IN MEMORIAM FLORiAN CAJORl ©■•n-^Kv-^^,:-^ . ^'S'-' '"i^' Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/eatonselementaryOObradrich V9 > \ ^ ^- : .-V.) >v : ^ \ v^ .^. ^.y - €uion anb ^irabburg's ^atljcmat'ttal ^txus EATON'S ELEMENTARY ALGEBRA DESIGNED FOR THE USE OF HIGH SCnOOLS AND ACADEHES. WILLIAM F. BRADBURY, A. M., ■OPKIXS MASTER Vf THE CAMBRIDGE niGH SCHOOL ; AOTUOR OF A TREATI8K OK TsmoNOiurKr Aia> soanrufG, and of an kuementart oecxetbt. BOSTON: THOMPSON, BIOELOW, AND BROWN 26 & 29 CORMBILU EATON AND BRADBURY'S used with unexampled success in the best schools and academies of the country. Eaton's Primary Arithmetic Eaton's Intellectual Arithmetic. Eaton's Common School Arithmetic. "•: .•'•.EatoSj's" ffi^fa* School Arithmetic. ►V. '» './* ^iTok's'.ELi^^i^Ta £)p Arithmetic. Eaton's Grammar School Arithmetic. Bradbury's Eaton's Elementary Algebra. Bradbury's Elementary Geometry. Bradbury's Elementary Trigonometry. Bradbury's Geometry and Trigonometry, in one volnme. Bradbury's Trigonometry and Surveying. Keys of Solutions to Common School and High School Arithmetics, to Elementary Algebra, Geom- etry, AND Trigonometry, and Trigonometry and Sur- veying, for the use of Teachers. Entered according to Act of Congress, in the year 1868, BY WILLIAM F. BRADBURY AND JAMES H. EATON, in the Clerk's Office of the District Court of the District of Massachusetts. University Press : Welch, Bigelow, & Co., Cambridge. CAJORl '^ PREFACE It was the intention of the author of Eaton*s Arith- metics to add to the series an Algebra, and he had com- menced the preparation of such a work. Although its completion has devolved upon another, the author, as far as practicable in a work of this character, has followed the same general plan that has made the Arithmetics so popular, and spared no labor to adapt the book to the wants of pupils commencing this branch of mathematics. A few problems have been introduced in Section II., to awaken the pupiPs interest in Algebraic operations, and thus prepare him for the more abstract principles which must be mastered before the more difficult problems can be solved. Special attention is invited to the arrangement of the equations in Elimination ; to the Second Method of Completing the Square in Affected Quadratics ; and to the number and variety of the examples given in the body of the work and in the closing section. The Theory of Equations, the Explanation of Negative Results, of Zero and Infinity, and of Imaginary Quantities, are omitted, as topics not appropriate to an Elementary Algebra. It may also be better for the younger pupils to Qi 1 ?59 7 IV PREFACE. pass over the two theorems in Art. 74, until they become more familiar with algebraic reasoning. While the book has not been made simple by avoiding the legitimate use of the negative sign before a parenthesis or a fraction, the difficulty which is caused to beginners by the introduction of negative indices in simple division has been obviated by deferring their introduction to the section on Powers and Roots, where they are fully ex- plained. The utmost conciseness consistent w^ith perspicuity has been studied throughout the work. It is hoped the book will commend itself to both teachers and pupils. W. F. B. Cambbidge, Mass., May 17, 1888. CONTENTS. SECTION I. Taom Defixitioxs A!n> Notation . 7 The SigoA 7 I Axioms 9 SECTION II. Alqeobaic OpzKAnoxs 10 SECTION III. Dfn5Tno:fs asi> Notahon (Contioaed from Section I.) 14 SECTION IV. Additiox 19 SECTION V. SCBTRACnOM 25 SECTION VI. MoiTirucATiox 81 SECTION VII. Dnnsiox 87 SECTION VIII. Dbxoxstkatiox or Tokobzms 43 SECTION IX. Factorixo 47 SECTION X. OuATBST Comiox DirisoR 54 LXAST COXXOK MOLTtPlI SECTION XI. Pa.^cTioxs 8BCTI0N XII. fl.«n«ral Prinriple* 6.3 To inultiply a Fraction by an Integral Picrn« of Fr.irtions 04 QimnMty 78 To n'liKc a Frirtion toit^lnwpst tprm« 05 To multiply an Int«^n^l Quantity by To re lure FrartionB to equivalent Frac- a Kmrtlon . 75 tioti* hiving a ('omnion Denominator C* To or <, the smaller quantity always standing at the vertex ; thus, 8 > 6 or 6 < 8 signifies that 8 is greater than 6. 11. Three dots .•. are sometimes used, meaning hence, therefore. 12. A Parenthesis ( ), or a Vinculum , indicates that all the quantities included, or connected, are to be considered as a single quantity, or to be subjected to the same operation ; thus, (8 + 4) X 3 = 12 X 3, or = 24 + 12 = 36 ; 21 — 6 -=- 3 = 15 -f- 3, or = T — 2 = 5. Without t^e parenthesis, these examples would stand thus :8 + 4X3 = 8 + 12 = 20;21 — 6-^3 = 21 — 2 = 19; the sign X. in the former, not affecting 8j nor the sign -i-, in the latter, 21. Examples. 1. g + t — 3 + 4 = how many? 2. (9 + 15) -r- 3 = how many? 3. — — — X 14 = how many ? 4. (14 + 13) X (5 — 2) = how many? 5. 10 + (7 — 4) -^ 3 X 4 = how many ? 6. 25 — (6 + T) = how many? T. 150 -^ (18 — 11) = how many? DEFINITIONS. 9 8. Prove that 175 + 8 — 49 = 14 + 190 — 64 — 16. 9. Prove that 216 — 44 + 14 > 144 + 13 — T5. 10. Place the proper sign (=, >, or <) between these two expressions, (247 + 104) and (546 — 195). 11. Place the proper sign (=, >, or <) between these two expressions, (119 — 47 + 16) and (317 — 104). 12. Place the proper sign (=, >, or <) between these two expressions, (417 + 31) — (187 — 72) and (127 + 179). AXIOMS. IS. All operations in Algebra are based upon certain self-evident truths called Axioms, of which the following are the most common : — 1. If equals are added to equals the sums are equal. 2. If equals are subtracted from equals the remainders are equal. 3. If equals are multiplied by equals the products are equal. 4. If equals are divided by equals the quotients are equal. 6. Like powers and like roots of equals are equal. G. The whole of a quantity is greater than any of its parts. 7. The whole of a quantity is equal to the sum of all its parts. 8. Quantities respectively equal to the same quantity are equal to each other. a* 10 ELEMENTARY ALGEBRA. SECTION II. ALGEBRAIC OPERATIONS. 14. A Theorem is something to be proved. 15. A Problem is something to be done. 18. The Solution of a Problem in Algebra consists, — 1st. In reducing the statement to the form of an equa- tion ; 2d. In reducing the equation so as to find the value of the unknown quantities. Examples for Practice. 1. The sum of the ages of a father and his son is 60 years, and the age of the father is double that of the son ; what is the age of each ? It is evident that if we knew the age of the son, by- doubling it we should know the age of the father. Sup- pose we let X equal the age of the son; then 2x equals the age of the father; and then, by the conditions of the problem, a:, the son's age, plus 2^7, the father's age, equals 60 years; or 3a: equals 60, and (Axiom 4) x, the son's age, is ^ of 60, or 20, and 2x, the father's age, is 40. Expressed algebraically, the process is as follows : — Let X = son's age, then 2x =z father's age. a; + 2 a; = 60, 3 a: = 60, X = 20, the son's age. 2 X = 40, the father's ago. DEFINITIONS. 11 2. A horse and carriage are together worth $450 ; but the horse is worth twice as much as the carriage ; what is each worth? Ans. Carriage, S150; horse, $300. All problems should be verified to see if the answers obtained fulfil the given conditions. In each of the pre- ceding problems there are two conditions, or statements. For example, in Prob. 2 it is stated (Ist) that the horse and carriage are together worth $450, and (2d) that the horse is worth twice as much as the carriage ; both these statements are fulfilled by the numbers 150 and 300. 3. The sum of two numbers is 12, and the greater is seven times the less ; what are the numbers ? 4. A drover being asked how many sheep he had, said that if he had ten times as many more, he should have 440 ; how many had he ? 5. A father and son have property of the value of $8015, and the father's share is four times the son's; what is the share of each ? Ans. Father's, $6412; son's, $1603. 6. A farmer has a horse, a cow, and a sheep ; the horse is worth twice as much as the cow, and the cow twice as much as the sheep, and all together are worth $490 ; how much is each worth ? OPERATION. Let X = the price of the sheep, then 2 a: = " " " " cow, and 4 a; = ^* '' " *' horse ; and their sum Tar = 490, a; = 70, the price of the sheep, and 2 a: = 140, " " " " cow, and 4x = 280, " '* " " horse. 12 ELEMENTARY ALGEBRA. 7. A man has three horses which are together worth $540, and their values are as the numbers 1, 2, and 3; what are the respective values ? Let a:, 2 a:, and 3x represent the respective values. Ans. $90, $180, and $2*70. 8. A man has three pastures, containing 360 sheep, and the numbers in each are as the numbers 1, 3, and 5 ; how many are there in each ? 9. Divide 63 into three parts, in the proportion of 2, 3, and 4. Let 2x, 3 a;, and 4 a: represent the parts. 10. A man sold an equal number of oxen, cows, and sheep for $ 1500 ; for an ox he received twice as much as for a cow, and for a cow eight times as much as for a sheep, and for each sheep $ 6 ; how many of each did he sell, and what did he receive for all the oxen ? Ans. 10 of each, and for the oxen, $ 960. 11. Three orchards bore 812 bushels of apples ; the first bore three times as many as the second, and the third bore as many as the other two ; how many bushels did each bear? 12. A boy spent $4 in oranges, pears, and apples; he bought twice as many pears and five times as many apples as oranges ; he paid 4 cents for each pear, 3 for each orange, and 1 for each apple ; how many of each did he buy, and how much did he spend for oranges ? how much for pears, and how much for apples ? . (25 oranges, 50 pears, and 125 apples. ( Spent for oranges, $0.75 ; pears, $2 ; apples, $1.25. 13. A farmer hired a man and two boys to do a piece of work ; to the man he paid $ 12, to one boy $ 6, and to the other $ 4 per week ; they all worked the same time, and received $ 264 ; how many weeks did they work ? Ans. 12 weeks. DEnXITIOXS. IS 14. Three men, A, B, and C, agreed to build a piece of wall for S 99 ; A could build 7 rods, and B 6, while C could build 5 ; how much should each receive ? 15. Four boys, A, B, C, and D, in counting their money, found they had together $1.98, and that B had twice as much as A, C afi much as A and B, and.D as much as B and C ; how much had each ? Ans. A 18 cents, B 36, C 54, and D 90. 16. It is required to divide a quantity, represented by a, into two parts, one of which is double the other. OPERATION. Let X = one part, then 2x = the other part . Sx = a, X = -, one part, o 2x = -—, the other part. o 17. If in the preceding example a = 24, what are the required parts ? ^""- 3 == T = ^' ^""^ T = T = •"• 18. It is required to divide c into three parts so that the first shall be one half of the second and one fifth of the third. . c 2r , •'>^ Ans. -, — , and — . 19. Divide n into three parts, so that the first part shall be one third the second and one seventh of the third. 20. A is one half as old as B, and B is one third as old as C, and the sum of their ages is p ; what is the age of each ? ^^^ ^,^ P g,^ 2p_ ^^^ ^'s '^. 14 ELEMENTARY ALGEBRA. SECTION III. DEFINITIONS AND NOTATION. [Continued from Section I.] 17. The last letters of the alphabet, x, y, z, &c., are used in algebraic processes to represent unknown quanti- ties, and the first letters, a, b, c, &c., are often used to represent known quantities. Numerical Quantities are those expressed by figures, as 4, 6, 9. Literal Quantities are those expressed by letters, as a, X, y. Mixed Quantities are those expressed by both figures and letters, as 3 a, 4 a;. 18. The sign plus, -f-' i^ called the positive or affirm- ative sign, and the quantity before which it stands a 'pos- itive or affirmative quantity. If no sign stands before a quantity, -|- is always understood. 19. The sign minus, — , is called the negative sign, and the quantity before which it stands, a negative quantity. 20. Sometimes both + and — are prefixed to a quan- tity, and the sign and quantity are both said to be am- biguous; thus, 8 zi= 3 = 11 or 6, and a z^h ::= a -\-hf or a — 6, according to circumstances. 21. The words plus and minus, positive and negative, and the signs -f- and — , have a merely relative signifi- cation ; thus, the navigator and the surveyor always rep- resent their northward and eastward progress by the sign +, and their southward and westward progress by the sign — , though, in the nature of things, there is nothing to prevent representing northings and eastings by — , and southings and westings by +• So if a man's prop- DEFnaiiONS. 