained in business as follows : The first year $ 400 less than his original capital ; the second year $ 300 less than the original capital ; and the third year $ 200 less than the first year. The gains of the three years amounted to f 700 more than the capital. What was the original capital ? 19. A mason, 5 carpenters, and 3 assistants receive together f 191 for working a certain number of days. The mason receives f 3 per day, the carpenters f 2^, and the assistants $ 1^. How many days do they work ? 20. A man spent $ 300 more than i of his earnings each year. In 5 years he had saved f 1000. How much did he earn each year ? 21. 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 ? 22. A boy spent ^ of his money and 2 cents more. He then spent i of what was left and 2 cents more, when he found that there remained 12 cents of his money. How much had he at first ? 23. A man bought a horse and wagon, paying twice as much for the horse as for the wagon. He sold the horse at a gain of 50 per cent, and the wagon at a loss of 10 per cent. He received for both $ 195. What did he pay for each ? 24. A person engaged to reap a field of grain for $ 1^ per acre ; but leaving 6 acres not reaped, he received f 12. How many acres of grain were there ? 128 ELEMENTS OF ALGEBRA. 142. Three unknown quantities. 2x-^ .V + 3;2=13] 3x-^2y -{- z=z \0 )■ to find x, y, and z. 2:2=13 I 1. Given < x-^3y-\-\ J Solution. 2x + 2/ + 3« = 13 (1) 3x + 2!/ + z = 10 (2) x + Zy + 2z = \Z (3) (2) X 3, 9s6 + 6j^ + 32 = 30 (4) (1), 2x+ 2^ + 3^ = 13 (4)-(l), 7x + 5y==n (5) (2)X2, 6x + 4^ + 2a = 20 (6) (3), x + 3y + 2z = 13 (6) -(3), bx + y-l (7) (7)x5, (5). 25 X + 5 8* = 35 7x+5j/ = 17 (8) (8) -(5). 18x = 18 (9) x = l (10) Substituting (10) in (7), 5 + ?y = 7 (11) y = 2 (12) Substituting (10) and (12) in (l),2 + 2 + 3« = 13 (13) z = Z (14) The student will observe that by combining two of the given equa- tions one of the unknown quantities is eliminated, and that by com- bining another pair of the given equations the same quantity is eliminated. We have thus two equations ' containing two unknown quantities which may be readily solved. Find the value of each unknown quantity in the following: i» + y -\- 25 = 6 ^ x-\-2y -^^ z = 7 x-{- y-\-2z = 9 2x-{-2y+ 2; = 15" 3x-h y + 2z = 23 x-3y-^2z = ll EQUATIONS. 129 { x-{-2y-{-z = 16] 4:. ^ 4:X — oy — z= GJ> [2x-\-2 y-z= 11 ] f5x- y^2z = 3S^ ' 2x-\- y — z= 4: y a; — 32/4-52; = 44 J ^-\- y-\- z = lo 6. ^ 2x + 2y— z= 6 X— y-\-4:Z = 37) r x + 2y- z= 6"j 7. ^ 4.X- y-{-2z = 27 [> 2x-\-2y-{- 2; = 3lJ 8. ^ 5a;4- 2/— -"^2;= 14 [4a;+22/-42;=- 4 a;— 1/ -4- 2;= 17 9. 4^/+ 2 = 31J a; 4- y—14: 2y-Sz= 9 5 a; — z = 5 j 2x- 2/ = 9] x-22J = 3 [. i/-2^ = 5j milne's el. of alg. — 9. 4: 10. { 11 12 13. 14. 15. < 0,^-12?/ 4-22;= -20 1 2a;— y— z= 7 V X-}- z= 19 J 6x- y-\- z = 2S 5x-\-2y-2z = 12 2x + 2y- z= 8 4 8 -4-^ = 5 4 10 y^ I 14 r X +?+^= 9l 16.^^ + 1+^=10^ a^ + .V + |=19 ( x-\-y= 9 17. ^ a; +2; = 12 U + ^ = 15. 18. 19. 2/ — a;-f2; = — 5 z — y — x = — 25 ^* + 2/ + ^ = ^^ 2aj-32/= 1 3y — 4:Z= 7 [42-5a; = -32 130 ELEMENTS OF ALGEBRA. 20. f - 4- ----- ^ X y z 3 X y z 3 ' 1+1-1=0 ^ y z X J X y 5 3_4_ 1 ^ y z 5 ► 3 4__1 .21 a; 2 . 2x- I ^ = G 22. < x + y — 2z = — 2 4aj — 32/+ z = 11 ' X z o ' 2 15 ~ 23. < 5 5 U 6 J > 21. 25a; -201/ + 152:= 80" 24. ^ 15a; -252/ + 202;= <50 20a; + 152/ -25:2 = -40 25. A farmer has sheep in three pastures. The number in the first plus \ of the number in the second plus \ of the number in the third equals 70. The number in the first plus \ of the number in the second minus \ of the number in the third equals 30. The number in the second plus ^ of the number in the first plus y^ of the number in the third equals 61. How many sheep are there in each pasture? 26. Henry, James, and Ralph together have 50 cents. Henry's and Ralph's money amounts to 35 cents ; James' and Ralph's amounts to 40 cents. How many cents has each? 27. A farmer has wheat, oats, and barley. The number of bushels of wheat and oats is 200 ; the number of bushels of wheat and barley is 190 ; and the number of bushels of oats and barley is 90. How many bushels are there of each kind of grain? INVOLUTION. 143. 1. What is a power of a quantity ? An exponent? 2. How many times is a quantity used as a factor in producing the second power? The third power? The fourth power? The fifth power? Any power? 3. What sign has the second power of -|-a? The third power? The fourth power? Any power of a positive quantity ? 4. What sign has the second power of — a? The third power? The fourth power? The fifth power ? The sixth power? Any even power of a negative quantity? Any odd power of a negative quantity ? 5. Which powers of a negative quantity are positive? Which are negative ? 6. What is the third power of a^? The fourth power? The fifth power? The sixth power? The nth power? 7. What is the third power of a^? The fourth power? The fifth power ? The sixth power ? The nth power ? 8. How is the exponent of a power of a quantity deter- mined from the exponent of the quantity ? 144. The process of finding a power of a quantity is called Involution. 131 182 ELEMENTS OF ALGEBRA. 146. Principles. 1. All poivers of a jjositive quantitr/ are positive. 2. All even poicers of a negative quantity are positive, and all odd powers are negative. 3. The exponent of the power of a quantity is equal to the exponent of the quantity muUiplied by the exponent of the power to ivhich the quantity is to be raised. 146. Involution of monomials. 1. What is the fourth power of 3a^ar^? Solution. (3 a'^x^y = 3 a-x^ • 3 a'^x^ • 3 a'^x^ • 3 a%3 =: 81 a^x^'^. Rule. Raise the numerical coefficient to the required power; multiply the exponent of each literal quantity by the exponent of the p)ower to which it is to be raised, and prefix the proper sign to the residt. 2. What is the third power of 2a&% Solution. ( ^ab^V = ^ah^ ^ 2al^ ^ 2a^ ^ i V3r.V/ SxV 3x'V Sx'Y 2 27 x^y^' In raising a fraction to a power, both numerator and denominator must be raised to the required power. Mnd the values of the following : 3. (2a^yy. 10. {lox^y 17. {-2aj'by. 4. {Axyzy. 11. (Sa'bcy. 18. {-lOa^by. 5. {-Sa^zy. 12. (-2m'ny. 19. {a^bh'dy. 6. {2a%y. 13. (-4 amy. 20. (-a^b'my. 7. (-Aab^xy. 14. (2 ab'dy. 21. (-2a'b'dey. 8. (-sc'd'y. 15. (-6a'my. 22. {-x^fzy. 9. {-^yzy. 16. {2x'yy. 23. {x^y^zy. INVOLUTION. 133 24. (:2xY^T. 32. f_J^y^]. 37. (■ 25. (-2&2c-d2")^ V 2amV V^'^' 26. (a-^6^c^)l 33 r a'bW ^^ /8 g^ftV 27. (5a'%V)^ ' V^"/^>'* * V^yj' 28. (_aV"2r*^")-l 3^ /5_arxy\\ 3^ /aVV". 29. {2a'hxy)\ ' V^mn'zJ ' \byj 30. f^^]\ 35. r^^V. 40. (-., -'y 3mxJ \xYzy \ 10 a'm 31. r^^Y. 36. r^^^Y. 41. f ^''^' 4aVy V4m^nW ' \mVaf 147. Involution of polynomials. (a + &)- = (a + 6) (a + ^) = a- + 2 a& + ft^. (a - by = {a -h) {a - h)= or -2 ah -\- h\ (a + by = {a + &) (a + &) (a + 6) = d' -\-3o?b + Saft^ + 6\ (a _ bf = (a-b){a- b) (a -b) = a^-^ Sa'b + 3ab^ - b\ (a - bY=-a' - 4.a'b + 6a'b' - 4a6'^ + b\ By examining carefully the letters, the exponents, the coefficients, and the signs of the above powers, the student will be able to formu- late laws for writing out the powers of quantities. 15. (a' -{-by. 16. (a + & + c)l 17. (a-^b — cy, 18. (x-y-zy. 19. {2xi-2y — zy. 20. (a" + ^>" + c")-. 21. (a"^-6"^-c2-)2. Raise to the required power : 1. (x-yy. 8. {7n-3ny. 2. {3a-\-2by, 9. {x + 2yy. 3. (5a + 4c)2. 10. {2a-oby. •4. (x-i-yy. 11. {3x + 2yy. 5. {X - yy. 12. {Ax~3yy. 6. {a + by. 13. (a + by. 7. (x-yy. 14. (x + iy. EVOLUTION. 148. 1. What are the tmo equal factors of 25 ? Of 36? Of 64? 2. What is one of the two equal factors, or what is the second or square root of 25 ? Of 81 ? Of 100 ? 3. What is one of the three equal factors, or what is the third or cube toot of a" ? Of 8 a^ ? Of 27 a^? Of 8 a« ? 4. What is the sign of any power of a positive quantity ? Siiice any power of a positive quantity is positive, what sign may one of the equal factors, or a root of a positive quantity have ? 5. What is the sign of the second power of a negative quantity ? The fourth power ? The sixth power ? Any even power ? 6. Since any even power of a positive or negative quan- tity has the positive sign, what is the sign of the everi root of any quantity ? 7. What sign has the second power of — 3 ? The third power? The fourth power? The fifth power? What powers of a negative quantity are negative ? 8. Since a power having the negative sign is the product of an odd number of equal negative factors, what is the sign of the odd root of a negative quantity ? 134 EVOLUTION. 135 9. What kind of quantity used twice as a factor will give a product with the negative sign? What kind, used four times ? Any even number of times? 10. Since no quantity used an even number of times as a factor will give a negative product, what may be said regarding an even root of a negative quantity ? 149. One of the equal factors of a quantity is called the Root of the quantity. Thus, 3 is a root of 9, of 27, etc. ; a is a root of a^, of a*, etc. 150. Roots are named from the number of equal factors into which the quantity is resolved or separated. Thus, one of the two equal factors is called the second or square root ; one of the three equal factors, the third or c^ihe root ; one of the four equal factors, the, fourth root, etc. 151. The root of a quantity is indicated by placing the sign -yj, called the Radical Sign, before the quantity. A number or quantity called the Index is frequently written at the opening of the radical sign, to show what root is sought. Thus, y/a shows that the third root of a is sought ; V«, the fifth root, etc. When no index is w^ritten in the opening of the radical sign, the second or square root is indicated. 152. The process of finding the root of a quantity is called Evolution. 153. Principles. 1. An odd root of a quantity has the same sign as the quantity itself. 2. An even root of a positive quantity is either positive or negative, 3. An even root of a negative quantity is impossible or imaginary. 136 ELEMENTS OF ALGEBRA. 154. Evolution of a monomiaL 1. What is the square root of 16a^x^? PROCESS Explanation. Since in squaring a monomial we square the coefficient and a/16 aV = + 4 a^x^ multiply the exponents of the letters by 2, to extract the square root we must extract the square root of the coefficient, and divide the exponents of the letters by 2. The sign of the root is either plus or minus (Prin. 2). Hence the square root of the quantity is ± 4 a^x^. Rule. Extract the required root of the numerical coeffi- cient ; divide the exponent of each letter by the index of the root sought ; and prefix the proper sign to the result. The root of a fraction is found by taking the root of the numerator and of the denominator separately. Find the values of the following : 2. V4a26l 13. -Vx'^fz'^ 3. ^8Sy. 14. V-64a^Y^'^ 4. V25¥y\ ^'' ViOOW^ 6. V--27m'V. , 17. ^x'^y'^z'''. 6. a%-\-3aW^b\ 5. a^ + 6a;2 4-12a; + 8. 4. x^'-^x^y + ^xy^^f, 6. 21 a^ + 21 a^ + ^a^-l. EVOLUTION. 145 7. 2lQ(?