16 erty is considered positive, his gains should also be con- sidered positive, while his debts and his losses should be considered negative; thus, suppose that I have a farm worth $6000 and other property worth $3000 and that I owe SIOOO, then the net value of my estate is S5000 + $3000 — $1000 = $7000. Again, suppose my farm is worth $5000 and my other property $3000, while I owe S 12000, then my net estate is worth $5000 _|_ S3000 — $12000 = — $4000, i. e. I am worth — $4000, or, in other words, I owe $4000 more than I can pay. From this last illustration we see that the sign — may be placed before a quantity standing alone, and it then merely signifies that the quantity is negative, without determining what it is to be subtracted from. 22. The Terms of an algebraic expression are the quan- tities which are separated from each other by the signs + or — ; thus, in the equation 4a — b =i Sx -{- c — 1 y, the first member consists of the two terms 4a and — 6, and the second of the three terms 3 x, c, and — 7 y. 23. A CoEFFiCTEXT is a number or letter prefixed to a quantity to show how many times that quantity is to be taken; thus, in the expression ix, which equals x -\- x -\- X -{- X, the 4 is the coefficient of x ; so in dab, which equals ab -\- ab -\- ab, 3 is the coefficient of a 6; in 4 a 6, 4 a may be considered the coefficient of b, or 4 6 the co- efficient of a, or a the coefficient of 45. Coefficients maybe numerical or literal or mixed; thus, in 4 a 6, 4 is the numerical coefficient of ab, a is the lit- eral coefficient of 4 6, 4 a is the mixed coefficient of 6. If no numerical coefficient is expressed, a unit is un- derstood ; thus, X is the same as Ix, be as 16c. 24. An Index or Exponent is a number or letter placed after and a little above a quantity to show how many times that quantity is to be taken as a factor; thus, in the ex- 16 ELEMENTARY ALGEBRA. pression b^, which equals 6 X & X &, the 8 is the index or exponent of the power to which b is to be raised, and it indicates that b is to be used as a factor 3 times. An exponent, like a coefficient, may be numerical, lit- eral, or mixed ; thus, ar^, x^, x^"^, &c. If no exponent is written, a' unit is understood ; thus b = b^, a =z a\ &c. Coefficients and Exponents must be careful)}'- distin- guished from each other. A Coefficient shows the num- ber of times a quantity is taken to make up a given sum ; an Exponent shows how many times a quantity is taken as a factor to make up a given product ; thus 4:X'=x-\-X'\-x-^x, and x^=zxy,xy^xy(^x. 25* The product obtained by taking a quantity as a factor a given number of times is called a power, and the exponent shows the number of times the quantity is taken. 26. A Root of any quantity is a quantity which, taken as a factor a given number of times, will produce the given quantity. A Root is indicated by the radical sign, \/, or by a fractional exponent. When the radical sign, \/> is used, the index of the root is written at the top of the sign, though the index denoting the second or square root is generally omitted ; thus, ^/ X, or x^y means the second root of x ; ^Ic, or a:^ " '' third '' " x, &c. Every quantity is considered to be both the first power and the first root of itself. 27t The Reciprocal of a quantity is a unit divided by that quantity. Thus, the reciprocal of 6 is -, and of a;, -. DEFINITIONS. 17 28« A Monomial is a single term; as a, or 3a*, or 5bxy. 29. A Polynomial is a number of terms connected with each other by the signs plus or minus ; sls x -\- y, or da -\- 4:X — *lahy. 30* A Binomial is a polynomial of two terms ; as 3x + 3y, or a: — 2/. 31* A Residual is a binomial in which the two terms are connected by the minus sign, as a: — y. 32t Similar Terms are those which have the mme powers of tJie same letters, as x and 3 a:, or 5ax^ and — 2ax'^. But X and ar, or 5 a and 5 b, are dissimilar, 33. The Degree of a term is denoted by the sum of the exponents of all the literal factors. Thus, 2 a is of the first degree ; 3 a^ and 4 fl 6 are of the second de- gree ; and 6 a' x* is of the seventh degree. 34. Homogeneous Terms are those of the same degree. Thus, 4a^ar, 3a6c, xry, are homogeneous with each other. 3^. To find the numerical value of an algebraic expres- sion when the literal quantities are known, we must sub- stitute the given values for the letters, and perform the operations indicated by the signs. The numerical value of 7 a — b* -\- c^ when a = 4, 6 = 2, and c = 6 is 7 X 4 — 2* + 6^ = 28 — 16 + 25 = 37. Examples. Find the numerical values of the following expressions, when rt = 2, 6=13, c = 4, d = 15, m = 5, and n = 7. 1. a + b — c + 2d. Ans. 41. 2. a* + Sbc — 2cd. Ans. 40. 3. i^^lt^* Ans. 219. 18 ELEMENTARY ALGEBRA. 4. m^ — 2mn + n^ 6. ^ - (^ - ^')- 6. (a" _ c + 6) (wi + n). Y. ^' ~ "•'' X {d — m + n). Ans. 86Y. \/c 4" V^^ + c — \/bm. Ans. 1. 9. 3a\/& — c X 4.71 \/2bm, 10. , ^ . Ans. 4. d — 2 c 11. \^d — n + ^/1 n. 12. (6 — fl) (cZ — c) — m. Ans. 116. 13. 13 (4.d) + Ad — la. 14. 4a6 + \/lOOc — ^d — n. Ans. 122. 15. 4a s^QOl + ba'b^ 16. 6 — a — (d — n). Ans. 3. 17. b — a — d — n. Ans. — 11. 18. (b — a) {d — n). Ans. 88. 19. (b — a) d — n. Ans. 158. 20. a + b^lO {d — m) + 14 V'c. 36* Write in algebraic form : — 1. The sum of a and b minus the difference of m and n. (m > n.) 2. Four times the square root of the sum of a, b, and c. 3. Six times the product of the sum and difference of c and d. {c >- d.) 4. Five times the cube root of the sum of a, m, and n. 5. The sum of m and n divided by their difference. 6. The fourth power of the difference between a and w, ADDITION. 19 SECTION IV. ADDITION. 37. Addition in Algebra is the process of finding the aggregate or sum of several quantities. For convenience, the subject is presented under three cases. CASE I. 38. When the terms are similar and have like signs. 1. Charles has 6 apples, James 4 apples, and William 5 apples ; how many apples have they all ? 6 apples, 4 apples, 6 apples. OPERATION. or, letting a represent - one apple, 6 a 4 a 5 a It is evident that just as 6 apples and 4 apples and 6 apples added together make 15 apples, so 6 a and 15 a cr naake 15 a. 15 apples, In the same way — 6 a and — 4 a and — 5 a are equal together to — 15 a. Therefore, when the terms are similar and have like RULE. Add the coefficients, and to their sum annex the common letter or letters, and prefix the common sign. (2.) (3.) (4.) (5.) (6.) 0') b ax 3a2 4x 6y — Zj^ — bby Sax 4a« X lOij — 2ar' — 26y 4ax la' 5x y — Ix" - hy 2ax 3a« 3x 2y — 4.x' - hy 19 ax 13 X -^16x» 20 ELEMENTARY ALGEBRA. 8. What is the sum of arc^, Saa^, 2ax^, and 4:asP? Ans. 10 a x^. 9. What is the sum of 3 5a:, 4 5a:, 6bx, and bx? 10. What is the sum of 2a:y, 6xy, lOxi/, and Sxy? 11. What is the sum of — Ixz, — xz, — 4a:z, and — xz? Ans. — 13x2;. 12. What, is the sum of —2 5, —3 5, —6 5, and — 35? 13. What is the sum of — a5c, — 3a5c, — 4a5c, and — a be? CASE II. 39. When the terms are similar and have unlike signs. 1. A man earns T dollars one week, and the next week earns nothing and spends 4 dollars, and the next week earns 6 dollars, and the fourth week earns nothing and spends 3 dollars ; how much money has he left at the end of the fourth week ? If what he earns is indicated by -[-, then what he spends will be indicated by — , and the example will appear as follows : — OPERATION. Earning 7 dollars and then spending 4 dollars, the man would have 3 dollars left^ then earn- ing 6 dollars, he would have 9 dollars; then spending 3 dollars, he would have left 6 dol- lars ; or he earns in all 7 dollars -j- 6 dollars =13 dollars ; and spends 4 dollars -|- 3 dollars = 7 dollars; and therefore has left the differ- ence between 13 dollars and 7 dollars == 6 dollars; hence the sum of -f- 7 d, — 4 d, -{- G d, and — 3 d is -\- 6 d. Therefore, when the terms are similar, and have unlike signs : + 7 dollars, -\-*ld • — 4 dollars. or, letting d — 4:d + 6 dollars, . represent ^ + 6d — 3 dollars, one dollar, — 3d + 6 dollars. + ed ADDITION. 21 RULE. Find the difference between the sum of the coefficients of the positive terms, and the sum of the coefficients of Uie neg- ative termSf and to this difference annex the common letter or letters, and prefix the sign of the greater sum. (2.) (3.) (4) (5.) 3xy ^y" 13o6c2 U^y xy -2i/» 6a6c» — 13x2 1/ — 5a:t/ ly" — ahc" lOx^y Ixy -31/^ — Sabc' — ^r^y — 2x1/ Uy" 4a6c2 24x^3/ 4x1/ Uabc" (6.) (T.) 25x1/2 8(x + i/) — - 60 X 1/ z -4(x + i/) 10x1/2 n^ + y) — -6txi/2 -3(x + i/) 8x1/2 - (^ + 2/) — 74x1/2 T (x + y) 8. Find the sum of 8 xV, — Hx^y", ITx^y*, and — x^^. 9. Find the sum of T(x+y), 8(x + y), — (x + y), and 4 (x + y). Ans. 18 (x + y). 10. Find the sum of — a x^, + a x*, — 10 a x^ + 25 a x^, and — 13ax2. 11. Find the sum of 21 a b, — 34 a i, — 150 ab, 21 a b, and — 13 a i. Ans. — 143 a b. 12. Find the sum of ax*, — 14 a x", IT ax*, — ax*, 44 a X*, and — a x*. 13. Find the sum of 17 (a + b), — (a + b), (a + b), and — 13 (a + 6). Ans. 4 (a + 6). 22 ELEMENTARY ALGEBRA. CASE III. 40. To find the sum of any algebraic quantities. The sum of 5 a and 6 & is neither 11a, nor \lh, and can only be expressed in the form of b a -\- Qb, or 6 5 -j- 5 a ; and the sum of 5 « and — 45 is 5a — 45; but in finding the sum of 5 a, 6 5, 5 a, and — 4 5, the a's can be added together by Case I., and the 5's by Case II., and the two results connected by the proper sign ; thus, 5 a 4-65 + 5a — 45r=10a + 2 5. 1. Find the sum of 6 c?, —25, x, Bi/, 5 x, —3 5, 35c + 4 c?, 5x, T 5 + 2 X, and — 3 5 c. OPERATION. Yor convenience, simi- ^d — 25-1- X -\- Sy -{- Sbc lar terms are written un- 4.d—Sb^6x —3 be ^^^ ^^ch other ; then by + Y5-|-5a: -\-2x Case I. the first column at the left is added; the second by Case II., and 10c?-l-2 5-f 13a; + 3^^ so on ;+ 35c and — 35c cancel. This case includes the two preceding cases, and hence to find the sum of any algebraic quantities : RULE. Write similar terms under each other, find the sum of each column, and connect the several sums with their proper signs. (2.) 4a;— Ya-t-3y — 45-t-3z 6a— ij -\- 4:b — 2z 4:a — 2i/-\-Sb— z — 3a — 85 — 10 c 4x —10c ADDITION. 23 (3.) — 36+ 3c — n \/x + y — 10c-|-8\/^ — y 7 a 4- 6— 10c + 6\/x 4. Add together *l s/~x, — 8.r, 7 x'*, — 6\/^ 4a:*. — 8 a:, 4 a:, and 7 a:'*, — ^ \^ x. Ads. 18 ar*— \2x — *l\/x. 5. Add together Zax — 4a6 + 2a7y, *l ah -\- bx — 4a, *l xy — 3 a + 4 X, and -{- ahc — ax -\- 6a: y. 6. Add together 7a: — 3ay — 5a6-|-4c, 3aa: + 4ar -}- 5 a i — 5 c, and 3c — 3ax-f- *?y4"<^- Ans. llx — 3ay + 3c-|-7y. 7. Add together 5 a — 32 + 7x + 4aa: — 3a^ bah — 5a + 22 — 4aa: + 4, and 6 — 2 a6 + 3a: + 4y -f 4 ax. 8. Add together ^xy -\- Qxz — 6mn-4-4n, 4mn — 3xy-|-2n — 8m n, — 6a:z + 47i — 3a:y4"6, and 10 win — lOn + 3 — 9. Aim. 0. 9. Add together 8am+19na:— 55 3 + c, — 19u + 14 6 — 16c+y, and 18 n a: — 44 am + 15 u — 4y. 10. Add together 17 aa:« + 19 aar^ — 14 aa:* + 16 aa:^, 13 ax' —5 ax* + 6 aar» — 10 ax\ and Uax* + 17 a x^ — 3ax'^+15ax^ Ans. 71 ax^ + lOax"** — 5ax\ 11. Add together m -^ n — 4a + 6c — 7y, 8c — 4m + 3n — 5a + 3c, 7a— 17c + 7y— 10m— 6n, and 14n — 8a — 7c+ lOy — 8 m. 12. Add together 8axy+17 6xy — 16cxy — 9axy, 16 6xy — 18 cxy + lOaxy — 14 ax 2, 16cxy + 25axy — 7 6xy + 25cxy, and lOaxar + Zhxy — lOcxy + 4ax2. Ans. 34axy + 296xy — Sexy. 24 ELEMENTARY ALGEBRA. 13. Add together S{x -\- i/), — 4:{x -\- y), and 1 (x -{- y). Ans. 6 (x -f- y). 14 Add together 5 (2 x + y — 3 z), and —2(2x-\-y — 3z). Note. — If several terms have a common letter or letters, the sum of their coefficients may be placed in parenthesis, and the com' mon letter or letters annexed; thus, 6a:-f8a; — 5ic=(6 + 8 — 5)a:; ax -^ dh X — 2 c X = (a -\- ^h — 2c)x; hcxy-\-adxy — ac xy = {h c -\-ad — a c) xy. 16. Add together ax — bx -\- Zx, and 2ax -\- 4:bx — x. Ans. (3a + 36 + 2) x. 16. Add together by — Scy -\-6ay, and cy -\-4:by — 2 ay. 11. Add together 2xy — axy, and 6xy — Saxy. Ans. (8 — 4 a) xy. 18. Add together 1{Bx-{-5y)-\-Sa — 6x + Sab, 3x + 5 (3a: +5y) -{-la —bab, and 8x +2(Sx -\- 5y) ^la — Sab. Ans. 4Yx+ TOy + 3a. 41. From what has gone before, it will be seen that addition in Algebra differs from addition in Arithmetic. In Arithmetic the quantities to be added are always con- sidered positive ; while in Algebra both positive and neg- ative quantities are introduced. In Arithmetic addition always implies augmentation ; while in Algebra the sum may be numerically less than any of the. quantities added ; thus, the sum of 10 a: and — 8 a; is 2x, which is the numerical difference of the two quantities. SUBTRACTION. 25 SECTION V. SUBTRACTION. 42. Subtraction in Algebra is finding the difference between two quantities. 1. John has 6 apples and James has 2 apples ; how many more has John than James ? Let a represent one apple, and we have 6a 2a 4a ., or 6a — 2a = 4a. 2. During a certain day A made 9 dollars and B lost 6 dollars ; what was the difference in the profits of A and B for the day ? If gain is considered -|-> ^^^^ loss must be considered — , and letting d represent one dol- lar, it is required to take — 6 c? from 9 d. OPERATION. 9rf — 6d It is evident that the difference be- tween A's and B's profits for the day is 9 c? + 6 c/ = 15rf; that is, 9 d — I5d (-6<0 = 9rf+6rf=15rf. Hence it appears that, as addition does not always im- ply augmentation, so subtraction does not always imply diminution. Subtracting a positive quantity is equivalent to adding an equal negative quantify; and subtracting a negative quan- tity is equivalent to adding an equal positive quantity. Suppose I am worth $1000; it matters not whether a thief steals $400 from me, or a rogue having the author- ity involves me in debt $400 for a worthless article ; for 2 26 ELEMENTARY ALGEBRA. in either case I shall be worth only $600. The thief sub- tracts a positive quantity ; the rogue adds a negative quan- tity. Again, suppose I have $1000 in my possession, but owe $400 : it is immaterial to me whether a friend pays the debt of $400 or gives me $400 ; for in either case I shall be worth $1000. In the former case the friend subtracts a negative quantity ; in the latter, he adds a pos- itive. Or, to make the proof general : 1st. Suppose + 5 to be taken from a -\- h the result will be a ; and adding — h to a -\- h v7Q have a -\- h — h, which is, as before, equal to a. 2d. Suppose — h to be taken from a — b the result will be a ; and adding -\- b to a — b we have a — b -\- b, which is, as before, equal to a. 3. Subtract b -\- c from a. OPERATION. I subtracted from a gives a — (b -X- c) :=. a — b — c a — b; but in subtracting b we have subtracted too small a quantity by c, and therefore the remainder is too great by f, and the remainder sought is a — b — c. 4. Subtract b — c from a. OPERATION. In subtracting b from a we a — (b — c)=a — 5-j-c subtract a quantity too great by c ; therefore the remainder (a — b) would be just so much too small, and the remainder sought is a — 6 -|- c. 43* By examining the examples just given it will be seen that in every case the sign of each term of the. subtrahend is changed, and that the subsequent process is precisely the same as in addition ; hence, for subtrac- tion in Algebra we have the following SUBTRACTION. 