-21a^y-Jt^xy''-f. 8. x^^l2x^-\-i:%x'-^^L 9. ^m^ — 12 m'^n-{-Qmn^ — n\ 10. x^-{-^x^-[-^x^^:x^. 1 1 . a?m^ + 6 a^m^^ + 12 amft^ + 8 h\ 12 . 8 a^' - 60 aV + 150 aV - 125 c\ 13. 27 + 54& + 3662 4.8^^ 14. l + Sa + Sa^ + a^. 15. m« + 18 m* + 108 m^ + 216. 16. 8.r^ + 36a;V + 54a;^2^ 272/3^ 17 . 64 0^9 + 576 it-V + 1728 :x^y' + 1728 y\ 18. ft;'''-3aa^4-5aV-3a^aj-a«. 19. 8.^*^-84 052 4- 294 a; -343. 20. 343a^ + 588a;22/ + 336ajz/2 + 642/^ 21. a^ + 9 a«63 + 27 «'W' + 27 &^. 22 . 8 a;« 4- 72 xY + 216 o^y + 216 y\ 23 . a^^^ - 9 a'W + 27 a-6 - 27. 24. l + 9a5 + 27a;2_^27ar^. 25. m^2 — 3 mhi + 3 mhi^ — nl 26. a;«-6.^'^' + 15a;^-20a^ + 15a;2-6a; + l. 27. a«-9a^ + 33a^-63a'^ + 66a--36a + 8. 28. a;6-3a^ + 5a;3_3^_-j^ 29. 27 m« - 135 m^ + 171 m* + 55 rrv" - 114 m^ _ 60 m - 8. 30. 8a;« + 36 aj^.y + 42 xY^ -^o^rf-21 ci^y* + 9 xy' - y\ 31. 1 - 9.T + 39ic2 _ 99^3 ^ -(^55^4 _ -^44^5 ^ ^4^6 32. 1 + 12aj + 60 0^ + UOx^ -f- 240x^ + 192 o^ + Ux\ milne's el. of alg. — 10. 146 ELEMENTS OF ALGEBRA. 160. To extract the cube root of numbers. 13= 1 10'3= 1000 100^= 1000000 3^= 27 36^= 4:6656 361^= 47044881 9^ = 729 99^=970299 999^ = 997002999 The student should observe carefully the number of figures required to express the cube of units, tens, and hundreds in the above examples. The cube of the smallest, of the largest, and of an inter- mediate number of each order is given, consequently the number of figures required to express the cube of any num- ber may be readily discovered. 1. How many figures are required to express the cube of any number of units? 2. How does the number of figures required to express the cube of any number between 9 and 100 compare with the number of figures expressing the number ? 3. How does the number of figures required to express the cube of any number between 99 and 1000 compare with the number of figures expressing the number ? 4. If, then, the cube of a number is expressed by 4 figures, how many orders of units are there in the root ? If by 5 figures, how many ? If by 6 figures, how many ? If by 8 figures, how many ? 5. How may the number of figures in the cube root of a number be found ? 161. Principles. 1. The cube of a number is expressed by three times as many figures as the number itself or by one or two less than three times as many. 2. The orders of units in the cube root of a number corre- spond to the number of periods of three figures each into which the number can be separated, beginning at units. The left-hand period may contain one, two, or three figures. EVOLUTION 147 162. If the tens of a number are represented by t, and the units by u, the cube of a number consisting of tens and units will be the cube of {t + u) or ^ + 3^-i^ + (^tiv^ + u^. Thus, 35 = 3 tens + 5 units, or 30 + 5, and 35^ = 30^ + 3 (302 x 5) + 3(30 X 52) +53 =^42875. 1. What is the cube root of 74088 ? Trial divisor Complete divisor PROCESS. 74.088 I 40 + 2 ^= 64 000 3^2^4800 ^tu= 240 ?r= 4 = 5044 10088 10088 Explanation. According to Prin. 2, the orders of units may be determined by separating the number into periods of three figures each, beginning at units. Separating 74088 thus, there are found to be two orders of units in the root ; that is, it is composed of tens and units. Since the cube of tens is thousands, and the thousands of the power are less than 125 or 5-^ and more than 64 or 4^, the tens' figure of the root must be 4. 4 tens, or 40, cubed is 64000, and 64000 subtracted from 74088 leaves 10088, which is equal to three times the tens2 X the units + 3 times the tens x the units2 + the units^. Since three times the tens2 is much greater than three times the tens multiplied by the square of the units, 10088 is a little more than three times the tens squared multiplied by the units. If, then, 10088 is divided by 3 times the square of the tens, or 4800, the trial divisor^ the quotient 2 will be the units of the root, provided proper allowance has been made for the additions necessary to obtain the complete divisor. 4800 is contained in 10088 twice ; consequently the second figure of the root or the units is 2. Since the complete divisor is found by adding to three times the square of the tens, three times the tens multiplied by the units, and the square of the units, the complete divisor will be 4800 + 240 + 4 or 5044. This multiplied by 2 gives as a product 10088, which, sub- tracted from 10088, leaves no remainder. Therefore, the cube root of 74088 is 42. 148 ELEMENTS OF ALGEBRA. Since any number may be regarded as composed of tens and units, the process given above has a general application. 2. What is the cube root of 1860867 ? SOLUTION. Trial divisor = 3(10)2 = 300 3(10 X 2) = 60 22 Z3 4 Complete divisor = = 364 Trial divisor = 3(120)2 = z 43200 3(120 X 3) = = 1080 32 z 9 1.860-867 IJ23 1 860 728 Complete divisor = 44289 132867 132867 EuLE. Separate the nitmher into periods of three figures each^ beginning at units. Find the greatest cube in the left-hand period, and ivrite its root for the first figure of the required root. Cube the root, subtract the result from the left-hand period, and annex to the remainder the next period for a dividend. Take three times the square of the root already found ivith a cipher annexed, for a trial divisor, and by it divide the dividend. The quotient or quotient diminished will be the second figure of the root. To this trial divisor add three times the product of the first part of the root with a cipher annexed, multiplied by the second part, and also the square of the second part. Their sum will be the complete divisor. Multiply the complete divisor by the second part of the root, and subtract the product from the dividend. Continue thus until all the figures of the root have been found. Decimals are pointed off into periods of three figures each, by beginning at tenths and passing to the right. EVOLUTION. 149 3. What is the cube root of 95256152263? Solution. 95-256. 152. 263 1 4567 64 4800 31256 60C 25 5425 27125 4131152 607500 8100 36 615636 3693816 437336263 62380800 95760 49 62476609 437.336263 An abridged method of extracting the cube root of a number is pre- sented on page 339 of the author's Standard Arithmetic. Extract the cube root of the following : 4. 571787. 5. 148877. 6. 262144. 7. 250047. 8. 704969. 9. 912673. 10. 314432. 11. 614125. 12. 16581375. 13. 44361864. 14. 100544625. 15. 5545233. 16. 34965783. 17. 41063625. 18. 68417929. 19. 743677416. 20. .015625. 21. 43614208. 22. 13312053. 23. .004019679. 24. .000166375. 25. 28.094464. 26. 130323.843. 27. 48.228544. QUADRATIC EQUATIONS. PURE QUADRATIC EQUATIONS. 163. An equation of the second degree is called a Quad- ratic Equation. 164. An equation which contains only the second power of the unknown quantity is called a Pure Quadratic Equation. 165. A value of the unknown quantity in an equation above the first degree is called* a Root of the equation. 1. Given 2 x^ — 5 = 45, to find the value of x. Solution. 2 x^ — 5 := 45 2 x-2 = 45 + 5 2 x2 = 50 x-2 = 25 Extracting the square root, x = ±b 2. Given ax' -|- c = &x- + c?, to find the value of x. Solution. ax^ + c = hx"^ -\- d ax?- — bx'^ = d — c (a - b)x^ = d- c 2 — d — C a — b Extracting tlie square root^ x= ± \/— ^ ^a ■ 150 QUADRATIC EQUATIONS. 151 Find the values of x in the following equations : 3. x^-4. = 12. 12. 15x^-6 = 7 x^ + 194.. 4. a^ + l = 10. 13. x(x-4.) = -x^-4:X-\-8. 5. 2x'-5 = x'-\-20. 14^ a?-2 ^ 12 6. 3x'-15 = 57 + x^. ^ ^ + ^ X' 15. 2a:2_8 = -4-7. 7. a:2 4.i^:^_{_4, 3 4 8. 5a;2-2 = 2a;'^ + 25. 16. ir(2a^+3)=.'K2^3a;-f 81. 19. -=4a. 9. a^4-18 = 22-3a;2^ ^^- — 3— + ^ = 42. 10. 3a:2_29 = - + 510. ^^^ ^'-9 = cr-6a. 4 11. 8aj2-f 2 = 42-2a;-. "'" a 20. x(x — 2a) = 2xr — 2ax — a^. PROBLEMS. 166. 1. If to four times the square of a certain number 3 is added, the sum is 19. What is the number ? . 2. Two numbers are to each other as 2 to 5, and the sum of their squares is 261. What are the numbers ? Suggestion. Let 2 x and 5 x represent the numbers. 3. If a certain number is increased by 1 and also diminished by 1, the product of the sum and difference is 48. What is the number ? 4. The area of one square field is twice that of another, and both together contain 867 square rods. What is the length of a side of the smaller ? Suggestion. Let x represent length and x^ area of smaller field. 5. A lady paid f 36 for a certain number of yards of cloth. She paid i as many dollars per yard as there were yards in the piece. How many yards were there ? 152 ELEMENTS OF ALGEBRA. 6. What number is that whose square plus 18 is equal to half its square plus 30|- ? * 7. There is a rectangular field whose breadth is f of its length. After laying out -^^ of the field for a garden, there remain 54 square rods. Find the length and the breadth of the field. 8. Two young men were conversing about their ages. The younger said, ^^I am 25 years of age." "Then/' said the older, " the sum of our ages multiplied by the difference equals 51." What was the age of the older man ? AFFECTED QUADRATIC EQUATIONS. 167. 1. What is the square oi x + 1? Oix-{-2? Of x + 3? 2. How may the first term of a binomial be found from its square ? 3. Since the second term of the square of a binomial is twice the product of both terms, how may the second term of the binomial be found when the first term of the binomial is known ? 4. Add a quantity to x^'\-2x that will make it a perfect square. How is the term found ? 6. Add a quantity to o;^ + 10 a; that will make it a perfect square. How is the term found ? 168. An equation which contains both the first and second powers of an unknown quantity is called an Affected Quadratic Equation. Thus, x2 -f 2x = 8, 8x2 4- 6ic = 16, and ax'^ -\- bx = c are affected quadratic equations. QUADRATIC EQUATIONS. 153 169. First method of completing the square. 1. Given x^-\-8x= 20, to find the value of x. PROCESS. aj2 + 8.^^ + 16 = 20 -hie a^ 4- 8a; + 16 = 36 a; + 4 = ±6 x = 6 — 4: or —6 — 4 x = 2 or -10. Explanation. The first member of the equation will be made a perfect square by adding the square of one half the coefficient of x to both members. One half of 8 is 4 and the square of 4 is 16. This added to each member gives a^ -|- 8 x -f 16 = 36. Extracting the square root of each member, we have iTH- 4 = ± 6. Using the first value of X, x = 2 ; using the second value of x, x = — 10. 2. Given aj^ -j- 3ic = 28, to find the value of x. Solution. x^ + .3 x = 28 Completing the square, x^ + 3 x + f = 28 + f or if i- Extracting the square root, x + i = ± V- Transposing, x = | or 4 X = - V- 0] 3. Given 3 a;- -f 4 a; = 20, to find the value of x. Solution. ' 3 x2 + 4 x = 20 Dividing by the coefficient of x'-^, x^ + | x = 2^*^- Completing the square, x'^ + | x + | = 2^0 + * or -^^ Extracting the square root, x + f = ± f .-. x = 2 or --V>- 154 ELEMENTS OF ALGEBRA. Find the values of aj in the following equations : 4. x^ + 4:X = 12. 16. cc^-7x = -6. 5. aj2-2a^ = 15. 17. x'-{-10x = 56. 6. 0^ + 80^ = 33. 18. 0)2 + 1205 = 13. 7. 0)2 + 6a; = 40. 