27 RULE. Change (he sign of each term of (he subtrahend from -f- io — , or — to -f-> or suppose each to he changed, and then proceed as in addition. (!•) (2.) (3.) (4.) (5.) (6.) 0-) Min. 9 9 9 9 9 9 9 Sub. 9 6 3 — 3 — 6 — 9 Rem. 3 6 9 12 15 18 In examples 1-7, the minuend remaining the same while the subtrahend becomes in each 3 less, the remainder in each is 3 greater than in the preceding. (8.) (9.) (10.) (11.) (12.) (13.) (14.) Min. 9 6 3 0—3—6—9 Sub. 9 9 9 9 9 9 9 Rem. —3 —6 —9 —12 —15 —18 In examples 8-14, the minuend in each becoming 3 less ■while the subtrahend remains the same, the remainder in each is 3 less than in the preceding. (15.) (16.) (17.) (18.) (19.) (20.) (21.) Min. 9 6 3 —3 —6 —9 Sub. 9 6 3 —^ — ^ —9 Rem. In examples 15-21, both minuend and subtrahend de- creasing by 3, the remainder remains the same. (1.) (2.) (3.) (4.) (5.) (6.) Min. 26x 27axy — 13a6 —18c 49xy —4386 Sub. 10a: — 4axy 4a6 — 6c — 25xy 27 6 Rem. 16x Zlaxy —llab — 12tf 28 ELEMENTARY ALGEBRA. (1.) (8.) (9.) (10.) (11.) (12.) Min. lOx — 4:axy 4,ah — 6c — 2bxy . 21h Sub. 26x 21axi/ —ISab — > 18 c 4:9x1/ —438^ Rem. — 16a: — Slaxi/ 11 ab 12c (13.) (14.) Min. 6 a:— 14^ + 3 2r 1 a+18b — 10c Sub. 3a:+3y+;2 -_25 5+6c — 82 _|. 4 ^3 J3 _ g ^2 54^ 8. Multiply x^ + 2x^ + Sx''-]-2x+lhjx'' — 2x+l. 9. Multiply x^-{-y^-\-z^ — xy — xz — yz hy x -\-y -\-z, Ans. x^ + 5/^ + 2'^ — 3 X y z. 10. Multiply 4 a;5 — T ic^ + 10 ar' — 13 a;2 by 3 a: — 2. 11. Multiply a^ + a2 — 1 by a" — 1. 12. Multiply x^+lax—Ua^ by a.- —la. Ans. ar^ — 49a2a:— 14a3a; + 98a*. 13. Multiply X -\- y — a by x — y -{- a. 14. Multiply a" + 5'* by a"* — b"". Ans. a "'+" + a*" 6" — a" 6"* — 5'" + ^ 15. Multiply Ixy— 14. x" y"^ -\- 21 x"" f by Gary — 3. Ans. — 21 xy + Ux"" f — 14.1 a? f + l2Qx^y\ 16. Multiply 6 an — 9 a S2 __ 12 aH^ by 2 a & — 3 5^. Ans. 12 a^ }? — 36 a^ 6^ _ 24 a^ Z.» + 2Y a i* + 36 a" b\ V\. Multiply x^ — x^ -\- x^ — x -\- \ by a; + 1. Ans. x^ + 1. 18. Multiply x^ — x^-\-x} — x-\-\ by a: — 1. Ans. x^ — 2x''^2x^ — 2i?-\-2x—\. 19. Multiply X* + ar^ + a;2 + a: + 1 by ar + 1. 20. Multiply a;* + a^ + a:^ + a; + 1 by cc — 1. DIVISION. 87 SECTION VII. DIVISION. 53. Division is finding a quotient which, multiplied by the divisor, will produce the dividend. In accordance with this definition and the Rule in Art. 48, the sign of the quotient must be -|- when the divisor and the dividend have like signs ; — when the divisor and the dividend have unlike signs ; i. e. in di- vision as in multiplication we have for the signs the fol- lowing RULE. Like signs give -f- > unlike, — . CASE I. 54* When the divisor and dividend are both monomials. 1. Divide 6ab by 2b. orERATiON. Thg coefficient of the quotient must 6ab-7-2b = Sa be a number which, multiplied by 2, the coefficient of the divisor, will give 6, the coefficient of the dividend ; i. e. 3 : and the literal part of the quotient must be a quantity which, multiplied by &, will give ab\ i. e. a: the quotient required, therefore, is 3 a. Hence, for division of monomials, RULE. Annex the quotient of the literal quantities to the quotient of their coefficients, remembering that like signs give -\- and unlike, — . 38 ELEMENTARY ALGEBRA. 2. Divide a^ by a\ OPERATION. a« -T- a^ = a^ For a' X a' =^ a^ (Art 50.) Hence, Poivers of the same quantity are divided by each other by subtracting the exponent of the divisor from that of the dividend. (3.) 21x^y'' 9 xy (5.) •^2lQa?y 4.Qx''y 3xy = — 6a; (4.) 4:8a^xy — 16aa; ^-S ay (6.) — 16a2 x^yz__ — 4 axyz * f- Ans. — ■Sxy. 2iahx. Ans. 6ab. 1. Divide S4:aby^ by 2 ay. 8. Divide 297 icV by — 99a:y2. 9. Divide — 14,xy^z by 2a:2/. 10. Divide —UiaH^x by — 24«5a:. 11. Divide ax* by czar'. 12. Divide 8 a:« by — 8 x\ 13. Divide — 210a;''y by ^2 x^ y. 14. Divide —210abx by — 135 a i a;. 15. Divide —4t14:aH^c^ by 158a53c. 16. Divide x"" by ar". Ans. a:"""". 11. Divide 14 a"* a:" by — 1 a"" x. Ans. — 2 a"* ""a;" 18. Divide — UT a^ b* c^ d' by SSaHc d\ 19. Divide 12 (x -\- yf by 4.{x-\-y). Ans. 3 (a: + 2^). 20. Divide —21 {a — by by 9(a — 5)^. 21. Divide {b — cy by (b — cy. Ans. (5 — cf. 22. Divide 14 (x — yf by t (a: — yy. — 1 DIVISION. 39 CASE II. 55» When the divisor only is a monomial. I. Divide ax -{- at/ -{- az by a. OPBRATiox. In the multiplication of a poly- a)ax-^ay-\-az nomial by a mononiial, each term jp I -. I ^ of the multiplicand is multiplied by the multiplier ; and theiv fore we divide each term of the dividend ax -\- ay -\' az by the divisor a, and connect the partial quotients by their proper signs. Uence, RULE. Divide each temi of the dividend by the divisor, and con- nect the several results by their proper signs. (2.) (3.) 3 a ) 6 a x^ — 24: aa? — 5 a:« y ) — 15 ai^y — 25 ar^y 2 2r»— 8ar» 3x + 5 4. Divide 12 a ar* — 24 a a:^ + 42 a a:y by 3 ax, 6. Divide — 4:a^x-\-Sabx — 4 6^ar by — 4a:. 6. Divide 6 a^x* — 12 a^x^ + 15 a*ar« by 3 a^x\ Ans. 2 a;* — 4 a x -[- 5 a* a:*. 1. Divide 12a*y« — 16 a^y* + 20 a«y< — 28 aV by 4aV. 8. Divide — 6 ar» + 10 x^ — 15 x by — 5 x. 9. Divide 273 (a + xy — 91 (a + x) by 91 (a + x). Ans. 3(a + x) — l = 3a + 3x— 1. 10. Divide 20a6c — 4ac -f 8acd— 12 a^ c^ by -^ 4: a c. II. Divide 16 a^x*— 32«2x^ + 48 «*x* by 16 a* x*. 12. DivideT2x«y« — 36x«y»~64x»y«2«by — 18x2y^ Ans. --4 + 2x?/+3x2«. 13. Divide 18ax»y— 54x^2/*+ 108cx^y« by 9x*y. 14. Divide 40 a*b^ + 8 a' 6^ — 96 a» i«x« by 8 a* b\ 15. Divide 39x*2« — 65aa:«2* + ISx'z* by — ISx^z^. 40 ELEMENTARY ALGEBRA. CASE III. 56. When the divisor and dividend are both polyno- mials. 1. Divide :t? — Zx'y -\-Zxy' -^ f by x^ — 'lxy-^- y^. OPERATION. 7? — 2 a; ^ -f- ^^) ic^ — Zx^ y -\- ^xy^ — ^ (a^ — y a^ — 2 x'^ y -\- X y"^ — x"^ y -{- 2 X y^ — if -- x''y + 2xy^^f The divisor and dividend are arranged in the order of the powers of X, beginning with the highest power, a;^, the highest power of x in the dividend, must be the product of the highest power of x in x^ the quotient and x^ in the divisor ; therefore, —^ = x must be the highest power of x in the quotient. The divisor x^ — 2xy -\- y^ mul- tiplied by X must give several of the partial products which would be produced were the divisor multiplied by the whole quotient. When (x' — 2xy -\- if) X = x^ — ^x"^ y -\- xif is subtracted from the dividend, the remainder must be the product of the divisor and the remaining terms of the quotient ; therefore we treat the remain- der as a new dividend, and so continue until the dividend is ex- hausted. Hence, for the division of polynomials we have the fol- lowing RULE. Arrange the divisor and dividend in the order of the powers of one of the letters. Divide the first term of the dividend by the first term of the divisor ; the result will be the first term of the quotient. Multiply the whole divisor by this quotient, and subtract the product from the dividend. Consider the remainder as a new dividend, and proceed as before until the dividend is exhausted. DIVISION. 41 Note. — If the dividend is not exactly divisible by the divisor, the remainder must be placed over the divisor in the form of a fraction and connected with the quotient by the proper sign. 2. Divide a:* + 4y* by a:« — 2ary + 2y'. x*—2x'y + 2^1/' 2x>y 4y ixf 2x»y' — 2r'y» — ixy' + 4xy» + 4y 4y Note. — By multiplying the quotient and divisor together all the terms which appear in the process of dividing will be found in the partial products. 3. Divide a:* — 1 by a: — 1. X — 1) X* — 1 (x^' + x' + x+l x» — 1 • 3^ — 7^ x" — 1 3? — X X — 1 X 1 4. Divide ax — ay -{-hx — by -\- z by x — y, ax — ay ax — ay x^y)ax-^ay-\-hx^hy-\-z{a + h-\- j-^ hx — hy hx — hy 5. Divide 2by^2}ry — Zlryz-\-Qb^y-\-byz—yz by 2 6 — 2. Ans. 3 h^y — by-{-y. 42 ELEMENTARY ALGEBRA. 6. Divide c^ -\- x^ by c -\- x. Ans. c^ — ex -\- x^. 1. Divide a^-{-a^-{-a^x-\-ax-\-Sac-\-Schya-\-l. Ans. a^ -\- a X -{- 3 c. 8. Divide a -\- b — d — ax — bx-\-dx by a-{- h — d. 9. Divide 2 a* — 13 a^ 2/ + 11 a^y^ _ g « ?/ -f 2 ?/ by 2 or — ciy -\- y'^. Ans. a^ — 6 a y + 2 ?/l 10. Divide a^ — Z an -\- Z ah'' — b^ hy a^ — 2 ah -\- h\ 11. Divide 2 a;3— 19x2+ 26a:— 19 by a: — 8. Ans. 2a;2 — 3a;4-2 ?-r ' X — 8. 12. Divide ar^ + 1 by a: + 1. Ans. X* — x^ -\- x' — a: -|- 1. 13. Divide a;« — 1 by a; — 1. Ans. a:^ 4" ^* ~1~ ^^ + ^^ "t" ^ ~l~ 1- 14. Divide x^ — 1 by a: -j- 1. 15. Divide -x^ — ^ by x — y. Ans. x^ -\- x^ y -\- x^ y'^ -\- X ^ -\- y^. 16. Divide a^ — x^ by a — x. 17. Divide m^ — n^ hy m' -\- m n -\- n^. Ans. w^ — m^n -\- mn^ — n*, 18. Divide 4 a* — 9 a^ + 6 a — 3 by 2 a^ + 3 a — 1. 19. Divide a:* + 4x2^2 _|_ 3^4 |3y ^_^ 2 y. 20. Divide a^ — a^ x"" -\- 2 a a? — x"" by a^ — ax-{- x\ 21. Divide x^ + 2 x^ y^ -\- y^ hy x'^ — xy + y\ 22. Divide 1 — a^ by 1 + « + a^ + al Ans. 1 — a. 23. Divide 10 x^ — 2Qx''y-\- 30 ^ by a; + y. 24. Divide 7 a a;* + 21 a x^ + 14 a by a: + 1. Ans. 7 a a;^ + 14 a ar^ — 14 a a: -|- 14 a. 25. Divide 27 a^ y^ — 8 a^y by 3^/^ — 2 ay. DEMONSTRATION OF THEOREMS. 4S SECTION YIII. DEMONSTRATION OF THEOREMS. 57. From the principleB already established we are pre- pared to demoustrate the foUowiug theorems. THEOREM I. The sum of two quantities plus their difference is twice the greater ; and the sum of two quantities minus their dif- ference is twice the less. Let a and b represent the two quantities, and a ^ & ; their sum is a -\-b] their difference, a — b. PROOF. 1st. {a + b) + {a — b)=a-{-b-\-a^b = 2a; 2d. (a-\-b) — (a — b)=a + b — a+ b = 2b. Therefore, when the sum and difference of two quanti- ties are given to find the quantities, RULE. Subtract the difference from the sum, and divide the re- mainder by two, and ice shall have the less ; the less plus the difference will he the greater. In the following examples the sura and difference are given and the quantities required. 1. 16 and 12. Ans. 14 and 2. 2. 272 and 18. 3. 456 and 84. 4. Sum 2 X and difference 2 y. Ans. x-\-y and x — y. 5. Sum Ix'^ + Zy and difference bx'^ — Zy. 6. Sum 2 a — 8 ft and difference 10 a + 14 b. Ans. 6 a + 3 6 and — 4 a — 11 ft. 44 ELEMENTARY ALGEBRA. THEOREM II. 58. The square of the sum of two quantities is equal to the square of the first, plus twice the product of the two, plus the square of the second. Let a and h represent the two quantities ; their sum will be a-\-b. PROOF. a +h a +b a''+ ah + ab + h'' a^-\-2ab + b'' According to this theorem, find the square of 1. X -{- 1/. Ans. a:^ -j- 2 a: y -f- y • 2. 2x + 2y. Ans. 4a:2_|-8x^/ + 4/. 3. a:+ 1. 4. 4: + X. 5. 2x + St/. Ana. 4:x^+l2xy + 9f. 6. Sa-\-b. THEOREM III. 59. The square of the difference of two quantities is equal to the square of the first, minus twice the product of the two, plus the squar-e of the second. Let a and h represent the two quantities, and a ]> 6 ; their dif- ference will be a — h. PROOF. a — b a — b a^ — ab — ab + b^ a2 _ 2 a 6 + 63 DEMONSTRATION OF THEOREMS. 45 According to this theorem, find the square of 1. X — y. Ans. x^ — Ixy '\- y^. 2. 2x — 4y. 3. X— 1. Ans. x^ — 2a:+ 1. 4. Tx — 2. THEOREM IV. 60. The product of the sum and difference of two quan- tities is equal to the difference of tJieir squares. Let a -^ b he the sum, and a — b the difference of the two quantities a and b. PROOF. a +b a —b a-' + ab --ab — b' According to this theorem, multiply 1. X -\- y hy X — y. Ans. x^ — y^. 2. 2 3:+ 1 by 2a:— 1. 3. ar» + t/2 by x* — y^. Ans. x* — y*. 4. 3 X + 4 by 3 a: — 4. 6. 3xy-["4a6by3xy — 4a J. 61 • This theorem suggests an easy method of squaring numbers. For, since a^ = {a — b) (a -\- b) -\- If^y 992 = (99 — 1) (99 + 1) + 1* = 98 X 100 + 1 = 9801. In like manner, 96«= 92 X 100 + 16 = 9216. 998^ = 996 X 1000 + ^ = 996004. 4972 = 494 X 500 + 9 = 247 X 1000 + 9 = 247009. 46 ELEMENTARY ALGEBRA. In accordance with this principle find the square of 1- 98. 4. 493. 7. 888.- 2. 89. 5. 789. 8. 999. S. 45. 6. 698. 9. 1104. Miscellaneous Examples. 1. Find the square of 3 a: — 6y. Ans. 9x^ —Mxy-\-S6f. 2. Find the square of 4:axy -\- 7 abx. Ans. 16a^x^y^-[- 5Qa^bx'^i/ + 4:9an^x^. 3. Multiply 1 x-\-l by T:r— 1. Ans. 49 x^ — 1. 4. Required those two quantities whose sum is Bx-\-2a and difference x — 2 a. Ans. 2 a; and x -\- 2 a. 6. Expand (rr^ — 4)2. 6. Multiply 4:ab -}- S by 4aJ— -3. T. Find the square of 14: aH^ + 10 a?^ 2^. 8. Find the square of 4 a — b. 9. Multiply 10 a: 4- 2 by 10 a; — 2. 10. Find the square ofSaa; — ^axy. Ans. 9a'x^ — 4.^a''x'y-\-Ua^x^f. 11. Find the square of 2 a + 5. 12. Find the value of (6 a + 4) (6 a — 4) (36 a^ + 16). Ans. 1296 a^ — 256. 13. Find the square of 10 a^ _ 5 52 14. Expand (3 a^a: + 4 bfy. Ans. 9 a* a:2 _|_ 24 ^2 j^y8 _j_ jg ^2^^«^ 15. Find the product of a}'' + 1, «» -f 1, a^ _|_ j^ ^2 _j_ |^ a+l'^nda— 1. An^. a^^ — I. 16. Find the product of a -[- i, a — 5, and a^ — 5^ FACTORING. 47 SECTION IX. FACTORING. 62. Factoring is the resolving a quantity into its fac- tors. 63. The factors of a quantity are those integral quanti- ties whose continued product is the quantity. Note. — In usin^ the word factor we shall exclude unity. 64* A Prime Quantity is one that is divisible without remainder by no integral quantity except itself and unity. Two quantities are mutually prime when they have no common factor. %5» The Prime Factors of a quantity are those prime quantities whose continued product is the quantity. 66. The factors of a purely algebraic monomial quan- tity are apparent. Thus, the factors of arbxyz are 67. Polynomials are factored by inspection, in accord- ance with the principles of division and the theorems of the preceding section. CASE I. 68. When all the terms have a common factor. 1. Find the factors of ax — ah -\- ac. OPERATION. As a is a factor of (ax — ah -\- ac) z= a {x — 6-j-c) each term it must be a factor of the poly- nomial ; and if we divide the polynomial by a, we obtain the other factor. Hence, 48 ELEMENTARY ALGEBRA. RULE. Write the quotient of the poly-nomial divided by the com- mon factor^ in a parenthesis, with the common factor pre- fixed as- a coefficient. 2. Find the factors oi ^ xy ^12 xf -\- l^ ax^ f. Ans. Qxy{l — l2y-\-^axy^). Note. — Any factor common to all the terms can be taken as well as 6 a; y ; 2, 3, a:, ?/, or the product of any two or more of these quan- tities, according to the result which is desired. In the examples given, let the greatest monomial factor be taken. 3. Find the factors of a: -|- x"^. Ans. x {\ -\- x). 4. Find the factors of 8 a^ a;^ + 12 a^ ar^ — • 4 a a:y. Ans. 4 a a; (2ax -\- S a^ x^ — y). 6. Find the factors of 5 x^ y'^ + 25 ax^ — 15 x^f. Ans. 5 a^ (xy"^ -\- 5 ax^ — 3 y). 6. Find the factors of 7 a a: — %hy -^-l^x^. T. Find the factors of 4 x^ y^ — 28 x'' y'^ — 44 ?:* y'^. 8. Find the factors of 55 a^ c — 11 a c -\- ^^ a^ c x. 9. Find the factors of ^^ a" x^ — 2^1 a^ x"" y\ 10. Find the factors o^ Ibd'^ cd — ^ ah'' d^ -\-l%a^(?d\ CASE II. 69. When two terms of a trinomial are perfect squares and positive, and the third term is equal to twice the product of their square roots. 1. Find the factors oi a" + 2 ah -\-h\ OPERATION. Y^Q resolve this into a^ -\- 2 ah -\- y^ z= {a -\- h) {a -\- h) its factors at once by the converse of the principle in Theorem II. Art. 58. FACTORING. 49 2. Find the factors of a' — 2 a J + ^. OPERATION. 'VVe resolve this into a* — 2 a i + 6' = (a — b) (a — b) its factors at once by the converse of the principle in Theorem IIL Art 59. Hence, RULE. Omitting the term thai is equal to twice the product of the square roots of the oUier two, take for each factor the square root of each of the other two connected by the sign of the term omitted. 3. Find the factors of x* — 2a;y +5^. Ans. (x—y) (x — y). 4. Find the factors of 4: a^ c^ -{- 12 acd + 9 d\ Ans. (2ac + 3d) (2ac-}-3d). 6. Find the factors of 1 — 4: x z -\- 4: x^ z^. Ans. (1 — 2xz) (1 — 2x2). 6. Find the factors of9a:2_g^_j_l Ans. (3a:— 1) (3a:— 1). 1. Find the factors of 25 a:* + 60 a; + 36. 8. Find the factors of 49 a^ — 14 a a: + ar». Ans. (la — x) (la — x). 9. Find the factors of 16^2—16 0*^ + 4 0*. 10. Find the factors of 12 ax + 4ar' -f 9 a^ 11. Find the factors of 6 x + 1 + 9 x». CASE III. 70. When a binomial is the difference between two squares. 1. Find the factors of a^ — ^. OPERATIOX. We resolve this into its fac- q2 if^ z=z (a -[- b) (a b) ^^ ** ^"^^ ^^ ^'*® converse of , the principle in Theorem IV. Art 60. Hence, S D 50 . ELEMENTAEY ALGEBRA. RULE. Take for one of the factors the sum, and for the other the difference, of the square roots of the terms of the hi- nomial. 2. Find the factors of oiP — y^. Ans. (x+y) (a; — y). 3. Find the factors of 4 a^ _ 9 b\ Ans. (2a + 3 52) (2 a — 3 J^). 4. Find the factors of 16 x"^ — c^. 5. Find the factors of a^^^c^ — oc^y^. 6. Find the factors of 81x^ — 49/. 1. Find the factors of 25 0^ — 4 c*. 8. Find the factors of m^ — w^*^. Note. — When the exponents of each term of the residual factor obtained by this rule are even, this factor can be resolved again by the same rule. Thus, x^ — ?/* = (oi^ -\- y^) {a? — y^) ; but 7? — ^ = (x -f- 2^) {x — y) ; and therefore the factors of x^ — y^ are a:^ -|- ^, X -\-y, and x — y. 9. Find the factors of a^ — h\ Ans. (a2 -f 6=2) (« + h) {a — h). 10. Find the factors of a;^ — y'^. Ans. {x^ + y') (x^ + f) {x + y) (a: - y). 11. Find the factors of a* — 1. 12. Find the factors of 1 — x^. Ans. (1 + x') (1 + a:2) (1 _|_ ^) (i __ ^y 13. Find the factors of a' — a^. Ans. a^{a-\- 1) (a — 1). 14. Find three factors of x^ — x^. 71. Any binomial consisting of the difference of the same powers of two quantities, or the sum of the same odd powers, can be factored. For . FACTORING. . 51 I. T?ie difference of the same powers of tv30 quantities is divisible by the difference of the quantities. Ivct a and b represent two quantities and a ]> 6, and by actual division we find ^ = -' + -^ + ^' and so on. II. The difference of the same even powers of two quan-- tities is divisible by the sum of the quantities. a -f 6 0* — 6* = a — h, a +6 ~~^ = a^ — aH -\- a^V" — a^l^ + ah'' — ¥, and so on. It follows from the two preceding statements that The difference of the same even powers of two quantities is divisible by either the sum or the difference of the quan- tities. III. TTie sum of the same odd powers of two quantities is divisible by the sum of tJie quantities. "^^^ = a* — a«6 + a«i» — a P+ 5*, a -{- b ' ' ' ^^ = a« — a* J + a*i^ — a« J^ + a*6* — a^ + 5^, and so on. 52 ELEMENTARY ALGEBRA. 1. Find the factors of oc^ — y^. OPERATION. (x» — 7/) --- (x -^ ij) = x^ + of'y + c^y^ -{- xf + y* By I. of this article, the difTerence of the same powers of two quantities is divisible by the difference of the quantities; therefore X — 7/ must be a factor of x^ — y^; and dividing x^ — 1/ by x — y gives the other factor x^ -^ x^ y -\- x^ y^ -\- x 1/ -\- y*. 2. Find two factors of c° — d^. OPERATION. (c^ — d') -i- (c + d) =zc'-^c^d + (^d^ — c^d^ + cd^-^d^ By 11. the difference of the same even powers of two quantities is divisible by the sum of the quantities ; therefore c -\- d must be a factor of c® — r/"; and dividing c® — r/* by c -\- d gives the other factor c' — c*d-}- c' d^ — c-'^ d' -^ c d' — d\ 3. Find the factors of m^ -\- n^. OPERATION. {m^ -|- n^) -T- {m -\- n) =:^'m^ — m^ n -\- m^'nP' — mn^ -\- n^ By III. the sum of the same odd powers of two quantities is divisible by the sum of the quantities ; therefore m-\- n must be a factor of m'^ -j- n^ ; and dividing irv' -\- n^ by m -\- n gives the other factor 771* 77l' 71 -|- 771^ f? — 771 71^ -|- 7i*. 4. Find the factors of c^ — 7?. Ans. (a — x) {c? -\- a X •\- :ji?) , 6. Find the factors of a^ -\- x^. Note. — In Example 2, the factors of c* — rf' there obtained are not the only factors; for by I. c' — d^ is divisible by c — c/; and dividing c* — d^ by c — d gives another factor, c^ -\. c' d ^ (^ d^ ■\- (? d' ■\- c d^ ■\- d'-, or by Art. 70, c« — d!« « (c« + rf») (c» — d»). FACTORING. 53 But c* — c*rf-[-c»6/«--c»(i»-t-crf* — rf», c^-^c*d-\-c'd'-{-c'd'-{-cd*-^d\ are not prime quantities ; for the first can be divided by c — c?, and the quotient thus arising can be divided by c* ± c J -j- t/' ; the second can be divided by c -j- 4 ELEMENTARY ALGEBRA. SECTION X. GREATEST COMMON DIVISOR.* 72. A Common Divisor of two or more quantities is any quantity that will divide each of them without remainder. 73. The Greatest Common Divisor of two or more quantities is the greatest quantity that will divide each of them without remainder. 74. To deduce a rule for finding the greatest common divisor of two or more quantities, we demonstrate the two following theorems : — Theorem I. A common divisor of two quantities is also a common divisor of the sum or the difference of any multiples of each. Let A and B be two quantities, and let d be their common di- visor ; d is also a common divisor of m ^ dz n B. Suppose A -^ d == p'^ i. e. A = dp, and mA = dmp, and B -^ d = q; I. e. B = d q, and n B == d n q\ then mA±nB = d7np±dnq = d (mp ± n q). That is, d is contained in m A -\- n B, m p -\- n q times, and in m A — n B, mp — n q times ; i. e. cZ is a common divisor of the sum or the difference of any multiples of A and B. Theorem II, The greatest common divisor of two quan- tities is also the greatest common divisor of the less and the remainder after dividing the greater by the less. Let A and B be two quantities, and A"^ B; and let the process of dividing be as appears in B) A (q the margin. Then, as the dividend is equal to qB the product of the divisor by the quotient plus ~ the remainder, A = r-^qB. (1) * See Prefia«e. GREATEST COMMON DIVISOR. 55 And, as the remainder is equal to the dividend minus the product of the divisor hy the quotient, r= A—qB. (2) Therefore, according to the preceding theorem, from (1) any divisor of r and B must be a divisor of A ; and from (2) any divisor of A and jS, a divisor of r ; i. e. the divisors of A and B and B and r are identical, and therefore the greatest common divisor of A and B must also be tlie greatest common divisor of B and r. In the same way the greatest common divisor of B and r is the greatest common divisor of r and the remainder after dividing B by r. Hence, to find the greatest common divisor of any two quantities, RULE. Divide Oie greater by the less, and the less by the remain- der, and so continue till the remainder is zero; the last dir visor is the divisor sought. Note 1. — The division by each divisor should be continued until the remainder will contain it no longer. Note 2. — If the greatest common divisor of more than two quan- tities is required, find the greatest common divisor of two of them, then of this divisor and a thirds and so on ; the last divisor will be the divisor sought. Note 3. — The common divisor of x i/ and z r is a: ; x is also the common divisor of x and x z, or of ax y and xz\ i. e. the common divUor of two quantities is not changed hy rejecting or introducing into either any factor which contains no factor of the other. Note 4. — It is evident that the greatest common divisor of two quantities contains all the factors common to the quantities. CASE I. 75» To find the greatest common divisor of monomials. 1. Find the greatest common divisor of S a"^ li^ c d, WaH'c\ and 2S aH* c. The greatest common divisor of the coefficients found by the gen- eral rule is 4 ; it is evident that no higher power of a than a*, of 66 ELEMENTARY ALGEBRA. h than 5', of c than itself, will divide the quantities ; and that d will not divide them ; therefore, the divisor sought is 4 a^ b^ c. Hence, RULE. Annex to the greatest common divisor of the coefficients those letters which are common to all the quantities, giving to each letter the least exponent it has in any of the quantities. 2. Find the greatest common divisor of 63 a^ b'' c^ d^, 27 a^ b' c^ and 45 a^ b"" c« d. Ans. 9 a^ h' c\ 3. Find the greatest common divisor of ^bx'y^z^ and l2babxUfz\ 4. Find the greatest common divisor of 99 a S^c^rf^a:^^ and 22 a" b^ c^ d^ xK Ans. II a b'^c'^d^ a^. 5. Find the greatest common divisor of 11 x^y"^, l^si^y^, and 2l2bx'y^^. CASE II. 76. To find the greatest common divisor of polynomials. 1. Find the greatest common divisor of a;^ — y^ and x"^ — 2xy -\- y"^. x'-^f)x^ — 2xy-{- f(l x^ — y"^ ^2xy + 2y^ Kejecting the factor 2 y X— y)x'^ — y^(x + y x^ — xy xy — y^ xy — y^ Ans. x — y. RULE. Arrange the terms of both quantities in the order of the powers of some letter^ and (lien proceed according to the general rule in Art. 74. Note 1. — If the leading term of the dividend is not divisible by the leading term of the divisor, it can be made so by introducing GREATEST COMMON DIVISOB. 67 in the dividend a factor which contains no factor of the divisor ; or either quantity may be simplified by rejecting any factor which contains no factor of the other. (Art. 74, Note 3.) Note 2. — Since any quantity which will divide a will divide — a, and vice versa, and any quantity divisible by a is divisible by — a, and vice versa, therefore all the signs of either divisor or dividend, or of both, may be changed from -|- to — , or — to -|-» "without changing the common divisor. Note 3. — When one of the quantities is a monomial, and the other a polynomial, either of the given rules can be applied, although gen- erally the greatest common divisor will be at once apparent. 2. Find the greatest common divisor of ax* — a^ar* — 8 a^x^ and 2car* — 2aca:" + 4a=^car* — 6a^cx — 20a*c. ax* — o'z* — Bc^x* DlTidlng by a Z* ar* — a z* — 8 a* 2a«x» — 3a»x — 2a* DlTidlng by a" 2z» — 3ax— 2a» 2 car* — 2ac3^-\-4a*ca^ — Sc^cx — 20a*c Dividing by 2 C x* — ax^-\-2a'x' — 3a*x—l0a* (1 a;* — gg' — 8a* 2a'z' — 30*3: — 2a* ntBem. 3^— ax*— 8a* Multiplying by 2 2 a;* — 2 a x» — 16 a* (a:" 2 a:* — 3 ga:* — 2 0*0:* a 3* -\- 2 a* x"— 16 a* Multiplying by 2 2ax»-f4a»x'— 32a* (aa; 2a3* — 3a*x* — 2a*z 7a*x»+ 2a»z — 32a* Multiplying by 2 14 a« a:* -f 4 o* a: — 64 a* (7 a« 14a'a:*— 21a'a:— 14 a* 25 a" ar — 50a* 25a'x — 50a* sdBem. DlTlding by 25 O* X— 2a)2a:' — 3ax— 2a'(2z-|-a 2a:'— .4ax ax— 2a" ax — 2a* Ans. x — 2a. 58 ELEMENTARY ALGEBRA. 3. Find the greatest common divisor of a* — x* and a^ -\- a^x — ax^ — x^. Ans. a^ — x^. 4. Find the greatest common divisor of a* — x* and a® — a^ x^. Ans. c^ — x^. 5. Find the greatest common divisor o£2ax^ — a^x — a^ and 2 x^ -\- S a X -\- a^. 