19. a;--30x = 64. 8. .T--6o; = 7. 20. a;- -f 14 a; = 32. 9. a:2_|. 10^.^11. 21. 3 a;^ + 6 a; = 504. 10. x2_8^. = 20. 22. ar-9a; = -8. 11. a;2 + 20x = 125. 23. a;^- 28a; = 60. 12. 3a;2 + 6a; = 24. 24. a;2^iii^. = 80. 13. a;- 4- 5a; =14. 25. a;^ — 15a; = — 36. 14. 2a;2_|_5^.^j[8 26. a;2 + 2a; = 99. 15. 3a;2 — 25a; = -50. 27. a;- -3a; = 4. 170. Other methods of completing the square. In the previous examples, when the square of the unknown quantity had a coefficient, the equation was divided by that coefficient so that the te^^m containing the square of the unknown quantity might be a perfect square. The coefficient of the second power may always be made a square, all fractions avoided, the square completed, and the values of the unknown quantity found as follows : KuLE. Multiply the equation by four times the coefficient of the highest power of the unknown quantity ; add to that the square of the coefficient of the first power of the unknown quantity, and then find the value of the unknown quantity by extracting the square root, etc. QUADRATIC EQUATIONS. 166 1 . Solve the equation 3x^ -\-5x = S. Solution. 3 x^ + 5 x = 8 Multiplying by 12, etc., 36 x2 + 60 x + 25 = 96 + 25 or 121 6x + 5:=± 11 6x=:6 or - 16 X = 1 or — I 2. Solve the equation ax^ -\-bx = G. Solution. '^ ax^ -^ bx= c Multiplying by 4 a, etc. , 4 a^x^ + 4 abx + b^ = 4: ac -\- b"^ 2ax-]-b=± Vi ac + 6^ 2ax = — b ± V4 «c + 6 x = -~ ± — V4 ac + b 2a 2a Solve the following equations : 3. 3x'-^x = 55. 14. x'-2bx = 3h\ 4. 2a;2 + a; = 21. 15. a;- + 3 aaj = 10 a^. 5. 4a;2 + 3a; = 7. le. 6a^-5a;=76. 6. 2ar^-9ro = -4. -^^ 3a^-x = 70. 7. 4a^4-2a;=110. 8. 2x2-6x' = 56. 18. 3x-^ = 26. 9. 5.'^-4. = 288. ^^ 20.-^ = 9. 10. 8ic--6a: = 2. ^ 11. 7a^-4a:=20. ^0. x{2x-A) = x^ + 12. 12. aj2^17aj-18 = 0. 21. aa;2_4^^^^4 13. 2a;2-18x = -40. 22. 2ax^ -bx = 2a + b. 23. a;(4a;+3) = a.-2 + 90. 24. (a;-4)(aj-h6) = 4aj + 66. 156 ELEMENTS OF ALGEBRA. 25. 2x^-5cx = -2c\ 26. x^-2ax = m'-a'. 27. {x-\-ay^ = 5ax — {x-'ay. PROBLEMS. 171. 1. Find two numbers whose sum is 10 and whose product is 24. Solution. Let x = one number. Then 10 - x = the other. 10 X - x^ = 24 x2 - 10 X = - 24 x2 - 10 X + 25 rr 1 x-5=± 1 x = 6 or 4 10 - X :zr 4 or 6 2. Divide 13 into two such parts that their product may be 36. 3. The difference between two numbers is 6, and their product is 160. What are the numbers ? 4. The difference between two numbers is 4, and the sum of their squares is 40. What are the numbers ? 5. An orchard containing 2808 trees had 2 trees more in a row than the number of rows. How many rows were there ? How many trees were there in a row ? 6. A rectangular piece of ground is 5 rods longer than it is wide and contains 500 square rods. What is the length of its sides ? 7. If four times the square of a certain number is diminished by twice the number, there is a remainder of 30. What is the number ? QUADRATIC EQUATIONS. 157 8. The difference between two numbers is 4, and the sum of their squares is 346. What are the numbers ? 9. If 6 times the square of a certain number is dimin- ished by 10 times the number, the result will be 304. What is the number ? 10. ♦ A sum of $ 224 is divided among a certain number of men in such a manner that each man receives $20 more than the number of men in the company. How many men are there ? 11. There is a certain number which, being subtracted from 20 and the remainder multiplied by the number, gives a product of 100. What is the number ? 12. A certain number of men pay a bill of $240. If each man pays $ 1 more than the number of men, what amount does each man pay ? 13. A lady paid $40 for some broadcloth. If the price per yard was $ 6 less than the number of yards she bought, how many yards did she buy ? 14. A man bought some cattle for $ 100. If the number had been one more, the price per head would have been $ 5 less. How many cattle did he buy ? 15. A merchant sold goods for $ 150, gaining a per cent equal to one half the number of dollars which the goods cost him. What was the cost of the goods ? Suggestion. Let 2x= the cost; then, — = the gain per cent, and — = the gain. 100 ^ 16. 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. How much did they cost ? 158 ELEMENTS OF ALGEBRA. SIMULTANEOUS QUADRATIC EQUATIONS. 172. The following solutions will illustrate some methods of solving Simultaneous Quadratic Equations. r a? -|- ?/ = 5) 1. Given < >■ to find the values of x and y. Solution. x + y = 5 (1) xy= 6 (2) (1) squared, a;2 + 2 5C?/ + 2/2 = 25 (3) (2) X 4, 4xy = 24: (4) (3) -(4), X2 - 2 iC?/ + ?/2 n: 1 (5) Square root of (5), x-y =± 1 (6) (l) + (6), 2 ic = 6 or 4 (7) ic = 3 or 2 (8) Substituting in (1), ?/ = 2 or 3 (9) f X 4-y = 2. Given - ^ ; ^, ix'-{-y^ = l to find the values of x = 52 ) and 2/. Solution. x-]-y= 10 (1) x^ + y'^= 52 (2) Squaring (1), ^2 ^2xy-\-y'^ = 100 (3) (3) -(2), 2xy= 48 (4) (2) -(4), x'^ - 2 xy -\- y^ = 4 (5) Square root of (5), x-y =±2 (6) W + (i), 2 a: = 12 or 8 (7) a- = () or 4 (8) Substituting in (1), ?/ = 4 or G (9) ( ^ +2/ 3. Given ] ^ ^ "^^ ^ >- to find the values of ■ = 17) x and 2/. Solution. x + y= 5 ^ (1) 2x2+ ?/2 = l7 (2) From (1), x= 5 — 2/ (3) QUADRATIC EQUATIONS. 169 Hence, 2 x'^ = 50 - 20 y + 2 ?/2 (4) Substituting (4) in (2), 50 - 20 ?/ + 2 2/2 + ?/2 = 17 (5) Collecting terms, 3 ?/2 - 20 ?/ = - 33 (6) Completing the square, etc. , j^ = 3 or 3| (7) Substituting (7) in (1) a^ = 2 or IJ (8) { X -{-y =11 ) 4. Given ] ^ „ ^^ [ to find the values of a; and ?/. t ar — 2/^ = 11 ) Solution. x + y = 11 (1) x^-y^ = ll (2) (2)-f-(l) (Ax. 6), x-y= 1 (3) (l) + (3), 2x = 12 (4) x= Q (6) Substituting (5) in (1), y= 5 (6) Find the values of the unknown quantities in the follow- ing equations : 6. ?/ = o ) ( X' — y (X +.V = 20| (3x + 2/ =18| (aj^-2/-^ = 120 J * ( x'-{-2y' = A3^ 1 4 aj + 4 ?/ = 5 ) I a? + .V = 11 > ' ( 8^77 = 2 ) ' lx'-f- = 33\ ..+,= 8. ..^ + ,-29. ,2. +,=11. .«^-,-102. i2a;2^y2^57; U^_2/= 3) 1 a. + .V=5| ^^ ra. + 2/ = 12| lx'-2xy-y' = 7 ) ' \ xy = 20\ 10. 160 17. 18. 19. 20. 21. 22. 23. ^ x-y= 1 \x'-xy-\-y'' ( aj2 _ 2^2 ^ 20 ELEMENTS OF ALGEBRA. f . I aj + 2/ = 10 j 2 a; H- 3 .?/ = IG ) 12 x"- 2/' = 46 ^ I a; -2/= 4 I ( 0^2/ = 90 I (a;24-2/^ = 181) (Ax''-2y'= 92) 1 xy = lo\ { 25. 26. 27. 28. ^ aj2 4-a'2; + 7/'^=13i' ( ic-?/ = 9) ^ + 2/^3 a h PROBLEMS. 173. 1. The sum of two numbers is 16, and their product is 63. What are the numbers ? 2. The sum of two numbers is 11, and the difference of their squares is 55. What are the numbers ? 3. A is 4 years older than B, and the sum of the squares of their ages is 976. What are their ages ? 4. The distance around a rectangular picture is 56 inches, and its square surface is 192 square inches. Find the length and the breadth of the picture. 5. There are two unequal square fields which together require 100 rods of fence to inclose them. If the sum of their areas is 325 square rods, what is the length of a side of each ? QUADRATIC EQUATIONS. 161 6. The difference between two numbers is 10, and the difference between their squares is 340. What are the numbers ? 7. The sum of two numbers is 7, and the sum of their squares is 29. What are the numbers ? 8. A and B travel a certain distance in 3 days. A travels 10 miles a day more than B, and the square of the distance B travels per day subtracted from the square of the distance A travels per day is 500 miles. What is the entire distance, and how much of it does each travel ? 9. A merchant received $10 for a certain number of yards of linen, and f 9 for 20 yards more of cotton at 10 cents less per yard. How many yards did the merchant sell of each? 10. The area of a rectangular field is 1350 square rods. If its length and breadth were each lessened 5 rods, the area would be 1000 square rods. Find the length and the breadth? 11. The sum of two numbers multiplied by their differ- ence equals 36, and the sum of the numbers is 18. What are the two numbers ? 12. A merchant has 200 yards of silk and velvet. If 50 times the number of yards of silk is subtracted from the square of the number of yards of velvet, the remainder is 400. How many yards are there of each ? 13. A rectangular and a square field joined each other. A side of the square field was half the length of the rectan- gular field, but less than its width, and the area of both fields was 2^ acres. What were the dimensions of each field, if it required 90 rods of fence to inclose them as one field ? milne's el. of alg. — 11. GENERAL REVIEW. I. 174. 1. When a = l, h = 2, c = 4, d = 6, what is the value of 2a + h'-ah + 2c_cd-bc^ d — c-\- ah cd — b 2. Find the sum of 3a{x — y), 4:a{x — y), 2h{x-'y), and ^{x — y). 3. Simplify (aj+ 5) - (x + 10) -[a^ - (3aj + 25) - 10]. 5. Add ax{a-l) + {W-2)^y\ 2(b'-2)-Sy' + 3aa;(a~l), Si/^- 6a.T((X - 1) - 6(62- 2), ^^^j subtract from the result 4 ax (a — 1) + (&- — 2) — 7 2/^. 6. What number added to three times itself equals ab? 7. A man had property costing 4 6 dollars, 7 c dollars, 5 6c dollars, which he sold for f 600. How much did he gain? 8. From 205 aj^ __ 74 / + 89 c^ subtract 35 a;^ - 4 2/' - c- + 15. 9. Simplify x'^ -f y'^ -\x^ - y^ -{2a^ - i f-) -2z\, 10. What number subtracted from a times itself equals a^-l? 11. Multiply a;^ + 2 0^2/ + 4 a^V _^ g ^j?/^ ^ 16 ?/^ by a; - 2 y. 12. Expand (a;+3) (a;- 4) {x -f 4) (a; - 3) {x - 1) (a^ + 1). 162 GENERAL REVIEW. 163 13. Factor 5 x^y — 10 axy -f- 25 xy" — 15 aVy\ 14. Factor it-- -f X — 156, a;- — 15 a? + 56, a;- — 3 a; — 70. 15. If 7 quarts of milk cost 1 cent less than a cents, what will b quarts of milk cost ? 16. With a dollars a man paid for 10 bushels of potatoes at c dollars per bushel, and received 2 dollars in change. Express this as an equation. 17. Divide x'^ + x^y^ + y^ by x^ -f- xy + y\ 18. Factor a*^ - ^>^ 125 a.-^ - 64 y^, a;^-81. 19. Divide 2 a;^ - 9 x^ - 8 a;^ - 1 by x^ + 3 a;^ _^ 3 .^ _|. j^^ 20. Find the value of x in the equation bcx — ab = — dx -1. 21. Factor 4a2 + 36a + 81, 100x^-25y^, 9x^-A2xy + 49/. 22. A merchant sold 6 pounds of coffee at b cents a pound, 5 pounds at c cents a pound, and b pounds at a cents a pound. What was the average price received per pound for the coffee ? 23. Find the value of x in the equation ax — a^ — b^ = 2ab- bx. 24. Factor 8 a;^ + 512, a'b^ - 1, m^ + 27 7i^ 25. Solve (a; - 5) (x + 4) = a;(a; - 5) + 8. II. 26. Write out the following squares : (x + 2yy, (4.a-3by, {2m-5)% (3ab-{-l)% (5 + 2ay, (i^x-ly)% (a2+10)2, {m'-4.ny, (x^-Syf, {3a + 5xy. 164 ELEMENTS OF ALGEBRA. 27. Write out the following products : (a; -\-7){x- 4), (x + 15) (x - 15), {x + 10) {x + 3), {x-12)(x-{-9), {2x+5){2x-o), {m' + 1) (m' ^ 1) , (aj_8)(.T-ll), (a; + 15) (a; -14), (3 a; + 7) (3 a; - 7). 28. Find the highest common divisor of 3a5_48a, 2a%-16b, and 5a'c-20c. 29. Find the highest common divisor of x'-\-6x + 9, x^-\-x~6, Sci^-]-7x-6. 30. Find the lowest common multiple of ar^_j_2a;-3, ar'-3a; + 2, x' + x-6. 31. Find the lowest common multiple of x^-9, a;2 + 9 a; + 18, x'-j-Sx- 18. 