6. Find the greatest common divisor of 6 a x — S a and Q ax^ -\- ax^ — 12 ax. Ans. Sax — 4a. v. Find the greatest common divisor of x^ — y^ and 8. Find the greatest common divisor of 3 ic^ — 24 aj — 9 and 2x^—16x — Q. 9. Find the greatest common divisor of x^ — y^ and x^ — y^. Ans. x — y. 10. Find the greatest common divisor of 10 x* — 20x'^y 4-30/ and x^ + 2x^y -\- 2xy^ + yK Ans. x + y. 11. Find the greatest common divisor of a* + "^ H~ ^^ + a — 4 and a^ + 2 a« + 3 a^ + 4 a — 10. Ans. a — 1. 12. Find the greatest common divisor of 1 ax^ -\- 21ax^ -f- 14 a and x^ -{- x^ -{- x^ — x. Ans. x^ I. 13. Find the greatest common divisor of 21 a^y^ — 8 a^y and 3/ — 2 a/ 4- 3 aV — 2 a^y\ Ans. By"^ — 2 ay. 14. Find the greatest common divisor of a^ -{- a — 10 and a'^ — 16. Ans. a — 2. Note 5. — The greatest common divisor of polynomials can also be found by factoring the polynomials, and finding the product of the factors common to the polynomials, taking each factor the least num- ber of times it occurs in any of the quaptities. (Art. 74, Note 4.) 15. Find the greatest common divisor of3a^^ — 4aar-f- Saxy — 4: ay and a^ x — x -\- a^y — y. Sax^ — 4:ax -{- Saxy — 4zay = a(x -\- y) (Sx — 4) a^x — X -{- a^y — yz=:(x-\-y)(a — I) (a^ -\- a -\- 1) Ans. so -\- y. LEAST COMMON MULTIPLE. 60 SECTION XI. LEAST COMMON MULTIPLE. 77. A Multiple of any quantity is a quantity that can be divided by it without remainder. 78. A Common Multiple of two or more quantities is any quantity that can be divided by each of them with- out remainder. 79. The Least Common Multiple of two or more quan- tities is the least quantity that can be divided by each of them without remainder. 80. It is evident that a multiple of any quantity must contain the factors of that quantity ; and, vice versa, any quantity that contains the factors of another quantity is a multiple of it : and a common multiple of two or more quantities must contain the factors of these quantities ; and the least common multiple of two or more quantities must contain only the factors of these quantities. CASE I. To find the least common multiple of monomials. 1 . Find the least common multiple of 6 a^b^c, Sc^lr^c^d, and 12 a* b ex. The least common multiple of the coefficients, found by inspection or the rule in Arithmetic, is 24 ; it is evident that no quantity which contains a power of a less than a*, of b leas than fc*, of c less than c\ and which does not contain d and a:, can be divided by each of these quantities ; therefore the multiple sought is 24 a* l^ c^ d x. Hence, iu the case of monomials, 60 ELEMENTARY ALGEBRA. RULE. Annex. (0 the least common multiple of the coefficients all the letters which appear in the several quantities, giving to each letter the greatest exponent it has in any of the quan- tities. 2. Find the least common multiple of 3 a* h^ c^ QaH*e d\ and IQahcx^. Ans. 30 a'' i^c^^^^^ 3. Find the least common multiple of IQahx, 80 a Z»^a:^ and ZbaHx\ Ans. 560a^Z»^ar^ 4. Find the least common multiple of Qa^b^, Iba'^bx^, and l^axy'^. Ans. ^Q a'^b^x^y^. 6. Find the least common multiple of l^a^bc^x, 2^ab^cx^y, and Z^aH'^xz. 6. Find the least common multiple of Ki^xyz, 4.5 a be, and 25 m n. 1. Find the least common multiple of 10 a^ by"^, 13 a^ b"^ c, and llan^c^ 8. Find the least common multiple of 14 a^ b'^ c*, 20 a^ b c*, 25a«6c», and 28 abed. CASE II. 81 1 To find the least common multiple of any two quantities. Since the greatest common divisor of two quantities contains all the factors common to these quantities (Art. 74, Note 4) ; and since the least common multiple of two quantities must contain only the factors of these quantities (Art. 80) ; if the product of two quanti- ties is divided by their greatest common divisor, the quotient will be their least common multiple. Hence, to find the least common multiple of any two q^uautities^ LEAST COMMON MULTIPLE. Oft RULE. Divide one of the quantities by their greatest common di- xrisor, and multiply this quotient by the other quantity, and the product unit be tJieir least common multiple. Note 1. — If the least common multiple of more than two quanti- ties is required, find the least common multiple of two of them, then of this common multiple and a third, and so on ; the last com- mon multiple will be the multiple sought. Note 2. — In case the least common multiple of several monomials and polynomials is required, it may be better to find the least com- mon multiple of the monomials by the Rule in Case I., and of the polynomials by the Rule in Case II., and then the least common multiple of these two multiples by the latter Rule. 1. Find the least common multiple of x^ — y^ and x' — 2xy + y\ OPERATION. Their greatest common X — y) x^ — 2xy-[-y divisor is x — y, with which we divide one of ""^ y the quantities ; and mul- (^ ir) (^ — y)> Ans. tiplying the other quan- tity by this quotient, we have the least common multiple (x* — ^ (x — y). 2. Find the least common multiple of 2 a* a:", ^ar^y, a* — X*, and a* — a' oc^. The least common multiple of the monomials is 4 a' x* y ; and the least common multiple of the polynomials is a' (a* — x*). The greatest common divisor of these two multiples is a' ; and dividing one of these multiples by a\ and multiplying the quotient by the other, we have 4 a? x* y (^a* — x*) as the least common mul- tiple. 3. Find the least common multiple of Sd^l^, 6a^by, a' — 8, and a^ — 4 a + 4. Ans. 6 a^ b^y («» — 8) (a — 2). 62 ELEMENTARY ALGEBRA. 4. Find the least common multiple of 3 a;^ — 24 a; — 9 and 2x^—16x — 6. (See 8th Example, Art. 76.) 5. Find the least common multiple of a* — x^ and 6. Find the least common multiple of a:* — 1, x^-\-2x-{-l, and (x — 1)^. Ans. x^ — x'^ — x^ -\- 1. t. Find the least common multiple of a:^ — y^ and x^ -j" y^* 8. Find the least common multiple of a^ -j~ ^ — ^^ ^^^ a* — 16. Note 3. — The least common multiple of any quantities can also be found by factoring the quantities, and finding the product of all the factors of the quantities, taking each factor the greatest number of times it occurs in any of the quantities. (Art. 80.) 9. Find the least common multiple of x^ — 2xy-{-y'^, x^ — y^, and [x -f- yy. x^—2xy-\-y'^={x — y){x — y) x'-y^= (x' + f) (X + y){x- y) (x + yy ={x + y) (x + y) Hence L. C. M = (a; — ?/) (x — y) (a;^ _(- y'^) (x -\-y) (x + y) = x^ — x^y^ — x'^,y^ + y^. 10. Find the least common multiple of Bax^ — 4aa?-(- S axy — 4: ay and a^ x — x -\- a^y — y. (See 15th Example, Art. 76.) Ans. a{x -\- y) (S x — 4.) (a'' + a + 1) (a — l)= Sa'x'' — 4:a^X -\- Ba^xy — 4a'*?/ — Sax'^ -\- 4,ax — Baxy + 4a?/. FRACTIONS. - 63 SECTION XII. FRACTIONS. 82. When division is expressed by writing the dividend over the divisor with a line between, the expression is called a Fraction. As a fraction, the dividend is called the numerator, and the divisor the denominator. Hence, the value of a fraction is the quotient arising from dividing the numerator by tlie denominator. X V Thus, - is a fraction whose numerator is z y and denominator y, and whose value is x. 83. The principles upon which the operations in frac- tions are carried on are included in the following THEOREM. Any multiplication or division of the numerator causes a like change in the value of (lie fraction, and any multiplica- tion or division of the denominator causes an opposite change in tlw value of (lie fraction. Let — ^ be any fraction ; its value == a: y. Ist. Changing the numerator. Multiplying the numerator by y, which is y times the value of the given fraction. Dividing the numerator by y, — *> y which is — of the . value of the given fraction. 64 ELEMENTARY ALGEBRA. 2d. Changing the denominator. Multiplying the denominator by y, which is — of the value of the given fraction. Dividing the denominator by y, which is y times the value of the given fraction. Corollary. — Multiplying or dividing both numerator and denominator by the same quantity does not change the value of the fraction. For if any quantity is both multiplied and divided by the same quantity its value is not changed. ^^' y ~ cy ~ I ~^' 84i Every fraction has three signs r one for the numer- ator, one for the denominator, and one for the fraction as a whole. Thus, +^- If an even number of these signs is changed from -\- to — , or — to -\-, the value of the fraction is not changed ; but if an odd number is changed, the value of the fraction is changed from -\- to — , or — to -\-. Thus, changing an even number, — xy_ -\-xy_ -\ry —y taking 1 +^y_- . jp +^^=+^; FRACTIONS. 6^r and changing an odd number, __ + ^?/ _ 1 — _5_y __, I +^y -_ __ — ^y = _ x 4-y "^ H-y — y — y The various operations in fractions are presented under the following cases. CASE I. 85t To reduce a fraction to its lowest terms. Note. — A fraction is in its lowest terms when its terms are mu- tually prime. 1. Reduce „:— ,— ^^ to its lowest terras. 24 a' z y" operation. Since dividing both terms 16 a' :r y 4xy 2 °^ * ^"^"'^'°" ^^^ ^^« '^°^« 24^T^ — 67y — ry quantity does not change its value (Art. 83, Cor.), we divide both terms by any factor common to them, as 4 a' ; and both terms of the resulting fraction by any factor common to them, as 2xy\ or we can divide both terms of the given fraction by their 2 greatest common divisor* 8 a' x y ; the resulting traction — is the fraction sought Hence, RULE. Divide both terms of the fraction by any factor common to them ; then divide these quotients by any factor common to them ; and so proceed till the terms are mutually prime. Or, Divide both terms by their ffi'eatest common divisor. 2. Reduce -~i to its lowest terms. Ans. x«y» xy 972 a' x*?/* 2x 3. Reduce T^^.— ^i to its lowest terms. Ans. ;— , — 408 a* ar y* 3 a' y 24 X 1/ z 2z 4. Reduce — - — ^— to its lowest terms. Ans. — 12a x y a 5. Reduce ^^-^,^'^ to its lowest terms. 51 crbxy 66 ELEMENTARY ALGEBRA. 6. Reduce 7^77— 7^-r^ to its lowest terms. 1. Reduce „ , , — ^—. — ^ to its lowest terms. x^-^2xy -{-f Ans. ^ a'—2ab-\-b' 8. Reduce -5 ^ ^ ,, , ,, to its lowest terms. 9. Reduce ^-— ^ ^ — to its lowest 2 ex* — 2acx^ -f- Aarcx' — 4 a^ c x terms. . aa^ ' ' J(2^'~^fT^' 10. Reduce — -^ — ^- — . to its lowest terms. gr — x* CASE II. 86. To reduce fractions to equivalent fractions having a common denominator. o c 1. Reduce j~ and j- to equivalent fractions having a common denominator. OPERATION. y^Q multiply the numerator and a g bx denominator of each fraction by the bij tt^xy denominator of the other (Art. 83, T ^ ,, Cor.). This must reduce them to c c y / j~ — ^r^ equivalent fractions having a common denominator, as the new denominator of each fraction is the product of the same factors. ORj In the second operation we find the a a X least common multiple, b x y, of the by hxy denominators by and fta:; as each de- c _ cy nominator is contained in this multi- Wx hxy pl®' 6^ch fraction can be reduced to a fraction with this multiple as a de- nominator, by multiplying its numerator and denominator by the quotient arising from dividing this multiple by its denominator. Hence, FRACTIONS. 6T RULE. Multiply all the denominators together for a common de- nominator, and multiply each numerator into the continued product of all the denominators, except its own, for new numerators. Or, Find the least common multiple of Oie denominators for the least common denominator. For new numerators, mul- tiply each numerator by tJie quotient arising from dividing this midtiple by its denominator. 2. Reduce — » ~. and —r- to equivalent fractions xy ab aby ^ having the least common denominator. . ahm nxy , '^ Ans. -T — » ^ > and a b xy ab xy a bxy 3. Reduce -— :. rr-r^* and — 7 to equivalent frac- 15 6 10 6 c 25 act/ ^ tions having the least common denominator. . 80 a' erf 45 ad xy , 126 a; ^^^' TbOabcd* IbO^bVd' ^ 150abcd' 4. Reduce — » » and z—, to equivalent fractions m nxy b a ^ having the least common denominator. 5. Reduce , . and —^t to equivalent fractions hav- ing the least common denominator. . a" — 2a6-f-6* , a*m -\- ahm 6. Reduce _ and — ^— r to equivalent fractions having the least common denominator. 7. Reduce -, v — i — » and to equivalent frac- 3^—y- x-\-y x — y ^ tions having the least common denominator. ; ELEMENTARY ALGEBRA. CASE III. 81 U To add fractions. 1. Find the sum of - and OPERATION. c X If anvthiner If anything is divided into equal 5 c h -\- c parts, a number of these parts rep- "^ I ^ ^ resented by 6, added to a number represented by c, gives h -\- c of these parts. In the example given, a unit is divided into x equal parts, and it is required to find the sum of h and c of these parts ; i. e. h ^, c h -\- c X ~^ X X It is evident, therefore, that fractions that have a common denom- inator can be added by adding their numerators. But fractions that do not have a common denominator can be reduced to equivalent fractions having a common denominator. Hence, RULE. Reduce the fractions, if necessary, to equivalent fractions having a common denominator; then write the sum of the numerators over the common denominator. r. A jj m X J a . bmy 4-bnx -\- any 2. Add -, -, and -r- Ans. ' , ' n y ony 3. Add -, -J, and ^. 4. Add ^^l m. and ^^. ^xy b ab 8 ab xy 30 a^ b^ - f l(^a?y^-\- 35 m 40 a b xy Ans. 5. Add -—;-, -— -y, and 3a^ ^c d 21 a c 6. Add -— 1 — and - — — • Ans Y. Add l-|-ti I — a I —a^ l_|_a ,1— a . 2+2 a" — ^ — and ■-— i — Ans. -— ^ ^• 1 — a \ -\-a 1 — a^ FRACTIONS. 8. Add i--t^ and ^. 9. Add ^(-;!^^-^ and ^^. 10. Add ^-S', 5^^^ and '-^ + " . AnB. 1. 11. Add -^^ and ^""^ a:* — y* a^ — 2/* Ans. ' + y 7 X 12. Add 7nx and -— -• lo a Note. — Consider m a: = — , and then proceed as before. . 