32. Eeduce — ^^ ± — to its lowest terms. a^ — b' 33. Eeduce — — ^^-^ — ^^ , to its lowest terms. Qir — 3xy — 10y' 4 2 ^ . 3x-y 34. Simplify /^—i_ + _?_y \x-\-y x-yj x-{-y o., ci- vi? 5 /2a;2 + 12 x''-{-9\ 35. Simplify -j-^x^—± t-j- 36. Simplify ^ ^ + ^ 4(1 + ^) 4(.v-l) 2(2/2-1) 37. Simplify ^(^-^) x ^'^^ + ^^> . 38. Simplify t^:^^ ^ ax -\- x\ 39. Simplify 3a- 4^> _2a - 6 - c 15a - 4c^ ^ ^ 7 3 12 GENERAL REVIEW. 166 40. A can do a piece of work in m days ; B can do it in 7?, days. Express the part that each can do in one day. Express the part that both can do in one day, and the num- ber of days in which both can do the work. 41. Solve ^^i+^+ 10 = '^-^ + ^^±1. 7 5 3 42. Solve x{x - 3) 4- ^ = a!(a; - 5) -f — . 43. Solve x^ — 4, ar— 4 x-\-2 44. Solve "(^^-^)-^(^ + '") = «;. h a 45. A farmer has one third as many horses as cows, and one half as many cows as sheep. If there are a animals in all, how many are there of each kind ? 46. Solve -^^ ^ = ^L^. X — a X — b X — c 47. Solve — ^ + - ■* ^ l-5x 2x-l ^x-1 48. The sum of two numbers is a, and the first divided by m equals the second divided by n. What are the num- bers ? 49. The mth part of a certain number plus 10 equals m times the number minus 5. What is the number ? 50. If a certain number is divided successively by a, 6, and c, the sum of the quotients will be 10. W^hat is the number ? 166 ELEMENTS OF ALGEBRA. IIL Solve the following equations : \l0x-9y = 2l 54. 55. 51. 52. ••^1= 6 x_±_l 4 X — + 2/ = 15 2 10-a^ _ 7/-10 ^ 53. ^ L^ y 2y + 4 2a; + y _ ^' + 13 r 8 56. < 4 ic 4-121/ = 5 6 a;- 3:2=2 [16a;- dz = l C^^'I^ ^ = 62 2 3 4 :17 ^ o ^'^ a^ — 10 Oo. -— — 4 5 12 I3 4"^ 5 ' x-\- y = a' 57. J aj+ 2; = 6 J/+ 2;= c 50- fo:^ 59. 5 2 1-aj l + i» 60. 0? 4-4 a; — 4 a; — 4 a; + 4 61. a^-3aj = 4. 15 10 3 * = ' 25 62. x''-{-15x=34:. 63. 4a;2-a; = 33. 64. a;^ — 2 aa? = m^ — a^. 65. 2aa^— 5a;=2a-|-5. 66. 5 X- 4- 4 a; = 9. GENERAL REVIEW. 167 Solve by factoring : 67. a;2-5a;-104 = 0. 68. a;2-18a; + 72 = 0. 69. x'--12x-\-30 = 5S, 70. What quantity multiplied by itself gives x'^-j-4:xy 71. Extract the square root of 60,025. Find the values of x and ?/ in the following equations : (20.^+ r = 59) Ix -y = 2\ ^3 |a;^ + r = 37| ^^ cx'^2xy-f- = 73^ ' \ Sxy =18) ' X x-y =5^ IV. 76. One half of Tom's money equals ^ of John's, and Tom has $12 more than John. How much money has each ? 77. At a certain election there were two candidates; the successful candidate had a majority of 60, which was 2V ^^ all the votes cast. How many votes did the defeated candidate receive ? 78. A lad spent on July 4th i of his money and 6 cents more for firecrackers, and ^ oi his money and 4 cents more for torpedoes. If that was all of his money, how much had he? 79. A man bought a number of sheep for $225 ; 10 of lem having died, he sold f of the re f 150. How many sheep did he buy ? 168 ELEMENTS OF ALGEBRA. 80. If to the numerator of a certain fraction 1 is added, the value of the fraction becomes 1 ; but if 3 is subtracted from the denominator, the vakie becomes 2. What is the fraction ? Suggestion. Let - represent the fraction. y 81. A man bought 20 bushels of wheat and 15 bushels of corn for $ 36, and 15 bushels of wheat and 25 bushels of corn, at the same rate, for $32.50. What did he pay per bushel for each ? 82. The sides of a rectangular court are to each other as 3 to 4, and their surface is 2700 square feet. What are the lengths of the sides? 83. If I of the value of a carriage is equal to f of the value of a horse, and the value of the carriage is f 20 more than the value of the horse, what is the value of each ? 84. The sum of two numbers is 34, and their product is 285. What are the numbers ? 85. A merchant, after selling from a cask of vinegar 15 gallons more than \ of the whole, found that he had left just 4 times as much as he had sold. How many gallons did the cask contain at first ? 86. The difference between two numbers is 6, and one half of their product equals the cube of the smaller. What are the numbers ? 87. A company of 20 men and women paid a bill of f 75. The men paid f of the bill, and by so doing each man paid f 1 more than each woman. How many men were there in the company ? How many women were there ? GENERAL REVIEW. 169 88. A and B own flocks of sheep. If A sells to B 10 sheep, they will each have an equal number ; but if B sells to A 10 sheep, A will have three times as many as B. How many sheep has each ? 89. What are the two numbers whose difference is 8, and whose sum multiplied by their difference is 240 ? Suggestion. Let x + 4 and x — 4 equal the numbers. 90. A grocer will sell 1 pound of tea, 2 pounds of coffee, and 1 pound of sugar for f 1.00 : or he will sell 2 pounds of tea and 6 pounds of sugar for $1.00; or he will sell 3 pounds of coffee and 2 pounds of sugar for f 1.00. What is the price per pound of each ? 91. How far may a person ride in a stage, going at the rate of 8 miles an hour, if he is gone 11 hours, and walks back at the rate of 3 miles an hour ? 92. I have two purses, one conta^ining gold coins, and the other silver coins. The money in both purses equals in value twice the value of the gold coins ; and i the value of the silver coins, increased by i the value of the gold coins, equals f 150. How much does each purse contain ? 93. A man bought a horse, a cow, and a sheep for a certain sum. The horse and the sheep cost 5 times as much as the cow, and the sheep and the cow cost f as much as the horse. How much did each cost, if the cow cost f 30 ? 94. Find two numbers, such that their product is 21, and their product added to the sum of their squares is 79. 95. A merchant sold some goods for f 96, thereby gaining as much per cent as the goods cost. What was the cost of the goods ? Suggestion. Letx = the cost ; then — = the gain per cent. 100 ^ ^ 170 ELEMENTS OF ALGEBRA. 96. A and B have the same income. A saves ^ of his, but B, by spending $ 100 each year more than A, at the end of 5 years hnds himself $240 in debt. What is the income of each ? 97. A farmer had his sheep in 3 fields. | of the num- ber in the first field was equal to f of the number in the second field, and f of the number in the second field was J of the number in the third field. If the entire number was 434, how many were there in each field ? 98. A boat's crew rows 12 miles down a river and back again in 2|^ hours. If the current of the river runs 2 miles per hour, determine their rate of rowing per hour in still water. 99. If the length and the breadth of a rectangle were each increased by 1, the area would be 120 ; but if they were each diminished by 1, the area would be 80. Find the length and the breadth. 100. A grass plot 9 yards long and 6 yards broad has a path around it. If the area of the path equals the area of the plot, what is the width of the path ? 101. A and B traveled toward each other from two towns 247 miles apart. A went 9 miles per day, and the number of days before they met was 3 more than the num- ber of miles B went per day. How far did each travel ? 102. Three equal square lots have an area of 193 square rods less than another square lot whose sides are each 13 rods longer than the sides of each of the three equal square lots. Find the length of a side of each. 103. A rectangular plot of ground is surrounded by a street of uniform width. If the plot is 50 rods long and 40 rods wide, and the area of the street is 784 square rods, what is the width of the street ? GENERAL REVIEW. 171 V. 104. Find the highest common divisor of o? + a^, (a + xy, and a- — x^. 105. Find the highest common divisor of 9 x- — 1, 9a;-f-3, and (3x + l)'. 106. Find the highest common divisor of x^ — ^x — 'S, a^ + 4 a; 4- 3; and x^ — bx — Q. 107. Find the lowest common multiple of 3{x^ -\-xy)^ 5(xy — y^), and 6(x- — y-). 108. Find the lowest common multiple of a^ + 11a; + 30, x" + 12a; + 35, and a;^ + 13a; -f- 42. 109. Find the lowest common multiple of 12xy{x^ — i/^), 2 x{x + yY, and ^y{x — y)\ 110. Reduce ^ !>^ ~~ „\ to its lowest terms. 4:(a^b — ab^y a^ 4-1 111. Reduce — — ^ „ ^ r to its lowest terms. a^ + 3a2 + 3a-f 1 Change to equivalent fractions having their lowest com- mon denominator : ,.o 8a;-f2 2a;~l , 3a;4-2 113. ^:=^, .--^. and --^ + 4 iB2_4 a.^ + a;-6 x'-^^x^^ 114. -A_, _ii^, and -^ a;-f 1/ iv- — 2/^ ic^ + 2/^ ,,.3 5 -, 2a; 115. J ? and 1 + a; 4-f-4a; 1 — ar^ 172 ELEMENTS OF ALGEBRA. Simplify : a — X a-\-2x (a — x)(a -\-2 x) 117. ^+ y 2a? x-y — xr y ^-\-y y(^^-y^) Solve : ■•■•g ^ — \^-xy-[-y^ = l^\ 165. The head of a fish is 8 inches long. The tail is as long as the head and ^ of the body, and the body is as long as the head and tail. What is the length of the fish ? 166. A tree is broken into three pieces. The part stand- ing is 8 feet long. The top piece is as long as the part standing and \ of the middle piece, and the middle piece is twice as long as the other pieces. How high was the tree ? GENERAL REVIEW. 175 167. A person, being asked the time of day, replied that it was past noon, and that ^ of the time past noon was equal to \ of the time to midnight. What was the time ? 168. A yacht, whose rate of sailing in still water is 12 miles an hour, sails down a river whose current is 4 miles an hour. How far may it go, if it is to be gone 15 hours ? 169. Three men engage to husk a field of corn. The first can do it in 10 days, the second in 12, and the third in 15 days. In what time can they do it together ? 170. Fifteen persons agree to purchase a tract of land, but three of the company withdrawing, the investment of each of the others is increased $ 150. What is the cost of the land ? 171. C and D have the same income. C saves -^j of his, but D, by spending f 65 more each year than C, at the end of 6 years finds himself $60 in debt. How much does each spend yearly ? 172. I sold a bureau to A for \ more than it cost me. He sold it for $ 6, which was | less than it cost him. What did the bureau cost me ? 173. A man agreed to work 16 days for $24 and his board, but he was to pay $ 1 a day for his board every day he was idle. If he received $ 14, how many days did he work? 174. A man can saw 2 cords of wood per day, or he can split 3 cords of wood after it is sawed. How much must he saw that he may be occupied the rest of the day in split- ting it ? 175. A carriage maker sold 2 carriages for $300 each. Did he gain or lose by the sale, if on one he gained 25 per cent, and on the other he lost 25 per cent ? 176 ELEMENTS OF ALGEBRA. ^ 176. A teacher agreed to teach 9 months for $ 562^ and ■ his board. At the end of the term, on account of 2 months' ■ absence, he received only $ 409^. What was his board ; worth per month ? 1 177. How many acres are there in a square tract of land ] containing as many acres as there are boards in the fence ] inclosing it, if the boards are 11 feet long, and the fence is \ 4 boards high ? : 178. The area of a square figure will be doubled if its ;j length is increased 6 inches and its breadth 4 inches. Find j the length of the side of the square. ] 179. A certain iron bar weighs 36 pounds. If the bar ; had been 1 foot longer, each foot would have weighed i a i pound less. Find the length of the bar and the weight per ; foot. - ■ 180. The fore wheel of a wagon makes 6 revolutions i more than the hind wheel in going 120 yards. But if the | circumference of each wheel were increased 1 yard, the fore ; wheel would make only 4 revolutions more than the hind | wheel in going the same distance. What is the. circumfer- ] ence of each wheel ? ■ 181. How many quantities each equal to a^ — 2a + l . must be added together to produce 5a^ — 6a^+l? \ 182. There is a cistern into which water is admitted by ; three faucets, two of which are of the same size. When ; they are all open the cistern will be filled in 6 hours, but \ if one of the equal faucets is closed the other two will \ require 8 hours and 20 minutes to fill it. In what time ; can each faucet fill the cistern ? ^ 183. When or -{- y- = 2i) and ocPy -f xy^ = 300, what are the ! values of x and y ? I QUESTIONS FOR REVIEW. 175. How does the algebraic solution of a problem differ from the arithmetical solution ? What are the letters used in algebra commonly called ? Define unknown numbers or quantities. What letters are used to represent unknown quantities ? What is an equation? Give the sign of equality ; the sign of deduction. What is an algebraic expression ? Illustrate. How do the uses of signs in algebra differ from their uses in arithmetic ? What is a power ? Give an illustration. What is an exponent ? Give an illustration. How are powers named ? What, also, is the second power called ? What, the third ? What is a coefficient ? When no coefficient is expressed, what is the coefficient ? Give an illustration. How should quantities inclosed in parentheses be treated ? What is an algebraic term ? Distinguish between positive and negative terms. Illus- trate each. Distinguish between similar and dissimilar terms. Illus- trate each. What is a monomial ? A polynomial ? A binomial ? A trinomial ? Give an illustration of each. Explain what is meant by using the signs + and — as signs of opposition. milne's el. of alg. — 12. 177 178 ELEMENTS OF ALGEBRA. When positive quantities are added, what is the sign of the sum ? What is the sign of the sum when negative quantities are added ? How is the sign of the result determined when both positive and negative quantities are added ? What kind of quantities can be united by addition into one term ? How may dissimilar quantities be treated in addition ? What are the two cases in addition ? Give the rule. Define known numbers or quantities. What letters of the alphabet represent them ? Instead of subtracting a positive quantity, what may be done to secure the same result ? Instead of subtracting a negative quantity, what may be done to secure the same result ? Give the three principles in subtraction. Give the cases in subtraction. When is it necessary in subtraction to inclose the coeffi- cient of the answer in parentheses ? Give the signs of aggregation. What do they show ? How may the subtrahend sometimes be expressed ? When the subtrahend is inclosed in parentheses and pre- ceded by the sign minus, what must be done when the subtraction is performed ? Give the two principles relating to parentheses, or other signs of aggregation. How does the sign plus before parentheses aft'ect the quantity inclosed ? When a quantity is changed from one member of an equa- tion to another, what change must be made in its sign ? What are the members of an equation? What is the first member ? The second member ? Illustrate. What is an axiom ? Give the six axioms. QUESTIONS FOR REVIEW. 179 What is transposition? Give the principle relating to transposition. Give the rule. How may the value of the unknown quantity obtained by solving an equation be verified ? Illustrate the process by an appropriate example. When a positive quantity is multiplied by a positive quan- tity, what is the sign of the product ? When a negative quantity is multiplied by a positive quantity, or a positive quantity by a negative quantity, what is the sign of the product ? When a negative quantity is multiplied by a negative quantity, what is the sign of the product ? Name the four ways of indicating multiplication. Give the three principles of multiplication. Give the cases in multiplication. Give the rules. What is the principle relating to the square of the sum of two quantities ? What is the square of a; -f- y ? What is the principle relating to the square of the dif- ference of two quantities ? What is the square of x — y? What is the principle relating to the product of the sum and difference of two quantities ? What is the product of (x + y){x-y)? What is the principle relating to the product of two binomials which have a common term ? What is the product of (x-\-3){x-{-2)? What are simultaneous equations ? Illustrate. Define elimination. Give the principle relating to elimi- nation by addition or subtraction. Give the rule. Illustrate the method of elimination by addition or sub- traction by the solution of a problem. What is the sign of the quotient when a positive quan- tity is divided by a positive quantity ? What is the sign of the quotient when a negative quantity is divided by a negative quantity ? 180 ELEMENTS OF ALGEBRA. What is the sign of the quotient when a positive quan- tity is divided by a negative quantity, or a negative quan- tity divided by a positive quantity ? How is the exponent of a quantity in the quotient found? Name and illustrate the two ways of indicating division. G-ive the principles relating to division. Give the cases in division. Give the rule under the first case. Give the rule under the second case. What is a factor ? Illustrate by giving a quantity and its factors. What is factoring ? Give the rule for factoring a poly- nomial all of whose terms have a common factor. How may a polynomial be factored when only some of its terms have a common factor? What is meant by " square root " ? Give the rule for separating a trinomial into two equal factors. When can a binomial be resolved into two binomial factors ? Give the rule for so factoring a binomial. What is a quadratic trinomial ? Illustrate, by solving an appropriate example, the method of factoring a quadratic trinomial. What quantity will divide the sum of two cubes ? How, then, may the sum of two cubes be factored? Give the factors of or^ -f if. What quantity will divide the difference of two cube^ ? How, then, may the difference of two cubes be factored? Give the factors of a^ — y\ Describe the ambiguous sign. What does it indicate ? Explain the solution of the four kinds of equations by factoring. What is a common divisor? Illustrate the use of the common divisor by giving two or more quantities and a common divisor or factor of them. QUESTIONS FOR REVIEW. 181 What is the highest common divisor or factor of several quantities? Give the principle relating to the highest common divisor. Solve an example in which the quantities are monomials. Solve one in which they are polynomials. What is a multiple of a quantity ? What is a common multiple of several quantities ? Illustrate. What is the lowest common multiple of several quantities ? Give the principle relating to the lowest common multiple. Find the lowest common multiple of two or more quanti- ties which are monomials. Find the lowest common multi- ple of two or more quantities which are polynomials. Define a fraction ; an entire quantity ; a mixed quantity. Illustrate each. What is meant by the sign of a fraction ? How should it be interpreted ? Give the principle relating to reduction of fractions. When is a fraction in its lowest terms ? Name the five cases under reduction of fractions, and illustrate by solving examples. Define and illustrate similar fractions ; dissimilar frac- tions. AVhat is meant by lowest common denominator ? Illus- trate. Give the principles relating to the lowest common denominator. Give the rule for reducing dissimilar to similar fractions. What effect has it upon the equality of the members of an equation to multiply both members by the same quantity ? How may an equation containing fractions be changed into an equation without fractions ? What is meant by clearing an equation of fractions ? Give the principle. Give the rule. If a fraction has the minus sign before it, what must be done when the denominator is removed ? 182 ELEMENTS OF ALGEBRA. If a fraction is multiplied by its denominator, what is the effect ? What kind of fractions can be added or subtracted ? What must be done to dissimilar fractions before they can be added or subtracted ? Give the principles. In what two ways may a fraction be multiplied by an ^entire quantity ? In what two ways may a fraction be divided by an entire quantity ? How should entire and mixed quantities be changed before multiplying ? How may an entire quantity be changed to ^ fractional form ? When may cancellation be used in multiplication ? Solve an example in multiplication, making use of factor- ing and cancellation. Solve an example in division, making use of factoring and cancellation. Explain the reason for inverting the terms of the divisor in division of fractions. How may an unknown quantity be eliminated from two simultaneous equations by comparison ? How by substitu- tion ? Give the rules. Illustrate by solving examples. Give the dehnition of : an equation ; members of an equa- tion ; transposition ; axiom ; simultaneous equations ; elimi- nation; clearing of fractions. What is meant by the degree of an equation ? Define and illustrate the term simple equation ; quadratic equation. What is the degree of the equation x -\- a = b? Of the equation x^ -{-2x = 4:? Of the equation x + xy = 6 ? Describe the method of solving simultaneous equations containing three unknown quantities. Solve a set of simul- taneous equations containing three unknown quantities. QUESTIONS FOR REVIEW. 183 What is a power of a quantity ? How many times is a quantity used as a factor in producing the second power? The third power ? The ?ith power ? What is involution ? Give the principles. Give the rule. How is a fraction raised to any power ? What is the sign of any power of a positive quantity ? What is the sign of any even power of a negative quan- tity ? Of any odd power of a negative quantity ? What is a root of a quantity ? How are roots named ? What is the radical sign ? What does it indicate ? What is the index of a root ? When no index is written in the opening, what root is indicated ? What is evolution ? Give the principles relating to evolu- tion. Illustrate their application. Name the cases of evo- lution. Give the rule under each case. How is the root of a fraction found ? Explain the method of extracting the square root of a polynomial by solving an example. Give the principles relating to square root of numbers. Give the rule and solve an example. How is the cube root of a polynomial found ? Illustrate by solving an example. Give the rule. Give the principles relating to cube root of numbers. Give the rule and solve an example. Define a quadratic equation; a pure quadratic equation. What is the root of an equation ? Give an example of a pure quadratic equation and solve it. What is an affected quadratic equation? Give an ex- ample. How many methods are given for completing the square ? Solve an example by the first method. Give the rule under the second method and solve an example. Give four types of simultaneous quadratic equations. ANSWERS. Page 8. — 1. 