18 a ma: 4- 7x A°«- — w^- 7 X 13. Add a? 4- y and — t-t- 14. Add a;2 + 2xy + y*and^-^. Ans. ^ + ^y-^y'-y' + i. % — y CASE IV. 88. To subtract one fraction from another. c b 1. Subtract - from -• X X OPERATION. If anj-thing is divided into x h c b — c equal parts, a number of these X X X pdrts represented by c, subtracted from a number represented by 6, leaves b — c of these parts ; i. e. = • Hence, *^ ' XX X RULE. Reduce the fractions, if necessary, to equivalent fractions having a common denominator; then subtract the numerator of the subtrahend from that of the minuend, and write the result over the common denominator. 70 ELEMENTARY ALGEBRA. 2. Subtract -r- from r — Ans. 4 8 c "'8 c 3. Subtract ;r- from ;r— • 7 X 6 ax 4. Subtract -z-ftzs from -r — 19 x^ Ida _ CI 1 ^ ^ 29 ac „ 39 a; . 273 ar* — 116 acy 5. Subtract — -— ^ from ^; Ans. -;— ^ ^• 14 a^ 8xy 56 xr y 6. Subtract ^ from :; — j — -' Ans. -s -• 1 — a 1 -\-a a^ — 1 T. Subtract *" , from — j— ,• X — 1 a; -}- 1 oot^x , ah 4-b c n ah — he 8. Subtract „,^^jrj from ^^— j,^- 9. Subtract r from — i-^- Ans. a — h ^^ a-\-h 6^ 10. Subtract -, r from „ ' • Ans. — 11. Subtract 16 from a;* _ 1 ""^" a;2 — 1 ar» -f- 1 a;3_7 1 4-a; 1 fi Note. — Consider 16 = — , and then proceed as before. . oi?-^lQx — 2Z Ans. --j 1 -\-x yZ g 12. Subtract , from xy. a — h ^ 13. Subtract x -{- b from , ; , ♦ Ans. , , , • CASE V. 89i To reduce a mixed quantity to an improper fraction. 1. Keduce x -\- - to an improper fraction. o OPERATION. As eight eighths make , a 8x ^^a 8a:-|-« 3- unit, there will be in •"S 8'8 8 X units eight times x . , ,, . 8a; j8a:,a 8a;-4-a ^t eighths; i. e. a: = — ; and y -f - = — -^ — . Hence, FRACTIONS. 71 RULE. Multiply the integral part by the denominator of the frac- tion ; to the product add the numerator if tlie sign of the fraction is plus, and subtract it if the sign is minus, and under the result write the denominator. Note. — By a change of the language, Examples 12-14 in Art. 87, and 11-13 in Art. 88, become examples under this case. Thus, Example 12, Art. 87, might be expressed as follows: Reduce tnx-f- _- to an improper fraction. 7 2. Reduce ar* -f- 4 to an improper fraction. AnB. tl+^l^. y a \ X 8. Reduce 25 a — 25 a: -I -I— to an improper frac- tion. 4. Reduce a — 1 -f- . T ^^ ^^ improper fraction. . a" — a Ans. — j— -. 6. Reduce y -| ^—^ — to an improper fraction. 6. Reduce — ^ (a -j- 6) to an improper fraction. . a* — ah Ans. —J 7. Reduce x — 1 J— to an improper fraction. Note. — It must be remembered that the sign before the dividing line belongs to the fraction as a whole. . , z»4-l a:"— i_x« — 1 —2 2 X — 1 -, = r— ; = — J—:' or r— -' Ans. x-\-l ^+1 ^+1 ^-f-l 8. Reduce x -(- 1 ^— to an improper fraction. o Ans 1 — X 72 ELEMENTARY ALGEBRA. 9. Reduce x^ — 2ax -\- c^ — - — to an improper fraction. Note. — According to the same* principle an integral quantity- can be reduced to a fraction having any given denominator, by multiplying the quantity by the proposed denominator, and under the product writing the denominator. 10. Reduce x -\- \ to a fraction whose denominator is X — 1 . * ^ — 1 • Ans. -• X — 1 11. Reduce x — 1 to a fraction whose denominator is a — h. 12. Reduce 4 a a: to a fraction whose denominator is a^ — z. CASE VI. 90. To reduce an improper fraction to an integral or mixed quantity. Cu -* -■ ^ (2 cc I 5 fl 1. Reduce ^ to an integral or mixed quan- X — z a tity. OPERATION. * a' X — 2 a) x^ — 4: a X -{- 5 a^ (x — 2 a -\ ^r— x^ — 2 ax — 2ax -\- ^ c? — 2 a ar -(- 4 a2 As the value of a fraction is the quotient arising from dividing the numerator by the denominator (Art. 82), we perform the indicated division. Hence, RULE. Dimde the numerator by the denominator ; if there is any remainder, place it over the divisor, and annex the fraction so formed with its proper sign to the quotient. FRACTIONS. 78 2. Reduce to an integral or mixed quantity. Ans. a — 4 6. 8. Reduce ""^^ "^"^^ to au integral or mixed quantity. 4. Reduce g^~^^ to an integral or mixed quantity. ^- Reduce ^^^~e^"^"^^ to an integral or mixed quantity. 7 Ans. 4y-2a + ^. 6. Reduce "^^ *o ^ integral or mixed quan- tity. >T P«^.,^^ Sax — 10 bx — 5rx ^- deduce ^- ±- to an integral or mixed quantity. ^- ^^^^^® 2a--2b to an integral or mixed quantity. a ^ ^ , a Ans. 2 a — 2 b a — 6 9. Reduce ^--^ to an integral or mixed quantity. 10. Reduce ^-— to an integral or mixed quantity CASE VII. 91. To multiply a fraction by an integral quantity. 1. Multiply ^ by c. OPKRAnoN. According to the theorem a?4-y w cx-f-cy ^° ^^ 83, multiplying the ab ^ ab numerator by c multiplies the value of the fraction c times. 74 ELEMENTARY ALGEBRA. 2. Multiply ^-i^ by a. OPERATION. According to the theorem , I in Art. 83, dividing the de- — — — — X 05 = — ^ — nominator by a multiplies the value of the fraction a times. Hence, RULE. Divide the denominator by the integral quantity when it can be done without remainder; otherwise, multiply the nu- merator by the integral quantity. « T»r li- 1 Sax -\- 4: xy , , 3. Multiply -^P^.~ by m + n. Ans. ^Zn ' 4. Multiply ^-^^j by ab. Note. — Any factor common to the denominator and multiplier may be cancelled from both before multiplying. ^ — a ..o ^* — «v^ ^*y — ^y A 36 + 3c X ^2^ = 6~+ c X ^ = -h^' ^^«- '' ^'^'''^'y ^^x ^^ ^ - 1. Multiply y^±^ by U (x^ ^ ^). Ans. 2{x^ — f) (a + x). 8. Multiply ^^^hyx—y. Note. — When a fraction is multiplied by a quantity equal to its denominator, the product is the numerator. Ans. FRACTIONS. 76 .9. Multiply ^y^^J^ by (x - a)«. 10. Multiply ^^-i^ by 2r» — 2 ary + y^. X y Ans. (a + h) {x — y). CASE VIII. 92. To multiply an integral quantity by a fraction. 1. Multiply ar* + 2xy + y^ by ^-. OPERATION. 4(ar« + 2xy + y«)-(x + y) = 4(x + y) We first multiply the multiplicand by the numerator 4; but the multiplier is 4 -^ (x -|- if) \ and therefore this product is a: -j- y times too great, and this product divided by x -j- .V must be the product sought. It is evident that the result would be the same if the division were performed first, and the multiplication afterward. Hence, RULE. Divide the integral quantity by the denominator when it can be done without remainder, and muUiply the quotient by the numerator. Otheruxise, multiply the integral quantity by the numerator, and divide live product by the denom- inator. 2. Multiply a» - 3 a» i + 3 a 4« - i^ by ^._//ft^y - Ans. Tx(a — b). 3. Multiply a* — x* by ^ a'-f X* g 4. Multiply 7 a* — 4xy by . « . 21a»— 12xy 5. Multiply 17(ar' — y') by ^i^- 76 ELEMENTARY ALGEBRA. Note. — Since the product is the same, whichever quantity is con- sidered as the multiplier, by considering the integral quantity as the multiplier, Case VIII. becomes the same as Case VII. CASE IX. 93. To divide a fraction by an integral quantity. According to the theorem in Art. 83, dividing the numerator by a de- creases the value of the fraction a times. According to the theorem in Art. 83, multiplying the denominator by c decreases the value of the fraction c times. Hence, RULE. Divide the numerator by the integral quantity when it loill divide it without remainder; otherwise, multiply the denom- inator by the integral quantity. 3. Divide 7-^— by a. Ans. ri— • 46c "^ 46c 4. Divide -y by Uf- Ans. ^yj- 5. Divide -r- — by Q abc. 6. Divide l^byQaJ^r^. Ans ^"^ 1. Divide - by a. OPERATION. a 1 6-^« = 6 2. Divide 7- by c. OPERATION. a a 3267 "■' -*• 32 6's' T. Divide ilg^J ^y 2 (« + ^) (X + y). Ans. 13 {X + yy FRACTIONS. 77 CASE X. 91 • To divide an integral quantity by a fraction. 1. Divide x by y OPERATiox. ^ -7- o = - ; but the divisor is not a, X but a-^h. Dividing by a, therefore, "^ a is dividing by a divisor h times too X hx great, and the quotient will be h times a^ 'a ^^^ small; therefore the quotient sought . X ^^ , hx -- 18 - X = — . Hence, a a RULE. Divide the integral quantity by the numerator, and mul- tiply the quotient by the denominator. « T^- .J . u 3x . 16a 2. Divide 4 a a: by — • Ans. -g-» 3. Divide 7x2 by ^_'. Ans. '^-^' ^ abc 3? 4. Divide a + 6 by -. Ans. ''^"^^^ > 5. Divide a2 + 2ax-fa:» by ^^. 6. Divide x* — hx^ by -• 7. Divide 2x' + 3y by ?^i^- 8. Divide 1 by -• Ans. -• Note. — Hence, the reciprocal of a fraction is the fraction inverted, CASE XI. 95. To multiply a fraction by a fraction. 1. Multiply T by — 78 ELEMENTARY ALGEBRA. OPERATION. We first multiply ^ by a: ; but the a ax multiplier is not x, but x ~- y^ there- h h fore the product is y times too great ; d X Q, X ax ax and -^ -^ y = r- (Art. 93) must be -J- -^ y -=2 — b y ^ ' "y the product sought. Hence, RULE. Multiply the numerators together for a new numerator, and the denominators for a new denominator. Note 1. — Common factors in the numerators and denominators may be cancelled before multiplication. Note 2. — Cases VIL and VIIL «an be included in this by writ- ing the integral quantity as the numerator of a fraction, with a unit as the denominator. Note 3. — Mixed quantities may be reduced to improper fractions before multiplying. ■ 2. Multiply J- by ^^. ^^^--j^d^- 3. Multiply ^4>y 1"^^. 4. Multiply —r- by ^ '' aoc '' mx 5. Multiply ^Sy'-=^. Ans.^^ 6. Multiply ""-f^ by -^^^. 1. Multiply -,-^"t-^ by ^. 8. Multiply ^^by ^'_f-;. 9. Multiply - — by ^ „ o * • Ans. ^-,~y.-^' ^ "^ 2Xy •'2a* — 8 a* 3 -f • « FRACTIONS. 79 10. Multiply ,^^_, by ,,^^^._,,^^ ' 11. Multiply YT-^-'iTy ^y T?' ^°«- ^"i^* 12. Multiply ^by^. 13. Multiply y + -^ by y "•' a-l-3/ Ads. -^ ^ a«-i^ 14. Multiply together ^^' 7^' ^^^ V^' Ads. 1. 15. Multiply together ~^» '2_/s' *°^ T" * 16. Multiply together a + - , 5 + - ' a°d ^ — u' Ans. a6v4-i^ — i 5* if ^ by y^ CASE XII. 96i To divide a fractioD by a fraction. 1. Divide - by -7 • y '' h OPERATION. X X - -f- a = — (Art. 93) ; but the - -4- a = — ... a y ay divisor is not a, but - ; we have used . . b _^ y, I __ ^ a divisor h times too great, and there- ^^ ^ fore the quotient — is 6 times too I. °^ X ox small, and the quotient sought is — X ^ ■=■ (Art. 91). It will be noticed that the denominator of the dividend is multiplied by the numerator of the divisor, and the numerator of the dividend by the denominator of the divisor. Hence, RULE. Invert the divutor, and then proceed as in multiplication of a fraction by a fraction. 80 ELEMENTARY ALGEBRA. Note 1. — All cases in division of fractions can be brought under this rule, by writing integral quantities as fractions with a unit for the denominator. Note 2. — After the divisor is inverted, common factors can be cancelled, as in multiplication of fractions. Note 3. — Mixed quantities should be reduced to improper frac- tions before division. 2. Divide - by — • Ans. c •' n cm x^ X* 4 3. Divide - by -• Ans. -^— • 4. Divide 3^ by ^^. . ^. ., 20^4- 2c , 3 a . 10 a«c 4- 10 c» 6. Divide ^,3 ', — by ^-. Ans. — — ,.. ^ , ■ 6. Divide i^^±# by -1- m — n 7. Divide ^ by ^+^?. Ans. ;|^ 8. Divide — ^—j- by — a — h *' 4 3 x* ar 3 T 9. Divide 3 . , by ■ , • Ans. 10. Divide ^"^7/ by ^-=1^^. x-}- 1 '^ 4 11. Dmde ^ by =t 12. Divide /"'•':-!"' . by '"' + '"" + "' ■ Ans. 3 (m« + n«). 13. Divide 1 + "^-^ by -4 1. Ans ^ <^^ + y) + ^^ +'^)' . c {c — x — y) 14. Divide x + y—^ by -4-a;-|-y. FRACTIONS. 81 15. Divide ^-^ + ^^ ^y r+^- Ans ^IL±^)* A^«- (i_x«j-«- Note. — The division of fractions is sometimes expressed by writ- a ing the divisor under the dividend. Thus, — . Such an expression y is called a Complex Fraction. A Complex Fraction can be reduced to a simple one by performing the division indicated. 16. Reduce y to a simple fraction. Ans. z— • 5 17. Reduce ^ to a simple fraction. c Ans. ' — i— • lex — bx x+l X 1 18. Keduce . to a simple fraction. 19. Reduce — , . to a simple fraction. \—x ^--^ 20. Reduce ^ . ? to a simple fraction. Ans. — i — 21. Reduce — ^— ^ to a simple fraction. a 4- 6 Ans. (x + y) (a + h). Note. — A Complex Fraction can also be reduced by multiplying its numerator and denominator by the least common multiple of the denominators of the fractional parts. Thus, if both terms of the fraction in Ex. 16 be multiplied by 5 a:, or both in Ex. 17 by ex, the result will be the same as above. 4* F 82 ELEMENTARY ALGEBRA. SECTION XIII. EQUATIONS OF THE FIRST DEGREE CONTAINING BUT ONE UNKNOWN QUANTITY. 97i An Equation is an expression of equality between two quantities (Art. 9). Tliat portion of the equation which precedes the sign = is called the first member, and that which follows, the second member. 98. The Degree of an equation containing but one un- known quantity is denoted by the exponent of the highest power of the unknown quantity in the equation. An equation of the first degree, or a simple equation, is one that contains only the first power of the unknown quantity. For example, 2 a; — a a; = 27. An equation of the second degree, or a quadratic equation, is one in which the highest power of the unknown quantity is the second power. For example, x^ — ax-=zh -\- c, or ax^ — 5 = IT. An equation of the third degree, or a cubic equation, is one in which the highest power of the unknown quantity is the third power, and so on. 99. The Reduction of an Equation consists in finding the value of the unknown quantity, and the processes involved depend upon the Axioms given in Art. 13. The processes can be best understood by considering an equation as a pair of scales which balance as long as an equal weight remains in both sides : whenever on one side any additional weight is put in or taken out, an equal weight must be put in or EQUATIONS OP THE FIRST DEGREE. 88 taken out on the other side, in order that the equilibrium may remain. So, in an equation, whatever in done to one side muat be done to tJie oifier, in order that the equality may remain. 1. If anything is added to one member, an equal quantity must be added to the other. 