30, John ; 10, James. 2. 20 bu. ; 40 bu. 3. $ 2000, A ; $ 1000, B. Page9. — 4. 5. 5. 30 yr., father; 10 yr., son. 6. 15 and 60. 7. ^200, horse ; $100, carriage. 8. 100 pear; 200 cherry ; 400 apple. 9. 80yr.,A; 40yr., B. ; 20yr.,C. 10. 7 and 42. 11. 5. 12. 3, Jane ; 6, Hannah ; 30, Mary. Page 10. — 13. 4 cents, ink; 20 cents, pens; 72 cents, paper. 14. $6, hat; -^ 12, trousers ; $42, overcoat. 15. $5, first; $10, second ; $ 15, third ; $20, fourth. 16. 20 sheep, 1st ; 40 sheep, 2d ; 60 sheep, 3d ; 80 sheep, 4th. 17. 7, 42, 70. 18. $1000, A ; $100, B ; $400, C ; $4800, D. 19. $12, silver watch ; $120, gold watch. 20. 360 votes, A ; 90 votes, B. 21. $25, cow; $50, wagon ; $ 100, horse. Page 11. —22. $2000, A ; $6000, B ; $6000, C. 24. 6 acres, potatoes ; 30 acres, oats ; 150 acres, wheat. 25. $ 1200. 26. 7 hens ; 14 ducks; 98 chickens. 27. 9 and 27. 28. 9 gal.; 36 gal. Pagel2. — 29. 700, smaller ; 5600, larger. 30. 7. 31. 5 days. 32. 11 yr., son; 55 yr., father. 34. 15. 35. 3. 36. 7 cows ; 3 horses. 37. 4 bu. ; 14 bu. Pagel3. — 38. 24. 39. 18. 40. 18 and 27. 41. 12. 42. 20. 43. 9 marbles. 44. 12. 45. 80, 1st; 160, 2d ; 20, 3d. 46. 6 and 30. 47. 20, 40, and 60. 48. $200, A; $100, B; $600, C. 49. 126. Page 14. — 50. 24, Henry; 8, James. 51. 12^, May; 4^, Julia; 2^, Hattie. 52. 30. 53. 500, history ; 1000, science, 400, fiction. 54. 70. 55. $15, A; $9, B; $3, C. 56. 40. Page 15. — 58. 10 apples. 59. $ 5, 1st ; $ 10, 2d ; $ 35, 3d. 60. 30, 1st ; 30, 2d ; 40, 3d. 61. $500. 62. 48. 64. 14. 65. 4, 16, and 14. 66. 5 and 20. Page 16. —67. $ 300, 1st son ; $ 100, 2d ; $ 200, 3d. 68. $ 125, 1st ; $ 125, 2d ; $62S 3d ; $ 187 J, 4th. 69. 5^, Emma ; 10 ^, Anna. 70. 10 mi. ; 8 mi. " 71. 60. 73. $ 500, 1st month. Page 17. — 74. llf^mi., stage; 34^ mi., steamboat; 277f mi., rail- road. 75. 12, Harry ; 4, Edward. ' 76. $ 100. 77. $ 2000, lot ; $ 4000, house ; $ 1000, furnishing. 78. 10 trees in each of the 5 rows ; 20 trees in each of the 8 rows. 79. $ 3 per day ; $ 1 per day board. 80. 40 yr. 81. $3. 184 ANSWERS. 185 Page 18. — 83. 4 quarter dollars ; 8 half dollars. 84. $ 1.25, man ; $ 1.25, wife ; $ .25 son. 85. 30 rods, length ; 15 rods, breadth. 86. 30 ft. left standing ; 15 ft. broken off. 87. $400, 1st ; ^ 200, 2d ; $ 200, 3d. 88. $ 2000, 1st yr. ; $ 8000, 2d yr. ; $ 32,000, 3d yr. Page 22. — 1. 72. 2. 480. 3. 72. 4. 360. 5. 13. 6. 10. 7. 5. 8. 768. 9. 8. 10. 9. 11. 20. 12. 21. 13. 82. 14. 78. 15. 25. 16. 16. 17. 45. 18. 8. 19. 4. 20. 18. 21. 586. 22. 35. 23. 37. 24. 26. Page 27. — 1. 18 x. 2. 24 ab. 3. - 20 mn. 4. -25xhj. 5. Wax. 6. -28 5c. 8. 10a. 9. xhf. 10. - mx. 11. 5?/. 12. 6 m. 13. Sbc. 14. 10 xy. 15. x^z. Page28. — 16. Snj^n. 17. 11 a-b. 18. 0. 19. 9(aby\ 20. 7{a-b). 21. 0. 22. 6(x + 2/). 23. 10(a-^by\ 24. 6cd. 25. (xi/)^. 2. 5a -7 6. 3. 8x4- 7?/ -2^. 4. S xy -\- 6 y'^ -\- S z\ Page 29. — 5. 4 c + 7 d. 6. 6 n - 2 m. 7. 4 a + 6 6 + c. 8. 3 x^^ + 10 xz' - 3 xy. 9. 5 ^^ - 4 ?/. 10. 4 a-6 ' - 5 c2 - a7. 11. 2a6-l. 12. 12a-2d. 13. 7 m + 6 ;i-x+5?/. 14. 6?/ + 4^ + 2?(7. 15. -x:hj-2xyK 16. llx'» + ?/". 17. 9n!/AV-3ed 18. 9mH3m«. 19. 22 ax^ - 28 a?/^ + 9 az^. 20. 8 a^b + 8 a^^^c - 20 5V/ + 8. 21. -10xH17ax-^-8a--'x + 9a^. 22. (24-a-2 5)(c+fZ). 23. 2m. Page 30. — 24. 10 ax + 11 x. 25. 5(x + ?/)+ 9^;- 4. I. (a+ 5 + 8) barrels. 2. (7>i + fZ+6) cents. 3. (2 c-f 2 n) dollars. 4. (a + 5 + 3) miles. .Pa?-: 31.— 5. (2c + 400) dollars. 6. 5 + 1; 5 + 2. 7. 4,1 ?7i years. 8. (c + fZ + 75) dollars. 9. 7 x ; ax. 10. (5 + c + d + 2) pounds. II. (m + n + _p + pr + r) dollars. 12. (a + 5 + d) dollars. 13. (a + 5 + c + 25) dollars. Page34. — 3. 15 x. 4. 4a5. 5. 5m2. 6. —4xy. 7. -2x^yH. 8. -bmyi. 9. 4x+4?/. 10. bab-c. 11. 45-4d. 12. 3x + 2i/. 13. x + ^. 14. 4a + 35 + c. 15. 2a- 5. 16. 2 x'^ - 5 ^/-^ - 4 x?/. 17. (Jxy-\-2z. 18. (a2 + 52)-3c-^ + 2(Z. 19. 3a(p + g)+5. 20. 4 a; - 3 y + 1. 21. 4 x' + ?/ + 5 ^ - 3 v. 22. 5 x - 3(^ + 2) + 2. 23. -2a- + 5-^- 3c-^+4d^. Page 35. — 2. a%. 3. IQx'^yz. 4:. y-6z. 5. 3x^+13z/^ 6. — 6 5 + 2 c. 7. 6 a5 - 8 c + 2 d 8. - x"^ - 8 xy + 2 x. 9. 5 ax -6y-\-2z\ 10. Sx^ -29x^y^ + Sxy. 11. - 2 rt5 + 5 5'^ - 8c4. 12. 6x-i/ — 6xy"^-3x?/-3. 13. 3x'»+4x'"?y"+4?/". 14. -4(m-\-n^) -4(m2 + n). 15. 6x^ + (yy^ - Sz^ + ivK 16. -3(p + g) -8(r+s)-10. Page 36. — 17. 20 mhix^- + 12 a25c2 - 4 abd. 18. - a^ + c^ - 6 5'^. 19. -Sxy. 20. x^ + 2xV-2?/*. 21. Ox^y^^ - 11 z/^^. 22. - x*" — 6 x"^'" + 5 2/"*. 23. 5 5x^ + 3 ay'^ + 9 — 2 cy. 24. 4 m'^ + mn -9n2-22. 25. - 2 a-'x - x^^?/ - 3. 26. llxV + 4?/2 - ^2 _ 5, 27. ic_?/_0_i<. 28. 5 a252- 17 5c + 952-4. 30. (a - 2)x + (5 + 2)y. 31. (2 c- 5)x- (a + c)?/ + (a + 3)^. Page 37. — 32. (c - 6)x - 2 a5^ + (4 a^ - 3)52. 33. (2 a - 3 c) (ic_i^) + (4a5_2a)x. 34. (a-5)x+(5-c)?/+(c-(Z)^. 35. (3-5c)x 186 ELEMENTS OF ALGEBRA. -^(' + a)y-\'iS-a)z. 36. 37. - bx + cp. 38. (7 39. (m— mn + n)x^ -i-(m — n + 4)1/ (X 4- ?/) + (« + 6) (X - y) + (c + d). (6a- c)i/ + (2c -^ a)z-}- (6 -\- d)x. '2a)x' + (5 a - 8)?/-^ + (H + ab)z\ )if'^+(l - a)z\ 40. (a- h) )x. 41. m — n. 42. X — 1, X 4- L 43. (125 - a) dollars. 44. 46. X = greater, x — b = less. 48. (a — m — n — 100) dollars. Page 38. — 49. (50 - 25 - a 51. (5 + d + c -/) dollars. (0 6 41. m — n. 25) cents. 47. (d (a-h) X — 1, X 4- 1. 45. 80 - X. /) — c) dollars. ■ c) dollars. 50. (8 a - 40) dollars. Page 39. — 2. 9. 8. 10. 16. Page 40. — 13. 20. llm-4 7i. 21 24. C-2& + 8. 28. 2 X + 2 ?/ + 2 ^. 32. 4x. 33. 8x4-45 Page 44. —2. x = 6. 3. 11 11. 4. 25. 14. 21. x'^ — 11 ax. 4. 4. 5. 6. 0. 7. 1. 8. 17. 15. 22. 25. la-hh. 26. 29. 4^2- 5a + ?/ + 1. 34. «-6. X = 7. 4. X = 10. 16. 17. 17. 8a- + 8?)' + 2a&. Zh~ a-Q. 30. 8w. 31. 35. 2a + 8c. = 2. 5. x = 8. 7. 19. Ux. 23. 8x?/-5. 27. 5x. 2 x'^ 4- 2 2/-^. 7. X = 2. 8. X = 7. 9. X == 4. 13. X = 5. 14. X = 3. 15. X =r 2. 19. x = -l. 20. x = 4. 21. 24. X = 6. 25. X = 4. 26. x = 7. 10. X- 8. 11. x= 5. 16. X = 1. 17. X = 2. x = 9. 22. x = 2. 27. X = 0. 28. X = 5. 6. 12. 18. 23. x = 3. X = 9. X = 2. x = 6. 9. X = 12. 30. x = 8. 31. x=:l.' 32. x = -8. 33. x = 5. 34. x = 4. 35. x=15. 36. x = 4. 37. x = 2. 38. x = 8. 39. x == 2. 40. x = 4. 41. x = 8. 42. x = 15. 43. x = 12. 44. x = 7. 45. X--4. 46. x = 5. 47. x = - 1. 48. x = 5. 49. x = 9. 50. X = 6. 51. X = 10. 52. x = ^. 53. x = 9. 54. x = 8. 55. x = 4. 56. X =3 10. 57. x = .3. Page 45.— 1. 4. 2. 12. 3. 2. 4. 14. 5. 5. 6. 87 cents, 1st ; 27 cents, 2d ; 26 cents, 8d. 7. 8 cents, youngest ; 7 cents, next; 10 cents, oldest. 8. 10 and 24. 9. ^2000, B ; f 6000, A. 10. % 12. 11. 5 miles, down hill ; 18 miles, level ; 9 miles, up hill. Page 46. — 12. $85, poorer horse ; % 115, better horse. 13. 100 gallons, first pipe ; 200 gallons, 2d pipe ; 800 gallons, 8d pipe. 14. 825 cattle. 15. % 5, chain ; % 55, watch. 16. 18 women ; 22 men; 50 children. 17. $42,800, cargo ; $77,200, steamer. 18. $600, 19. 8 7500, house ; $ 2500 lot. 4. -40 a. 5. -18 a. 6. - llOx. 7. 108 x. 10. -216x^. 11. lOx'vV. 12. 45 xV- 126m%3. 15. _ 125am 16. 4(a + ?)). 6 ax + 4 a?/ — 10 az. 19. 6 x^y — 6 xy^z. ~ x.y^. 21. a3& + 2 a-^?>2 _|_ ^53. 22. -18m% clerk; $1800, brother. Page 50. — 3. 40. 8. -42 6. 9. 51 c^. 13. - 64 cW^zK 14. 17. -20(x-?/). 18. 20. - 12 (f'xHi + 20 xhf- - 21 m'^n' — J5 mn^. ■15()a^6'^4-48a;^6' 23. 9x^-18x5 + 45x4 + 9x^^-27x2. 24. 108a7>^ 72 a262. 25. 2 ax^ — 6 ax-y — 6 ax2^ + 2 axy'-z^. '-h^. 27. 1 2 m.-^n - 40 m.^n^ - 82 mn"^. 28. — 5x^ — 5x^— 5x'^— 5x2— 5x. -60a55^c + 42a^&2. 29. Page 51.-4. + 8 a&2 + 6=i. 7 10. 9a2-49&?. ^ + 2 xy + ?/2. 5. 4a2-25 6^. 8. x2-6x 11. 6x2 -9x- 6. 18 a462 _ 54 a^b^ + 45 a262c2 m2 - n\ 6. a^ + S a^-b -40. 9. 6?/2 + 18?/^ + 602. 12. 6a2+rt&- 1562. ANSWERS. 187 13. 6 m^ + 17 mn+ 12 n^. 14. 20)/^-S2yz-\-V2z^. 15. 6 62+ 6c- 40 c-'. 16. 24 x^ - 148 a: - 80. 17. 20 m^ - It) mn +25 mi/- 20 ntj. 18. x^ + 7 .T?/ + 10 ij\ 19. 1 - x^. 20. .r^ - y^ -2ij -\. 21. x^ + 8 xy -Si/K 22. 2rt^ + a5 -4flc + 2?>c - 6^ 23. 12.r;-2 + 13.x ~ 14. 24. 24 rt V?», + 48 ac' + o ac'-m + c ^ 25. 1 2 x -?/- — 48 ?/ '. 26. x"^ + 2 x// + ,V^ — x^ — yz. 27. «■- — 2 rtc + c- —ah + 6c. 28. 2 a- — ax - at/ + xy - x\ 29. 2 x- + 13 x?/ - x - 30 // + 15 ?/2. Page 52. — 30. 24 x^z^ + 4 x//-^ - 8 yK 31. m^ + 2 ijj - ;//' + 1 . 32. 25x- - 25x- 30. 33. Dx^ - 33xV' + 28 ?/*. 34. a ^iah + 6rt'^62 + 4 a6H 6^. 35. 8 x^ - 30 x% + 54 xy^ -21 y^. 36. m' - n\ I. 2. 2. 4. 3. 40 yr., father; 10yr.,son. 4. 25 '/, Samuel ; 15^, John. 5. ^1500, A ; ,1^500, B. 6. 10 yr. hence. 7. .f 12. Page53. — 8. 2 or 50. 9. ^xhf-z\ 10. c?/ cents. 11. {a,b-\-i)a) miles. 12. (4 a —8) animals. 13. (20 a — 3 c?) dollars. 14. 12 r? days. 15. 2 h dollars, A ; lU dollars, B. 16. 20 m cents. 17. (12 6 - ah) dollars. 18. (600 - 40 c) gallons. 19. 10 ac dollars. Page 54. — 1. x^-\-2xy -\- if. 2. 6'^ + 2 5c + c^. 3. w'^ + 4 m + 4. 4. a^ + 2 ax + x-. 5. x"^ + 6 x + 9. 6. ?/'^ + 2 ?/ + 1. 7. 4 x- + 4 xy + ?/-^ 8. m'^ + 4 m;i + 4 7^■^ 9. a- + a6 + 6-. 10. 4 x'^ + 12 xy + 9 ?/2. 11. a' + 8 a6 + 16 h'^. 12. 4 jti'^ + 8 7>iH + 4 ii\ 13. 4 x>* + 20X+25. 14. 9??i*^ + 6m+l. 15. 4 a2+ 20a6 + 25 /A 16. a* + 2 a-7)2 + h^. 17. x^ + 6 X- + 9. 18. 4 m^ + 12 mhi^ + 9 «*. 19. a 62 + 4a6c + 4c2. 20. 4 xV"^ + 4 xy^- + ^^. 21. y^-\-Sy-z^-^l6z*. J32. x+16xH64. 23. 25aH70 a6 + 49//^. 24. 16a* + 24a'^//^ + 9 6*. Page 55. —1. a- - 2 ax + x^. 2. h- - 2 he + c^. 3. ?»"^ - 2 ??i;i + w^. 4. x- - 4 X + 4. 5. ?/- - 2 ?/^ + ^2. 6. a^ - 6 ah + 9 62. 7. 5-2 _ 4 ^^. _^ 4 (.-2. 8. 4 x2 - 8 x?/ + 4 ?/^. 9. 62 - 10 6 + 25. 10. ?/2-2?/ + l. 11. a6i-4a6+4. 12. x2-8x+16. 13. 4a2 - 12 a6 + 9 62. 14. m^ - 4 mn + 4 ?^2. 15. 4 62 - 16 hd + 16 (^2. 16. a^ - 4 a262 + 4 6^. 17. 6-c2 - 2 hcxy + X-Y2. 18. 4 x* - 20 x:^y'^ + 25 y». 19. 4 a- - 4 ac + cA 20. 9 ??i* - 6 m2 + 1. 21. 9 m2n2 - 24 mn +16. 22. ?/* - 12 ?/2 + 36. 23. 16 x* - 40 x^'i/^ + 25 yK 24. a262-4a6c2 + 4c^. Page 56. — 1. a2 - 6". 2. m^ - n^. 3. a- - x2. 4. 4 a2 - 62. 5. 4x2-?/-^. 6. a2-16. 7. 4 w2 - 9 ?22. 8. 2/2-1. 9. x2 - 2.'S. 10. 4 - ifK 11. a-62 - 9c2. 12. 4.m'^ -4 n^. 13. 6^ - 4 <•'-. 14. 9x2 -64 2/2. 15. X2-100. 16. lY^c' - e'p. 17. 9x^-4?/*. 18. 25a2-9x2. 22 16?/^ -49^2, Page 57. — 1. x 4. x' — X — 42. 5. + 100. 8. x2 + 4 II. x2- 12 x + 35. 14. x2-9x + 20. Page 58. — 15. x2 + 9 x - 22. 16. x2 - 29 x + 100. 17. x2 + 20 x + 75. 18. x2 - 9 X + 18. 19. x2 + 3 x - 18. 20. x2 + 21 x + 108. 21. x2 + 2 X - 120. 22. .x2 + 2 xj - 8 ?/2. 23. x2 - 8 ax + 7 a2. 24. x2 + 16x?/ + 60?/2. 19. a^ - hK 20. mhi^ - 16. 21. x2 - 3(J. 23. 9 X' - 16 yK 24. 4 a"62 - 25 C ^•- + 7 X + 12. 