2. If anything is subtracted from one member, an equal quantity must be subtracted from the other. 3. If one member is multiplied by any quantity, the other member must be multiplied by an equal quantity. 4. If one member is divided by any quantity, the other member must be divided by an equal quantity. 6. If one member is involved or evolved, the other must be involved or evolved to the same degree. TRANSPOSITION. 100. Transposition is the changing of terms from one member of an equation to the other, without destroying the equality. The object of transposition is to bring all the unknown terms into one member and all the known into the other, 80 that the unknown may become known. I. Find the value of x in the equation x-|- 16 := 24. Subtracting 16 from the first OPERATION. member leaves x ; but if 1 6 is sub- ^ ~r 16 = 24 tracted from the first member, it x = 24 — 16 = 8 must also be subtracted from the second. 2 Find the value of x in the equation x — b = a. OPERATION. Adding b to the first member J. ft = a gives x ; but if 6 is added to the __ I 1 first member it must also be added to the second. 84 ELEMENTARY ALGEBRA. 3. Find the value of x in the equation 2 x = a; -j- 16. OPERATION. Subtracting x from both mem- 2x = a: + 16 2a: — x= 16 bers, we have 2 a; — a; = 16, or x= 16. a: =16 It appears from these examples that any term which dis- appears from one member of an equation reappears in the other with the opposite sign. Hence, RULE. Any term may be transposed from one member of an equation to the other, provided its sign is changed. 4. Find the value of x in the equation Sx — 15 = 4x4-5. OPERATION. Sx — 15 = 4:X-\- 6 Transposing, Sx — 4x= 5 -\~ 15 Uniting terms, 4 a; = 20 Dividing both members by 4, x := 5 6. Find the value of x in 4 x -)- 46 = 5 x -|- 23. Note. — Reducing, we have — x = — 23. If each member of this equation is transposed, we shall have 23 = a: ; i. e. 23 equals ar, or x equals 23. Dividing both members by — 1 will give the same result. Hence, the signs of all the terms of an equation may be changed without destroying the equality. 6. Find the value of x in It ar + It = 19 x + 13. Ans. x = 2, 1. Find the value of a: in 8 a; — 14 = 13 a; — 29. 8. Find the value of x in 5 a: + 25 = 10 a: — 25. Ans. X =: 10. 9. Find the value of x in 24x — 17 = 11 x + 74. 10. Find the value of x in 37 x — (4 + 7) = 41 x — 23. EQUATIONS OF THE FIRST DEGREE. 85 CLEARING OF FRACTIONS. 101. To clear an equation of fractions. 1. Find the value of a: in the equation - — 2 = -+ 1. OPERATION. If the given equation is mul- X 2 -4-1 tipHed by 6, the least common 3 6 ' multiple of 6 and 3, it will give 2x — 12 = a:-|-6 2x — 12 = x-|-6, an equation X = 18 without a fractional term. Hence, RULE. Multiply each term of the equation by the least common multiple of the denominators. Note 1. — In multiplying a fractional term, divide the multiplier by the denominator of the fraction and multiply the numerator by the quotient. Note 2. — An equation may be cleared of fractions by multipljnng it first by one denominator, and the resulting equation by another, and so on, till all the denominators disappear; but multiplying by the least common multiple is generally the more expeditious method. "Note 3. — Before clearing effractions it is better to unite terms which can readily be united ; for instance, the equation in Ex. 1, X X by transposing — 2, can be written - = - -|- 3. o o Note 4. — When the sign — is before a fraction and the de- nominator is removed, the sign of each term that was in the nu- merator must be changed. 2. Given ^ _ ^ + 25 = 33 — ^^- operation. n -oc X X ^ X — 6 1 ransposing 25, - — I = ^ 5 — Multiplying by 20, 5 x — 4 x = 160 — 10 x + 60 Transposing and uniting, 11 x = 220 Dividing by 11, x=:20 86 ELEMENTARY ALGEBRA. Note. — The sign of the numerator of — - is -|-, and must be o changed to — when the denominator is removed ; for — (-}- 4 x) == — 4 a:; and so the sign of each term of the numerator of the fraction — must be changed when the denominator 2 is removed ; for — (-f 10 a: — 60) = — 10 a: -|- 60. 102i To reduce an equation of the first degree contain- ing but one unknown quantity, we deduce from the preced- ing examples- the following RULE. Clear the equation of fractions, if necessary. IVanspose the known terms to one member and the un- known to the other, and reduce each member to its simplest form. Divide both members by the coefficient of tlie unknown quantity. Note 1. — To verify an equation, we have only to substitute in the equation the value of the unknown quantity found by reducing the equation. For instance, in Ex. 2, Art. 101, by substituting 20 for X, in ^ — f -f 25 = 33 — ^ "~ , we have 4 5' 2 4 5' 2 6 _ 4 _|_ 25 = 33 — 7, 26 = 26. Note 2. — When answers are not given, the work should be veri- fied. 103* Since the relations between quantities in Algebra are often expressed in the form of a proportion, we intro- duce here the necessary definitions. EQUATIONS OF THE FIRST DEGREE. 87 104. Ratio is the relation of one quantity to another of the same kind ; or, it is the quotient which arises from di- viding one quantity by another of the same kind. Ratio is indicated by writing the two quantities after one another with two dots between, or by expressing the division in the form of a fraction. Thus, the ratio of a to b is written, a : b, or j ; read, a is to b, or a divided by b. 105. Proportion is an equality of ratios. Four quan- tities are proportional when the ratio of the first to the second is equal to the ratio of the third to the fourth. The equality of two ratios is indicated by the sign of equality (=) or by four dots (::). Q C Thus, a \ b =. c : dy ox a '. b '. : c : rf, or 7 = - ,; read, a to 6 ha equals c to d, or a is to b s^ c is to d, or a divided by b equals c divided by d. The first and fourth terms of a proportion are called the extremen, and tlie second and third the means. 106. In a proportion the product of the means is equal to the product of the extremes. Let a : b = c : d a c Clearing of fractions, ad = be A proportion is an equation ; and making the product of the means equal to the product of the extremes is merely clearing the equation of fractions. Examples. 1. Reduce 1^ + 10 = ^^ + 13. Ans. x = 30. 2. Reduce 17 « — 14 = 12 x — 4. Ans. x r= 2. 88 ELEMENTARY ALGEBRA. 3. Reduce 6 a: -— 25 + a; = 135 — 3 a: — 10. Ans. x-==. 15. 4. Reduce 3a:+5 — a! = 38 — 2a:. Ans. x = 8^. 5. Reduce ""-^ -j- ^ =, 30 — ^-i-^- Ans. x = 12. 6. Reduce a: — 7^ = — — • Ans. a; =: lly^. T. Reduce ^ + ^ + ^ + ^= 154. Ans. a: = 120. 2 o 4 o 8. Reduce f + |= 16 + f . Ans. a: = 24. 9. Reduce --|-a=7 \- d. h c OPERATION. 6 ' ^ c ' Multiplying byJc^, chx-\- ahch:=:zh ex — hhx-\-hcdh Transposing, chx — b c x-\-hhxr=:h c d h — ahch Factoring 1st mem., (c A — hc-\-hh)xr=hc dh — a hch Tx-.i. 1 m ' , /. b c dk — ah ch "^& "J ^" WiXJ.V>iVXJ.U VA ^, ch — hc-{-hh 10 Reduce X -\- mx =: c. A TTS '*» — — 1 4-m 11. Reduce 'T «-^- Ans. a:= ^^ 12. Reduce 1 , h - + - = a;. Ans. X = ' — a c 13. Reduce a; ' X Ans. X = — '— c 14. Reduce «=.-^+«- ^ns. X = 9. 15. Reduce 2 3a = c. XXX Ans. a: = — ^ — EQUATIONS OF THE FIRST DEGREE. 89 X , X , X 16. Reduce - + - + - = 39. i i \ 2 X X 17. Reduce - - — --\-c = d. 18. Reduce (a — 3) x + ^ = i /-r -r\ 19. Reduce x — / 1 — ^\ =z 5. Anfl. ar = 6. 20. Reduce 6 — ^^^tl __ a: ._ 4. 5 21. Reduce 2 a: ^ =18 { Ans. X = 9. ortr»j ^17 — X „ .a: — 95 22. Reduce - =3xH --• 23. Reduce 2 a: — ?^^ = 14 — —'^- Ans. x = 5. 24. Reduce 6 a: + T^ - | = 9^ - ^/ + ^. Note. — Before clearing of fractions, transpose 7^ and unite it X 1 1 X with 9^; also transpose — -, and unite it with -5-. 25. Reduce 4ar+?-±i = 5 + iliil X 3~ 26. Reduce ^—i — ^^ = 21 — -"ti. Ans. a: = 39. Jo G 27. Reduce - 4- -r 4- - = (/. Ans. x = r — ; ; — -.' a ^ b ^ c be -\- ac-\- ab 28. Reduce — ~^ — 6 = -7-? + 7. ftAOj ^ — 1 /» 22 — a: 3 + x . ^ 29. Reduce — ^ — = 6 ^- Ans. x =: 7. O 90 ELEMENTARY ALGEBRA. on r> J in I 2a: — 22 3 x — 75 . 284. — 4.x 30. Reduce 19 + —^ — = 1 ^ oi T>j 4:X-\-5 5x — 5 a- 4-1 , 31. Reduce — -^ — = — ^ 1. 5 4 b . Ans. X = 5. 32. Reduce 1^^ - '-^ = 4.-17+ '-^. o 4 '6 oo r> J A ar— 12 , _ 20a: 4- 21 1 33. Reduce 4:X [- 5 = } • o ' 4 4 34. Keduce = ~ » Ans. a; r=: a; c m ' hm-\-cd 35. Reduce 7 — — = 1 — 3 « c. oa x> A 5a:+3 3a:-j-15 . , 6a:4-10 36. Reduce — ^ h 6 ,' = 4 -] \ 2 ' 4 '4 OK -o J o 3a — 19 _ 23— a: , 5.r— 38 , -^ 31. Reduce Zx 8 =: — — - 4 \- 10. z 4 o Ans. X = 19» «« _,, 13 — 3x 3a:4-2 ^ „ ,8a:— 13 38. Reduce — ,^ J— - = T — 6 :r ^ 10 5 '0 on T> A 4.(x—7) 3(a:4-l) 7.t— 17 a; 39. Reduce ^ + 11 = —lo ^ 21 ' ,^ _., 4.r — 6,„ 19— 4a: 5.r — 6, 7 a: + 8 40. Reduce a: 1-3=:— ^ ^—] ^. 41. Reduce ^ + ^ + ^ + ^ = m. ,^_., 7x4-5, 6a: — 30 ,, 42. Reduce - J— + ~ — = x + I. 7 ' 7 X — 7 ' Note. — Multiply by 7, transpose, and unite. 43. Reduce 2 (3 4- a:) : 6 a: — 9 = 2 : 3. Ans. x= 6. 44. Reduce l + f: ^^^J-, - H -^ 45. Reduce b : c -\- d = - : n. X. ' a: EQUATIONS OF THE FIRST DEGREE. ftl PROBLEMS PRODUCING EQUATIONS OF THE FIRST DEGREE CON- TAINING BUT ONE UNKNOWN QUANTITY. 107. The problems given in this Section must either con- tain but one unknown quantity, or the unknown quanti- ties must be so related to one another that if one be- comes known the others also become known. 108. With "beginners the chief difficulty in solving a problem is in translating the statements or conditions of the problem from common to algebraic language ; i. e. in preparing the data, and forming an equation in accord- ance with the given conditions. 1. If three times a certain number is added to one half and one third of itself, the sum is 115. What is the number ? SOLUTION. Let X =. the number ; then by the conditions of the problem. Clearing of fractions, 18x + 3a; + 2a: = 690 Uniting terms, 23 ar = 690 Dividing by 23, ar=: 30 VERIFICATION. 3X30 + f + ^»=U5 115 = 116 In this problem there is but one unknown quantity, which we rep- resent by X. 2. There are three numbers of which the first is 6 more than the second, and 11 less than the third ; and their sum is 101. What are the numbers? 92 ELEMENTARY ALGEBRA. SOLUTION. Let X r= the first, In this problem then X — 6 = the second, there are three un- and a: + 11 = the third. known quantities; Their sum,3x+ 6 = loT 1^"^*^"^ "^" ^^ ^^ lated to one an- other that, if any x= 32, the first, one becomes known, X— 6 = 26, the second, t^g other two will a? + 11 = 43, the third. be known. VERIFICATION. 32 + 26 + 43 = 101 101 = 101 From these examples we deduce the following GENERAL RULE. Let X [or some one of the latter letters of the alphabet) represent the unknown quantity ; or, if there is more than one unknown quantity, let x represent one, and find the others by expressing in algebraic form their given relations to the one represented by x. With tJw data thus prepared form an equation in accord- ance with the conditions given in the problem. Solve the equation. The three steps may be briefly expressed thus : — 1 St. Preparing the Data ; 2d. Forming the Equation ; 3d. Solving the Equation. 3. The sum of three numbers is 960 ; the first is one half of the second and one third of the third. What are the numbers ? Ans. 160, 320, and 480. 4. Find two numbers whose difference is 18 and whose sum 112. Ans. 47 and Qb, EQUATIONS OF THE FIBST DEGREE. 98 5. A man being asked how much he gave for his horse said, that if he had given $ TO more than three times as much as it cost, he would have given S445. How much did his horse cost him ? 6. A man being asked how many sheep he had, replied that if he had as many more, and two thirds as many, and three fifths as many, he should have 8 more than three times as many as he had. IIow many sheep had he ? *?. Divide $675 between A and B in such a manner that B may have two thirds as much as A. Ans. A^s share, $ 345 ; B^s " $230. 8. A father divided his estate among his three children 80 that the eldest had $ 1440 less than one half of the whole, the second $500 more than one third of the whole, and the youngest $ 250 more than one fourth of the whole. What was the value of the estate ? SOLUTION. Let X •=. whole estate. Then ^ — 1440 = share of the eldest, 1+ 500= " " " second, 1+ 250= " '* " youngest, 13 X Their sum — 690 =: x, whole estate. i= 690 X = 8280, whole estate. 9. A gentleman meeting five poor persons, distributed $7.50, giving to the second twice, to the third three times, to the fourth four times, and to the fifth five times as much as to the first How much did he give to each ? 94 ELEMENTARY ALGEBRA. 10. Divide 195 into two such parts that the greater di- vided by 3 shall be equal to the less divided by 2. Note. — To avoid fractions, let 3 a: = the greater and 2 a; = the less. Ans. 477 and 318. 11. Divide a into two such parts that the greater di- vided by h shall be equal to the less divided by c. SOLUTION. Let X = the greater, then a — x = the less. . , ' X a — X And T = b c Clearing of fractions, cx^= ah — hx Transposing, hx -\- ex =. ab Dividing by 5 -f- c, x-= r-^-^ the greater, ah ac , , , a — x=:a — f—. — = 5— j — , the less. b-\'C b-\-c 12. What number is that which, if multiplied by 7, and the product increased by eleven times the number, and this sum divided by 9, will give the quotient 6 ? 13. If to a certain number 55 is added, and the sum divided by 9, the quotient will be 5 less than one fifth of the number. What is the number? Ans. 125. 14. As A and B are talking of their ages, A says to B, *' If one third, one fourth, and seven twelfths of my age are added to my age, the sum will be 8 more than twice my age.