2. X- + 4 X - 5. 3. x- - 5x + r. x' + 15 X + 50. 6. x2 - 10 X - 39. 7. x2 + 25x x-32. 9. x2+ 12X + 27. 10. X' - 6 X - 72. 12. x2+ lOx-56. 13. a:-2 + 7 X - 8. 188 ELEMENTS OF ALGEBRA. Page 60. —2. x = 4: -, y = 2, 3. x = 6; y = 1. 4. x -- 5 , y = 4. ^. x=l; y = S. 6. x=^S; y = 7. 7. x = 5;?/ = 7. 8. a;=:9; ?/ = 7. 9. ic = l;^ = 5. 10. x = 6; y = 1. 11. x =^S ; y -2. 12. x = 6; y = 13. IZ. x = S ; y = 91. U. x = 2 ^y = 2. 15. x = 8 ; ?/ = 1. 16. ?/ = 12 ; 2 = 2. 17. ?/ = 4 ; ^ = 5. 18. x = 5 ; ^ = 8. 19. x = 2; y=l{. 20. x=15;2/z=15. 21. x=4:;y = 2. 22. 2/ = 3 ; 2! = 13. 2Z. X = 2 ; y = - 2. 2^. x = S; y = 2, 25. x = 6 ; ?/ = L Page 61. — 1. X = 6 ; 2/ = 4. 2. x = 8 ; y == 6. 3.1^ 100, A ; $200, B. 4. 10 yr., son ; 40 yr., father. 5. 10 marbles in one ; 15 in other. 6. 60 A. at $ 30 ; 40 A. at $ 40. 7. 9 oranges : 4 pears. 8. $300), A; $2000, B. 9. 5 and 7. 10. $ 100, horse ;$ 8o, cow. Page 62. — 11. $ I \, man ; $ 1, wife. 12. 20 and 6. 13. 6 one- dollar bills ; 12 two-dollar bills. 14. $ 2, velvet ;$ 3, broadcloth. 15. 3 13, A ; .$ 11, B. 16. 5 cents, pad ; 3 cents, pencil. 17. $ i, men ; $ 1, boys. 18. $ U, sawing ; $ ^, splitting. Page65. — 3. 2. 4. 8. 5. -2. 6. -5?/. 7. 5 a6. 8. - 3 m?i. 9. 2 a/jx. 10. - 3 y'-z. 11. - 3 x^. 12. 3 c. Page6S. — 13. 5x'. 14. - xV^. 15. 2xV'- 16. -4x. 17. 4x3. 18. -DC<1. IS. -3X.2-. 2D. 8?7^^ 21. 60 xz'. 22. 5x^. 23. ax -2xy. 2i. 3xH-5^-^. 23. 2 a^h' -j- 1 a. 26. ^'^ - x^; + 3. 27. 2 c — Id. 23. 2 x^ - X — 3 V. 29. 4 7}i;i - 3 m — 2 ?i. 30. 5 bxy — 3 ax' -ley-. 31. X - 4 ?/^'^ H- 5 a^^ 32. Ox*^ - 2 x^ + 4x + 5. 33. 3 a 6c + 4c-2?>c\ 34. iJ9m^4- 20m'-28m - 23. 35. bxhjz'-z -2xz'^-(5x:\/\ 36. x-2xH3x5-4x'. 37. 2 x'^?/H 4 x^^ - ;j ?/i -f 5. 38. 3x - 2 ?/ + 4ax-^ — 1. 39. x^-^x^z — y^z-^xy'z^ — yz^ 40. 10 xV + 7 x?/ — 3 ax^;- — 4 /)?/^^. 41. 2 ?ix' — 3 x?/ — 4 m?/'^ + 3 nx. 42. 1 — 6 (x+ ?/)+ ab (X + ?/)-. Page 63. — 6. x + ?/. 7. ?>^ + n. 8. x"^ + 2 xy + 2/^. 9. x + 3. 10. x+ 5. 11. a -12 6. Page 69. —12. x-2. 13. x;2-7. 14. a^ + i. 15. 5x-3. + X + 3. 17. 4a-3 6. 18. a— ^/6 + 6-^ 19. x2+2x + 4. 20. 21. x-2. 22. x2+10x-Hl4. 23. 2x-4?/. 24. 17x-l. -2ax + 4x'. 26. 9 x- - (i xy -\- 4: y\ 27. 2x3-3x'^ + 2x. -Sx-y. 29. x2-5x + G. 30. 1 - 3x -f 2x2 - x^. 1. 2. — cents. 3. - apples. 4. -• 5. — cents. X 2 a 5 15 d Page 70.— 6. —days. 7. 3 a animals. 8. , James; a c + 1 ^^, Henry. 9. ^^^±^ dollars. 10. ^^l+1^ dollars. c 4- 1 ft c 11. ^JlA±-2 cents. 1. o(x+ ?/)+9;s-4. 2. 3^2 _ 2 62 _ 5^;,. o 3. 2x+2?/ + 2^. 4. 3 x"^ + 5 x?/" + 4 ?/»* - 4. 5. l^(m — n) -1(m + ?i)- 6- 7x + 11x2 + 9x3. 7. J a - ^ 6 + c. 8. Sx^y - 5 z^y' + 10 x"»2". Page 71.— 9. x2 - 4 5j2 _ (^ + 2/)2 -|- 8. 10. - a'^-x - 2 m - U. 11. m^+l. 12. a2+2(i6 + 62^2ac+2ae + 2 6c+c2+2 6e + 2c6-f62. 16. x2 a'-\-y\ 25. «2 28. x2 X + ?/ ANSWERS. 189 13. x^ - 32 r. 14. x^ -76x- 240. 15. a* - 2 a'^b'^ + bK 16. 16 a* -80a3 + 500a-625. 17. 2a^-\-Sab-\-b-^. 18. a*-a34-a2_^^ ^^ 19. m - c -V z. 20. 1 - 3ic + 2 x;^ - x\ 21. 2x^ + ioc^ -\. Sx -h 10. 22. x = 2. 23. x=:4. 24. x = 5. 25. a; =7. 26. x = 10. 27. X = 3. 2S. x = 9; y =2. 2d. x = 2; y = 4. 30. x = 8 ; w = 3. 31. X = 4 ; 2/ = 4. ' ^ Page 72. — 32. $250. 33. 40 mi., 1st day ; 55 mi., 2d ; 60 mi., 3d. 34. 7 days. 35. 10 days at $1 ; 10 days at $1]. 36. ^60, father- $40, David. 37. $25, overcoat; $30, clothes;' $5, boots. 38. $3. 39. 33A pounds at 5 cents ; (56,^ pounds at 8 cents. Page 74.-2. 9x(2x-Sy). 3. ox^y{^y -^ 4x). 4. I2m7i^ (m-4w)- 5. 5ac(6-c + 36--^). 6. 2x(Sx -\- 2y - ix"-). 7. 3A (32/ -2 + 4^). 8. Sa^b(2a^ + 7 -6ab'^). 9. ah(Ha~c-d). 10. 6a^b'\a¥ -\- 6 - 7b). 11. bab'^Sb-^ -^ 4ab^ - ^a'). 12. 10m2(2?7i - 5 4- 3m-0. 13. xy^(l -^ 9 x + 27 x'p). 14. bm{am -\- 2 n + Zn^). 15. 5x''?/^(9?/ + 12 i). 16. V6x;'y{;^y^ - 5x + 7?/-^). 17. ^a%\4 + 12 a^^^'^ - w^m. 18. 5a(5w -f 15 m- - ^n^). 19. 5x3 (3 a - 72/3 _ n ^^^ 20. 15x*(3&-^ + 4?/ - 2z). 21. 36a22/3(2a?/ - 1 - 3?/2). 2. (^ + 2/) (« + &). 3. (?> + 3)(a + 2). 4. (x + 3)(2/ + 3). 5. (a: + 2/)(a-6). 6. (a:?/ + 6)(1 - ?>). 7. (?/-! + 1)(2/ - 1). 8. (x2-2/-^)(a; + ?/). 9. (x^ - &2)(^-2 _^ 5), 10. (rr^ + 2)(2a - 1). 11. (X- ?))(«- 6). 12. (x2+«)(x+l). 13. (a-6)(x^4-?/"-^). 14. (l-a3)(l + a-0. 15. {\-x^){l-x). 16. x(x - y)(y - z). •17. «(l + 2/)(l + ^-). Page 75. — 1. (x -\- y) (x -\- y) . 2. (x + 2)(x + 2). 3. (2a~b) (2 a- b). 4. (3 m + n) (3 m + n). 5. (x - 2 ?/) (x-2y). 6. (2/ + !)(?/+ 1). 7. (a?>-4)(a6-4). 8. (2 w - 5)(2 ?i - 5). 9. (2x + 2?/)(2x + 2?/). 10. (x-5)(x-5). 11. (l + ^)(l + 2:). 12. (a^ 4- 2 52) (a2 + 2 &2). 13. (a; _ 3 0) (x - 3 ^). 14. (x + 10) (x+10). 15. (m + 4?i)(''^ + 4w). 16. (3x - 3?/)(3x - 3 2/). Page 76. - 17. (3 + a^) (3 4. «2). ig. (2 a^x - b'^) (2 ^23; _ 52). 19. (am-4jO(«m -4w). 20. (5x2 + 8)(5x2 + 8). 21. (^2 + 2x2) (a2 + 2x-'). '22. (2bc-Sd){2bc-'Sd). 23. (a'^ -\- 9)(a^ + 9). 24. (lOx- l)(10x- 1). 25. (mn-{-2)(mn + 2). 26. (x« + r) (X" + r)- 27. (4 + 2 m2) (4 + 2 m2). 28. (1-4 x"») (1-4 x"»). 1. (x + 2/)(x-?/). 2. (x2 + 2)(x2-2). 3. (a + 3 &)(« - 3 6). 4. (wi + l)(m-l). 5. (ab-\-c)(ab-c). 6. (x+5)(x-5). 7. (llx + 10z/)(llx-102/). 8. (4a + 3?>)(4a-36). 9. (x^ + 12) (x2-12). 10. (l + ^)(l-.e). Page 77. — 11. (x- + 2/2)(x + 2/)(a: - ?/). 12. (x + 4)(x-4). 13. (3x + 6)(3x-6). 14. (2x + 5)(2x-5). 15. (x2 + 9) (x + 3)(x-3). 16. (m?i + 8x)(mw-8x). 17. (2« + 4?>)(2a-4?>). 18. (x2 + 25)(x + 5)(x - 5). 19. (3x + 9)(3x - 9). 20. (5 a6 + 7 c^d^) {^ab-7 d^d'-). 21. {m' + 25) (?^ + 5) (m - 5) . 22. (15x+10y)(15x-10?/). 23. (4^/2+ 10)(2?/+4)(2?/-4). 24. (ac+3d)(ac-3c?). 25. (x + ll)(x-ll). 26. (cd+l)(c(^-l). 27. (5?}i + 15w)(5m - 15?^). 28. (x^ + l)(x3 - 1). 29. (x'"+2/«) (x'« - 2/"). 30. {x^ + 13) (x^'^ - 13). 190 ELEMENTS OF ALGEBRA. Page78.— 3. (a + 4)(a + 2). 4. (.■« + 7)(x - 2). 5. (x - 7) (x+l). 6. (w-0)(m-l). 7. (6 + 3)(6 + 2). 8. (?i - 2) (>i + l). 9. (/i + 2)(w-l). 10. (x-9)(x + 2). 11. (x + ()) (x-2). 12. (a-8)(a-f-3). 13. (x-h9)(ic-4). 14. (?/_2^)(2/ - ^). 15. (m-8;0(m + 2w). 16. (a + 8ic)(a +x). 17. (ic-20)(x+10). 18. {a-nx){a + bx). 19. (x-4)(x-10). 20. (x + 5?/)(x-4 ?/). at. (m + 6)0m + 4). 22. (a-75)(« + 36). 23. (x-8)(x-()). 21. (xH-9)(x-2). 25. (c + 21)(c - 20). 26. (a + 75)(a + 46). 27. (x«-5)(a^"-4). 28. (x-* - 14)(x + 11). 29. (?7i + 17)(m-15). 30, (x^ + 7)(x« + 5). Page79. — 4. lx-^y)(x'^-xij-\-if). 6. (a + ?>)(a"^ - «6 + ft"-^). 6. (m - /i)("*'^ + »^i« + »i'0- 7. (a -b)(a^ -^ ab + b-). 8. (jji + w) (m ■ - m;i + >i . 9. (x - y) (x- -\- xy + y'^) . 10. (s + (^'^ - ^t + ^■-) . 11. (c-t?)(c^ + C(? + (/-). 12. (?/ + l)(2/-^-2/+l). 13. (x-1) {x:'-\-x -\- I). 14. (a-c(^)(a"'H-rt^(? + c'-^t?-). 15. (m + s)(?>i-— 9ri+s-). 16. (x — 2/^) (.x' + xyz 4- ?/%'). 17. (a + 6c) («'^ — abc + ft'c^). 18. {mn-\-cd (m'hi^ — C(lmu-\-c-d'). 19. {ax — by){a-x^-\-abxy+b'^y'-). 20. (x + l)(a^^-.x+l). 2L (^-l)(2/-^ + 2/+l). PageSO. — 3. x = ±3. 4. x = ±^. 5. x=±7. 6. x=±2. 7. x = ±4. 8. x=di^. 9. x=±3. 10. x = ±4. 11. ^ = ±2. 12. ic = ±2. 13. ic=±9. 14. x = ±6. 15. ic=±5. 16. x = ±10. 17. x=±\. 18. x = ±4. 19. x-±5«- 20. 5c = ±2c. PageSl. — 23. ?/ = 9 or 1. 94. x = 13or-o. 25. x = 11 or 5. 26. = 2 or -14. 27. x = 3 or - 17. 28. x - 15 or 5. 29. y ■- 13 ' or 5. 30. X — 1 or -23. 31. x = - 1 or -5. 32. x = 3 or -27. 33. x = 5or-7. 34. ?/ = 10 or 4. 35. 0- = 10 or -40. 36. x=\ or -25. 37. a: = 10 or -50. 38. x==27or-l. 39. ?/ = 6 or -4. 40. X — 15 or —65. Page82. — 43. x = -5. 44. x^-Q. 45. x = -?>. 46. x = 9. 47. x = -8. 48. x=:-13. 49. x = 7. 50. x:=-15. 51. x = 14. 52. x = -10. 53. x = -12. 54. x=-]8. 55. x = 25. 56. x = -ll. 57. xr=50. 58. x=-22. 59. x == - 30. 60. x = -40. Page 83. — 63. x = 5 or - 3. 64. x = - 5 or - 1. 65. x = - 9 or — 1. 66. X = 4. 67. X =: - 8 or 6. 68. x = - 8 or - 5. 69. x=8or-3. 70 x=:-4or-3. 71. x = 2or-ll. 72. x=-5 or - 10. 73. X = 2 or - 1. 74. x = 7 or 11. 75. x = 10 or - 12. 76. xrn 25or - 3. 77. x=4a or 2 <7. 78. x =6 or -9. 79. x = 12« or - 8 «. 80. X = - 8 6 or - 3 h. Page 84. — 83. x = - 1 or - 9. 84. x = 2 or - 10. 85. x = 13 or 5. 86. x = l or -5. 87. x = 15or5. 88. x = lor-15. 89. x = — 2 or - 20. 90. x = 4 or - 8. 91. ?/ = 2 or - 18. 92. x = 17 or 7. 93. X = 24 or 6. 94. x = 5 or - 45. 95. x = 14 or - 0. 96. x = 1 or - 13. 97. X = 22 or 16. 98. y=\ or - 3. 99. x = 8 or - 2. 100. x = 2 or - 10. Page86. — 3. 7x^0. 4. ^xyz. 5. 5a6. 6. W mH'^z. 7. 20a6x*^. 8. Qbmhi. 9. x — 1. 10. x -\- y. 11. m + n. Page 87. — 12. a - h. ' 13. x + 1. 14. x - 2. 15. y - \. 16. \ + a. 17. x + 3. 18. x + 3 19.* x- 4. 20. x^-1. 21. x-1. 22. a + 6. 23. x+7. 24. x-2^. 25. x-5. 26. x-4. 27. x-*2y. ANSWERS. 191 28. x + 7 y. 29. x - y. 30. a: - 5. 31. x - 2 y. 32. x - 3. 33. a + 2x. 34. /> + c. 35. x - 1. 36. a + 3. 37. a-2y. 38. 2 X + 6 y. Page 89.— 3. T2a%'^d^. 4. 30a252a.-2. 5. 70 rr-^ftV^Sw. 6. 891 a27>i y2. 7. 105 x32/322. g. a;^ - 2 x^ - 4 x H- 8. 9. r^ + x'-^?/ - xy- - if. 10. «'2(x'2 - ?/2). 11. a^ - h^. 5age 90. — 12. .x^ - 11 x^ - 4 x + 44. 13. x^ + 9 x'^ 4- 26 x + 24. 14. xH2x^-16x-32. 15. c3-9c-' + 24c-16. 16. m^x(d^-b-^). 17. x\a^ - b^). 18. 16 rt^ - 8a3 + 2 a - 1. 19. a^-4n^-lla-^ 60. 29. xy{x'-y^). 21. a?;xy(x3 - 1). 22. x^ - a*. 23. (i^-6aV) 4.9ah-^-41A 24. x^ + Sa^^ _ 36x - 180. 25. x* - x^ 4 x - 1. 26. x^ + 8x-2 - X - 3. 27. aJfm{4-^ + a^/) - ab^ - b^). 28. 105 ab^ (a- - foO- 29. x^ + X-' - Ur - 24. 30. x^ - 7x2 + 7x + 15. 31. x^ - 2 x^«/ - 52 x^y^ + 98 xy^ -f 147 ?/i 32. x^ - 6 x-^ - 19 x + 84. 38. x^^-4x-^-47x + 210. 34. x^-\-xSf-xy^-yK 35. x*^-13x- 12. 36. x8 - 1. Page 93. -2. 1^- 3. i^^^. 4 l^J^- 5 '^l^llJ^^-V 30 24 28 * 45 62c3d^ g Pax — 15 to - 12??i — 9?^2 g 4 x'^y ^ 12afec 33 X ' 48 6ax + 4x ' 10 ex -6 c?/" ,Q 16 axy — 20 bxy 12 x?/ Page 94. — 11 14 ^>l + n ^ (m + 70'"^ 4. ^. 5. 7 acx 10. ' ■ 2x- 3a Page95. — 11. ^^- 12. — i— 13. — —- 14. ^ a — ?/- (« — Z> w + 7i a — b 15. '^~^. 16. ^L_. 17. 1_. 18. «i:/l 19. "^ + ^. x+?/ x+1 x+l a — 7 9>i — 5 20. ^-+^. 21. ^^-^- 22. ?^ 23. ''^^. X — 6 X — 1 tn- + m/i + w- x + 7 24. -^^^^ 25. ^~^ — 26. 5^ x'2 + xy -f t/'2 X- -x+1 9(x - y) 27. 28. ^ ~ ^ (2x + 32/)--2 * 2?/ Page96.-2. 1^^±A^. 3. ^l^'-^^ll. 4. '^^-±J^. 2 5 4 , 20x-6y g 9x- 1 - 5x -4 g 40x + 5y 10 * ' 7 * ' 3 * ' 6 * g 40aj-45;-x ,q 1 1 m - n ^^ a - ?> + ex ^g 18 x - ?/ - g 5 ' ' 3 * ' c ' * 9 192 elemp:nts of algebra. 13. ?A±_5j2. 14 ^ a; — 5 y -g 3 ax- + 6 g — x 2 7 ' rtx jg a-^ - 2 qc + c2 ^^ l + 2x ^g 3x2 ^^ 4a5-a a * 1 + X ' X ' - 1 * h ' 20. A«!_. 21. <^'^ + 2 gc - 2 c^ gg x- + x - 8 gg ?n-^ - 4 n^ + 14 « — X * a~ c *x + 2 ' m + 2n ' 24 m2 + 2 77171 + n^ - 5 gg 2rt2_5 g^ (m^ m+Ti + x *a + x Page97. — 2. ^a-\-~ 3. 2rt6 + 5. 4ac+^. 6. «+-• 7. 2x- 9 a 9. a + X. 10. x2 + X + ^ "^ ^ m - n 4a& 11 4. 3x?/2^. 6x 8. 2 ax - x3. 11. 2a -26- _ 2 62 X- 1 a 4- 6 12. 5z/-fa^ + -- 13. 2 6 ^. 14. 2x + 8+ ^^ a a + 6 X — 4 15. x2 + x?/ + 2/2 + -li^. 16. 2 -f 6 - ^. 17. X + 3 + ^^ + ^ . X — w h X'- — X — 1 Page98. — 18. 2a + 26. 19. x - 1. 20.' 2x + 6 + x-3 21. a + 6 + -?^. 22. X-2+ — 23. x-l+ — ^- a — 6 X — 2 2x — 3 24. 6x-2 + — 5 — 25. X2-X+1 —- X — 1 X + 1 Page 99. -2. ^, ^. 3. ^, ^^ 4. ^-^^', -^^. 6c he nx 7ix 3 6c2/ 3 bcij g 3a6?/ 4c(? g 2 6^ 3cdx ^ «^ ^2/ a6x « ox^ 12?/' 12?/ * x2^2' grV2 * dxy^ axy axy xyz bdz acy abx g 2a^ 2yz^ 2x ^ x?/;^' x?/;^' x?/.^ ax?/^' ax?/^' ax?/^ p ,QQ -Q 6c277^ a26c7i a-7nn -- 10 gxy^ 8 by-z aVc'^' a^l)h'- a-b'-c'^ ' x'^y^z ' x^y^z' 9x *2 12x 52 X 3xy 8?/ ^g 2 ay -\- 2 by 3 ax — 3 6x x-V'^* ' 78' 78' 78' 78* ' 6 x?/ ' 6xy ' a6xy 14 <^cx + <^cy 2 ax — 2 a y 2cx^ + 2c7/2 5 x — 5 .?/ 6 xj/ * 4 ac ' 4 ac ' 4 ac x^ — 7/2 7 X + 7 V x + y jg 3 a - 3 6 4a + 46 ^^ 3x2 -(i x^ x2 — y- ' x2 — 7/2* * a2 — 62 ' a2 — 62 ' x(x2 — 4) 5x2+ lOx 9x2-36 j^g a + 6 a2 + a6 + 62 a-b x(x^ - 4) x(x2 — 4) a-^ - 6> a^ - 63 a*^ - 6^ 4x-4 3x-3 4x ^ ^ '^ 4(x2-l)' 4(x2-l)' 4(x2-l)* 19 4x-4 3x-3 4x g^ 2 a2 -t- 2 6 ANSWERS. 193 a^ - b^ ' a^ — b^ ' x=^— x"^-4a:4-4' x^ — x^ — ix -{- i 22. ^^- 5 a; ax -a 2^ 4a; + 4 x3 — 3.T-^ — 13x+ 15' ic^ — 3a;'^ — 18x+ 15 * 4x(a;-2)' a;2-x 2x-4 2^ 5- lOx 8x + 0a:2 4 - 13a; 4a;(a;-2) 4a;(x-2) ' l-4x-' l-4x-=' l-4a:-^* gg a^ - ab + ab' - &? a^ + ^^^ 3a JlLzlVL a^-b^ ' a^ - 6' ' a^ - 6^* * y{x'^ + 2/3)' xy + y'^ x^ — x^y + x-y- y{x^ + y^y yix^ + y'O Page 103.— 4. a; = 8. 5. a: = 15. 6. a; = 4. 7. x = 16, 8. a: = 12. 9. a; = 3. 10. a: = 6. 11. a: = 12. 12. a;=14. 13. a; =9. 14. a; = 30. 15. x = 24. Page 104. — 16. x = 3. 17. x = 10. 18. x = 24. 19. x = 35. 20. a; = 15. 21. x = 10. 22. a; = 4. 23. a* = 44. 24. a; = 5. 25. x = 6. 26. a; = 12. 27. x = 22i\. 28. a: = 4. 29. x = l.^. 30. x = 2. 31. x = 8. 32. x = 13. 33. x = 5. 34. x = h 35. x = 3jV 36. x = 41. 37. x = 17. 38. x = 7. Page 105. — 39. x = 1. 40. x = 9. 41. x= 50. 42. x = 2^\. 43. x = 40. 44. x = 16. 45. x = 3. 2.12. 3.^7200. 4. 75 gal- lons. 5. 10 yeard. 6. 15 and 20. Page 106. —7. ^75, horse ; i$50, wagon. 8. 20 and 56. 9. 220. 10. 10 and 40. 11. 50 years, father ; 20 years, son. 12. 100 sheep. 13. 10 children, DO cents. 14. »^18, father; $12, oldest son; $7, youngest son. ,15. $4225. 16. $ 135, A ; $140, B ; $165, C. Page 108. -4. -^ • 6. ^^^ + ^^^. ab Sxy 7. 2 x2 + 3y ^ ^ 2a2 ^ ^a'^-b'^ ^^ ISa 6 ax ' aP- — x^ '^a — A^b * 12 c '"' x- — ?/2 2a^^ 14q + 26 ^g 6 x2-9x y + 4y2 ^^ 12m4-2n a^ + 8 a + 15 * 2 x^ — 3 x?/ + ?/^ * m- — n^ 2x2 + 2^ jg^ 2/2 -2xy ^^ ^»^L±1.2/l^ 18 2 g^ _ 6 (:r + 43 12. 15. 19. 27. x2-l 2 r?2 -f 2 62 «2 _ 62 3 a2 _ qa; a2 _ x2 ' cdx — 2bz xyz Page 109. — 28. 1 oa 3X+13 x2 - 3 X + 2 OA 4a6 gj X- - 9 X — 9 gg 3x — 11 milne's el, of alg, — 13. 194 ELEMENTS OF ALGEBRA. 83. zd«^ 34.^^±2i^dols. 35. ^dols. paid out, 2^ left. 36. ^^ilMdols. 37. 55^. 38. fx. 39. i|a6. 40. 2 105 ^ ^^ 6 41. 2.3 X 42. 39 a - 28 5 46. 20 x-\- y — 2 xy^ r^l _ y'l Page 110. —50. 63 47. 43. ^1 ax 44. 5a + 76 36 60 2a-2c \-a 6x-22 48. 49. x^-x-2 51. ac 6a;H 4a;-6 x3-7x-6 ' 45. ^-±^. y x-^ - 1 * 52. 2ic 53. 2a;2 54. 2a2 55 4 d;^ — 4 g'^x + 4 g-^g X* — 5x^-1-4 11a, x'2 _ 1 x^ — a-x a^ — x^ 57. ^^-^J^ dollars. 58. ^^ + ^^ miles. 5 3 Page 112.-3. "^-^^ 4. 1^^. 5. ^-^. 6. ex 9x2 4 5 8. ^Jll. 9. m- n ^Q 11 ^2^2 j^^^ - miles. 7. 12. 16. 21. 7m2 6bx 36 x + 2* a; — 5 x-f3* x + 6 13. 4. m — n 3(m + w) 14. 1 x-y 17. ic'(a + x) 5(x + 1) 3 ^a X-hS ,n 4 18. 5(x - 7) 19. x-S 2 6m 6 ^2?/^ 10 a25 a:2 - 2/2 ^g^ q(a-36) a^ — 6^ 6(a2-62) a 20. Page 113.— 22. 26. (x + 2). 27. f- 23. a. 24. a ^. 28. 29. -\ 32. —pounds. 6 a 3 ^22-4 ^: 33 3m_,K4n ^^11^,3^ 3 25. x-y 30. 1. 31. —dollars. a 34. ^^(^-^) dollars. 3 85. ^,^A',~'^— miles. b b b b Page 115. —3. ^. my 25 x3?/5^ a 1 4. 5. X2;. 6. 3a62 7. 8fl3 13. 18. 4a^bc X ■]- y dxy ,^ 2 14. 2 x^ 3 c?xaAr2. 28. x2 - x - 1. 29. 3m2-5m-2. 30. 2x2 + 3x?/-?/\ 31. l-3x + 4x2. 32. 1 + 4x44 x^. Pagel49. — 4. 8-3. 5. 5-3. 6. 64. 7. 6.3. 8. 89. 9. 97. 10. 68. 11. 85. 12. 255. 13. 354. 14. 465. 15. 177. 16. 327. 17. 345. 18. 409. 19. 906. 20. .25. 21. 352. 22. 237. 23. .159. 24. .055. 25. 3.04. 26. 50.7. 27. 3.64. PagelSl.- 3. x = ±4. 4. x = ±3. 5. x = i 5. 6. x = ±6, 7. x = ±2. 8. x = ±3. 9. x = ±l. 10. x = ±14. 11. x = ±2. 12. X = ± 5. 13. X = ± 2. 14. X =3 ± 8. 15. x = ± 3. 16. x = ± 9. 17. x = ±10. 18. x=±(a-3). 19. x = ±2a. 20. x = ± «• 1. 2. 2. 6 and 15. 3. 7. 4. 17 rods. 5. 12 yards. Page 152. — 6. 5. 7. 10 rods, length ; 6 rods, breadth. 8. 26 years. Page 154. —4. 2 or -6. 5. 5 or -3. 6. 3 or -11. 7. 4 or -10. 8. 7or-l. 9. lor -11. 10. 2 or -10. 11. 5 or -25. 12. 2 or -4. 13. 2 or -7. 14. 2 or -4\. 15. 5 or 3^. 16. 6 or 1. 17. 4 or -14. 18. 1 or -1.3. 19. 32 or -2. 20. 2 or -16. 21. 12 or -14. 22. 8 or 1. 23. 30 or -2. 24. 5 or -16. 25. 12 or 3. 26. 9 or -11. 27. 4 or -1. 198 ELEMENTS OF ALGEBRA. Page 155. —3. 5or-3i. 4. 3 or -3^-. 5. lor 7. 5or-5i. 8. 7 or -4. 9. 8 or -7i. 10. lor-}. 11. 2 or -If. 12. 1 or -"'is. 13. 5 or 4. 14. 3 & or - h. 15. 2 a or - 5 a. 16. 4 or -31. 17. 6 or -42. 18. 13 or - 4i. 19. 6 or - li. 20. 6 or - 2. 21. - 1 or ^^. 22. ?-^-±_^ or - 1. 23. 5 or - 6. 24. 10 or - 8. ^ ^ ^ Page 156. —25. 2 c or -• 26. a + m or a - m. 27. 2 a OT\a. 2. 9 and 4. 3. 10 and 16. 4. 2 and 6. 5. 52 rows ; 54 trees in a row. 6. 20 rods ; 25 rods. 7. 3. Page 157. — 8. 15 and 11. 9. 8. 10. 8 men. 11. 10. 12. $16. 13. 10 yards. 14. 4 cattle. 15. $100. 16. $30. Page 159. — 5. x = 2) y = l. 6. x = 13 ; ?/ = 7. 1. x = \ or J ; 2/ = J or 1. 8. x = db 4 ; ?/ = ± 4. 9. x = 2 or 5^ ; ?/ = 7 or i. 10. x = ± 4 ; y = 1 or 9. 11. x = ^) y = b, 12. x = 5 or O^^ ; 2/ = 3or-li2^. 13. x = 7;?/ = 4. 14. x = 5 or 2 ; 2/ = 2 or 5. 15. X = 18| ; 2/ = Vo\. 16. x = 10 or 2 ; ?/ = 2 or 10. Page 160. — 17. X = 5 ; ?/ = 4. 18. x = 6 ; ?/ = 4. 19. x = 5 or - 9f ; 2/ = 2 or llf-. 20. x = 5 ; 2/ = 1. 21. x = 10 or 9 ; 2/ = 9 or 10. 22. x = 11 or - 5 ; 2/ = 14 or - 2. 23. x = 2 or — If ; 2/=5 or — 9. 24. x=:l or 5; 2/=4or2. 25. x=4 or 3 ; 2/=— 3or— 4. 26. X = 8 or 1 ; 2/ = - 1 or - 8. 27. x = ^.±1^ ; y = ^Llilk. 2i 2 28. x = — ;2/ = |- 1.9 and 7. 2. 8 and 3. 3. 24 years, A ; 20 years, B. 4. 16 in., length ; 12 in., breadth. 5. 15 rods ; 10 rods. Page 161. — 6. 22 and 12. 7. 5 and 2. 8. 150 miles, whole dis- tance ; 90 miles, A ; 60 miles, B. 9. 40 yards, linen ; 60 yards, cotton. 10. 45 rods, length ; 30 rods, breadth. 11. 10 and 8. 12. 120 yards silk ; 80 yards velvet. 13. 20 rods, 15 rods, rectangle ; 10 rods, square. Page 162. — 1. 2jV 2. (7a + 25 + 5)(x - 2/). 3. 2x + 30. 4. 18x2-16x-7\/x. 5. 10?/— 6ax(a-l)-4(&2_2). 6. — • 4 7. (600 - 4 5 - 7 c - 5 ftc) dollars. 8. 170x2 - 7O2/2 + 90c2 - 15. 9. 2x2-22/2+2^. 10. a + 1. 11. x^ -Z2y^. 12. x6-26x* + 169x2- 144. Page 163.— 13. 5x2/(x-2 a+5 2/2-3a2ic2/). 14. (x+13)(x-12); (x-8)(x-7); (x-10)(x+7). 15. ^^ ~ ^ cents. 16. a- 10c = 2. 17. X2-X2/ + 2/2. 18. (a-62)(a2 4.^^2 + 54). (5jc_42,)(25x2 +20x2/+ 162/-^); (x2 + 9)(x+ 3)(x - 3). 19. 2x3 - 6x2 + 3x - 1. 20. -^^^zil-. 21. (2a + 9)(2a + 9); (lOx + 52/)(10x - 5?/)^ he -\- d (3x-7 2/)(3x-72/). 22. ?-^-±l^+-^ cents. 23. a + &. 11 + 5 24. (2x + 8)(4x2-16x+64); (a5 + l)(a6-l); (m + 3)(m2-3m + 9). 25. x = 7. 26. x2+4x2/ + 42/2; 16a2_24a5+9 62j 4wi2-20m+25; ANSWERS. 199 da^b'^+Qah + l, 25+20a4-4a2; S6x^—Sixy + 4dy\ a*-h20a^-hl00 ; m* - 8 m^n^ -\- Wn^; x* - 6 x'^y + 9y^; 9 a'^ -i- SO ax -\- 25 x-. Page 164.— 27. x2+3x-28 ; x2-225; xH13x+30; x^-3x-108; 4x2-25;m^-l;x"-19x+88;x2+x-210; 9x2-49. 28. a -2. 29. x+3. 30. x^- 7x2 + 6. 31. x3 4-6x2-9x-54. 32. ~ 33. ^~^^. a — b X — by 34. -^— 35. ^^=^. 36. ^ 37. — ^— . ^-2/ 3 2(^2 _i) ^2_a.2 38 ( «-^)(<^-a;) gg 85a -206 X * * 84 ' Page 165. - 40. 1 , A ; ^ , B ; '^^i^, both ; -??^, days. 41. x= 10. m n mn 'm-\-n 42. x = 2. 43. x = 3. 44. x = a - 6. 45. — , horses; ^, 10' ' 10' cows ; ^, sheep. 46. x = ^ 47. x = f . 48. -^^, 10 a + 5_c m + 7i an ^g 15 m -^ lOa&c m + n m2 — 1 a6 + ac + 6c Page 166. —61. x = i; y=l 52 x = 2}; y = 141. 53. x = 7 ; 2^=10. 54. x = --15--; 2/ = ,-^- 55. x = 2] ; y = ~A; oo — 2a ba — b 5? = 4J. 56. x = 24; 2/ = 60; = 120. 57. x = ^-±-^^^=-^ ; ^^«^^^,^^5^|+c. 53 ^^^, 59. x = ± 1 2 2 a — I 60. x = ±8. 61. x = 4or-l. 62. x = 2 or -17. 63. x = 3 or -2J. 64. x=a + mora- m. 65. x = 2«_±_5 or -1. 2a 66. x=lor -If. Page 167. — 67. x = 13 or - 8. 68. x = 12 or 6. 69. x = 14 or -2. 70. x + 2?/-3. 71.245. 72. x = 5 or -2f ; y=3 or 6J. 73. x = ±Q; y=±l. 74. x = lO; y = S, 75. x = ±7; y="2 or -12. 76. $96, Tom; $84, John. 77. 570 votes. 78. 60 cents. 79. 60 sheep. Pagel68. — 80. f. 81. $1.50, wheat; $.40, com. 82. 45 feet, 60 feet. 83. $ 180, carriage ; ^ 160, horse. 84. 15 and 19. 85. 262^ gallons. 86. x = 8 or 4} ; ?/ = 2 or — 1.}. 87. 15 men ; 5 women. Page 169.-88. 50 Wep, A ; 30 sheep, B. 89. 11 and 19. 90. $.35, tea; $.30, coffee; $.05, sugar. 91. 24 miles. 92. $150. 93. $140, horse; $30, cow; $10, sheep. 94. 7 and 3. 95. $60. Page 170. —96. $312. 97. 162 sheep, 1st; 144 sheep, 2d ; 128 sheep, 3d. 98 10 miles. 99. 11 and 9. 100. 1 J yards. 101. 117 mi., A ; 130 mi., B. 102. 12 rods. 103. 4 rods. Page 171. —104. a + x. 105. 3 x + 1. 106. x + 1. 107. 30 xy (x2 - 2/2). 108. x3 + 18 x2 + 107 X + 210. 109. 12 xy(x^ - y^y. 200 ELEMENTS OF ALGEBRA. 110 ^(<^ + ^) - 111 a^^-a+l . 112 l^Q^+'^^Q 10a; -5 4abia-b) ' a- + 2a-{-l ' 15(x - 2) ' 15(x - 2)' 9x + 6 , 113 x3+3x-^-9x-27 x3+3x^-4x-12 15(x-2) ' x*+6x3+5x^-24x-36' xH6xH5x^-24x-36' x^ — 16 --- x^ — x^y + a;-y^ — x^'^ x^y + xy^ a:* + 6x^ + 5x-2-24x-36" * x^ - 2/* ' x* - y* ' xV-yy 115 12 - 12 X 5-5x 8x x*-?/*' * 4(1 -xO' 4(1 -x2)' 4(1 -x2)' Page 172.— 116. — 117. ^ - 118. x = 7. (a — x)(rt + 2x) X -{- y 119. x=: 4. 120. x=-2i. 121. x = 2. 122. x==l|i. 123. x = to. 124. x = l. 125. x = ^^^^=J- 126. x = --. 127. x = oSa. hc + d abc 128. x = |. Page 173. —129. x = 3 ; y =2; z = 1. 130. x = 2 ; y = 4 ; ^ = 0. 131. x = «;2/ = 2. 132. x = ^' + ^'- ^^ y = ^' " ^' + ^^ 2a 26 133. X = a - _^-^; 2/ = — ^. 134. x =(« + 6)2 ; 2^ =(« - 6)2. ac + 6 «c 4- & 135. 94. 136. 75. 137. 327. 138. 3.24. 139. .0.321. 140. .0071. 141. 2187. 142. 6561. 143. 2x2 - x + 3. 144. 4^2 ^ 3^ 4. 10. 145. X + 2/ + 3 ^. 146. z + m - 1. 147. x-^ - 3 x2 4- 4 x - 5. Page 174. — 148. 709. 149. 805. 150. 40.8. 151. 9.30. 152.20.53. 153.1.111. 154. 2x2 + 4 ax - 3^2. 155. 3m2 - 4m-7. 156. a3 - a2 - a + 1. 157. x=±7; y=±l. 158. x = 8 or -3 ; y = l}jOY-4. 159. x = 3or-2; y = 2or-S. 160. x = ^-±-^; y = ^^. 161. x = G',y = i. 162. x = I ; y = l 163. x = f; y^\' 164. x = ±4;?/=±3. 165. 04 inches. 166. 90 feet. Page 175. — 167. 4 p.m. 168. 80 miles. 169. 4 days. 170. % 9000. 171. ^070, D; $005, C. 172. $8. 173. 12 days. 174. 11- cords. 175. #40, lost. Page 176. — 176. #14. 177. 92,160 acres. 178. 12 inches. 179. 8 feet; 4V pounds weight, per foot. 180. 4 yards, fore wheel; 5 yards, hind. " 181. 5a* + 4a'^ + 3a2 + 2a + l. 182. 211 hours, equal faucets ; 13 j^^. hours, the other. 183. x = 4 or 3 ; ?/ = 3 or 4. XO 86 1* RETURN TO the circulation desk of any University of California Library or to the NORTHERN REGIONAL LIBRARY FACILITY Bldg. 40a Richmond Field Station University of California Richmond, CA 94804-4698 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS • 2-month loans may be renewed by calling (510)642-6753 • 1-year loans may be recharged by bringing books to NRLF • Renewals and recharges may be made 4 days prior to due date. DUE AS STAMPED BELOW JAN 16 2001 RETURNED JAN18 12,000(11/95)