^' What was A's age ? 15. A farmer having bought a horse kept him six weeks at an expense of $20, and then sold him for four fifths of the original cost, losing thereby $ 50. How much did he pay for the horse? Ans. $150. 16. A man left $ 18204, to be divided among his widow, three sons, and two daughters, in such a manner that the widow should have twice as much as a son, and each son as much as both daughters. What was the share of each ? EQUATIONS OF THE FIRST DEGREE. 95 IT. If a certain number is divided by 9, the sum of the divisor, dividend, and quotient will be 89. What is the number? Ans. 72. 18. If a certain quantity is divided by a, the sum of the divisor, dividend, and quotient will be b. What is the quantity ? 19. Verify the answer to the preceding problem. 20. A farmer mixed together corn, barley, and oats. In all there were 80 bushels, and the mixture contained two thirds as much corn as barley and one fifth as much bar- ley as oats. How many bushels of each were there ? 21. Three men, A, B, and C, built 572 rods of fence. A built 8 rods per day, B 7, and C 5. A worked one half as many days as B, and B one third as many as C. How many days did each work ? 22. What number is as much greater than 340 as its third part is greater than 34 ? Ans. 459. 23. A man meeting some beggars gave 3 cents to each, and had 4 cents left. If he had undertaken to give 5 cents to each, he would have needed 6 cents to complete the dis- tribution. How many beggars were there, and how much money did he have ? SOLUTION. Let X = the number of beggars ; then, according to the first statement, 3 X + 4 = the number of cents he had, and, according to the second statement, bx — 6 = the number of cents he had. Therefore, 6a: — 6 = 3x + 4 2a:=10 ar = 5, the number of beggars, and 8x -[- 4 = 19, the number of cents he had. 96 ELEMENTARY ALGEBRA. 24. A boy wishing to distribute all his money among his companions gave to each 2 cents, and had 3 cents left ; therefore, collecting it again, he began to give 3 cents to each, but found that in this case there was one who had received none, and another who had only 2 cents. How many companions, and how much money had he? Ans. 7 companions, and IT cents. 25. What two numbers whose difference is 35 are to each other as 4 : 5 ? 26. A man being asked the hour, answered that three times the number of hours before noon was equal to three fifths of the number since midnight. What was the time of day ? SOLUTION Let X = the number of hours since midnight, i. e. the time ; then 12 — x = the number of hours before noon. Then . 36— 3^:=?^ 5 Clearing of fractions, 180 — 15 x =z^x Whence 18x = 180 X =z 10. Ans. 10 o'clock. 2Y. A gains in trade $ 300 ; B gains one half as much as A, plus one third as much as C ; and C gains as much as A and B. What is the gain of B and C ? Ans. B's, $375; C's, $675. 28. What number is to 28 increased by one third of the number as 2 : 3 ? Ans, 24. 29. What number is that whose fifth part exceeds its sixth b}^ 15 ? 30. Divide $3740 into two parts which shall be in the ratio of 10 : 7. 31. Divide a into two parts which shall be in the ratio of6:c, . ah , ac Ans. i— 1 — and 7—-, — h-\- c -\-c EQUATIONS OF THE FIRST DEGREE. 97 32. What number ia that the sum of whose fourth part, fifth part, and sixth part is 37 ? 33. What quantity is that the sum of whose third part, fifth part, and seventh part is a ? » 105 a -ft-ns. ^^ 34. A farmer sold IT bushels of oats at a certain price, and afterward 12 bushels at the same rate ; the second time he received 55 shillings less than the first. What was the price per bushel ? 35. A certain number consists of two figures whose sum is 9 ; and if 27 is added to the number, the order of the figures will be inverted. What is the number ? SOLUTION. liCt X =z the left-hand figure ; then 9 — x = the right-hand figure. As figures increase from right to left in a tenfold ratio, 10 x + (9 — x) = 9 a: -f- 9 = the number ; and when the order of the figures is inverted, 10 (9 — ar) -f- a: = 90 — 9 a: := the resulting nu mber. Therefore 9x+9 + 27 = 90— -9a:. Or 18 a: = 54 Whence ar = 3, the left-hand figure, and 9 — x = 6, the right-hand figure. Ans. 36. 36. A certain number consists of three figures whose sum is 6, and the middle figure is double the left-hand figure ; and if 198 is added to the number, the order of the figures will be inverted. What is the number ? Ans. 123. 37. Two men 90 miles apart travel towards each other till they meet. The first travels 5 miles an hour and the second 4. IIow many miles does each travel before they meet? 6 o 98 ELEMENTARY ALGEBRA. 38. A man hired six laborers, to the first of whom he paid t5 cents a week more than to the second ; to the second, 80 cents more than to the third ; to the third, 60 cents more than to the fourth ; to the fourth, 50 cents more than to the fifth ; to the fifth, 40 cents more than to the sixth; and to all he paid $68.15 a week. What did he pay to each a week ? 89. What number is that to which if 20 is added two thirds of the sum will be 80 ? 40. What number is that to which if a is added - of c the sum will be c?? \^„ ^^ ^ Ans. -T a. 41. A man spent one fourth of his life in Ireland, one fifth in England, and the rest, which was 33 years, in the United States. To what age did he live? 42. A post is one fifth in the mud, two sevenths in the water, and 18 feet above the water. How long is the post ? 43. What number is that whose half is as much less than 40 as three limes the number is greater than 156? Ans. 56. 44. Two workmen received the same sum for their la- bor ; but if one had received $15 less and the other $15 more, one would have received just four times as much as the other. What did each receive ? 45. Of the trees on a certain lot of land five sevenths are oak, one fifth are chestnut, and there are 32 less wal- nut trees than chestnut. How many trees are there ? 46. Divide 474 into two parts such that, if the greater part is divided by 7 and the less by 3, the first quo- tient shall be greater than the second by 12. Ans. 357 and 117. EQUATIONS OF THE FIRST DEGREE. 99 4Y. Two persons, A and B, have each an annual income of S1500. A spends every year $400 more than B, and at the end of five years the amount of their savings is $6000. What does each spend annually? Ans. A $1100, and B $700. 48. In a skirmish the number of men captured was 41 more, and the number killed 26 less than the number wounded ; 45 men ran away ; and the whole number en- gaged was four times the number wounded. How many men belonged to the skirmishing party ? Ans. 240. 49. A and B have the same salary. A runs into debt every year a sum equal to one sixth of his salary, while B spends only three fourths of his ; at the end of five years B has saved $1000 more than enough to pay A's debt What is the salary of each ? Ans. $ 2400. 60. A man lived single one third of his life : after hav- ing been married two years more than one eighth of his life, he had a daughter who died ten years after him, and whose age at her death was one year less than two thirds the age of her father at his death. What was the father's age at his death ? SOLUTION. Let X =: his age ; then - = his age at marriage, o ^ + f + 2 = his age at daughter's birth, 8 o and ar — ( 1 4- ^ -|- 2 ) = her age at his death. Then x-f-|-2+10 = '^-l Transposing and uniting, — - = — 9 X = 72, the father's age. 100 ELEMENTARY ALGEBRA. 51. Divide $ 864 among three persons so that A shall have as much as B and C together, and B $5 as often as C $ 11. 52. A father and sou are aged respectively 32 and 8. How long vi^ill it be before the son will be just one half the age of the father ? 53. A man's age was to that of his wife at the time of their marriage as 4:3, and seven years after, their ages were as 5:4. What was the age of each at the time of their marriage ? 54. One fifth of a certain number minus one fourth of a number 20 less is 2. What is the number? Ans. 60. 55. There are two numbers which are to each other as J : ^ ; but if 9 is added to each, they will be as i : ^. What are the numbers ? Ans. 9 and 6. 56. A person having spent $ 150 more than one third of his income had $ 50 more than one half of it left. What was his income ? 5T. A merchant sold from a piece of cloth a number of yards, such that the number sold was to the number left as 4 : 5 ; then he cut off for his own use 15 yards, and found that the number of yards left in the piece was to the number sold as 1:2. How many yards did the piece originally contain ? Ans. 45. 58. Four places. A, B, C, and D, are in a straight line, and the distance from A to D is 126 miles. The distance from A to B is to the distance from B to C as 3 : 4, and one third the distance from A to B added to three fourths the distance from B to C is twice the distance from to D. What is the distance from A to B, from B to C, and from C to D ? 59. A laborer was hired for 40 days ; for each day he wrought he was to receive $2.50, and for each day he was idle he was to forfeit $1.25. At the end of the time he received $58.75. How many days did he work? Ans. 29. A e.^,- EQfUATIONS OF THE HBST DEGREE. 101 60. A cask which held 44 gallons was filled with a mixture of brandy, wine, and water. 'T^crif) 'were '19: gal- lons more than one half as muct ,wine as brandy, ^nd , as much water as brandy and wJDje/'-Hpw* ulai^y*' g&>)[^V8' were there of each ? 61. Two persons, A and B, travelling each with S80, meet with robbers who take from A $ 5 more than twice what tliey take from B; then B finds he has $26 more than twice what A has. IIow much is taken from each ? Ans. From A, $69 ; from B, $32. 62. Four persons, A, B, C, and D, entered into part- nership with a capital of $84810; of which B put in twice as much as A, C as much as A and B, and D as much as A, B, and C. How much did each put in ? 63. In three cities, A, B, and C, 1188 soldiers are to be raised. The number of enrolled men in A is to that in B as 3 : 5 ; and the number in B to that in C as 8 : T. How many soldiers dught each city to furnish ? Ans. A, 288 ; B, 480 ; C, 420. 64. Divide $65 among five boys, so that the fourth may have $2 more than the fiftli and $3 less than the third, and the second $4 more than the third and $5 less than the first. 65. A merchant bought two pieces of cloth, one at the rate of $ 5 for 7 yards, and the other $ 2 for 3 yards ; the second piece contained as many times 3 yards as the first times 4 yards. He sold each piece at the rate of $6 for 7 yards, and gained $24 by the bargain. How many yards were there in each piece ? Ans. First, 84 ; second, 63. 66. A drover had the same number of cows and sheep. Having sold 17 cows and one third of his sheep, he finds he has three and a half times as many sheep as cows left. How many of each did he have at first? 102 ELEMENTARY ALGEBRA. 67 A flour dealer sold one fourth of all the flour he had aod one foilrth'. of: a barrel ; afterward he sold one third. c>f wli^t he had left .and one thh-d of a barrel; and ttlen. ofte Jigjf of tl^Q remainder and one half of a barrel; and had 15 barrels left. How many had he at first ? SOLUTION. Let X = number at first; 3 X 1 then — — - = number after first sale, 2 /3 X 1\ 1 a; 1 g{~l ;) — 3^^^2 — 2^^ number after second sale, and 2 §2 — 2/ — 2^^4 — 4^^ number after third sale. Then ^ -^ ? =:: 15 4 4 Clearing of fractions, x — 3 = 60 Whence x = 63, number at first. « 68. A merchant bought a barrel of oil for $50; at the same rate per gallon as he paid, he sold to one man 15 gallons ; then to another at the same rate two fifths of the remainder for $ 14. IIow many gallons did he buy in the barrel ? 69. Two pieces of cloth of the same length but dif- ferent prices per yard were sold, one for $5 and the other for St. 50. If there had been 6 more yards in each, at the same rate per yard as before, they would have come to $ 15.41^^. How many yards were there in each? Ans. 21. 10. A and B began trade with equal sums of money. The first year A lost one third of his money, and B gained $ 150. The second year A doubled what he had at the end of the first year, and B lost $150, when the two had again an equal sum. What did each have at first ? EQUATIONS OF THE FIRST DEGREE. 108 11. A man distributed among his laborers $2.50 apiece, and had $25 left. If he had given each $3 as long as his money lasted, three would liave received nothing. How many laborers were there, and how much money did he have? Ans. 68 laborers, and S195. 72. A man who owned two horses bought a saddle for S35. When the saddle was put on one horse, their value together was double the value of the other horse ; but when the saddle was put on the other horse, their value together was four fifths of the value of the first horse* What was the value of each horse ? T3. From a cask two thirds full 18 gallons were taken, when it was found to be five ninths full. How many gallons will the cask hold ? H. A farmer had two flocks of sheep, and sold one flock for $60. Now a sheep of the flock sold was worth 4 of those left, and the whole value of those left was $8 more than the price of 8 sheep of those sold, and the flock left contained' 40 sheep. How many sheep did the farmer sell, and what was the value of a sheep of each flock ? Ans. Number sold, 15; value, $4 and $1. 75. A man has seven sons with 2 years between the ages of any two successive ones, and the sum of all their ages is ten times the age of the youngest. What is the age of each ? 76. Divide 75 into two parts such that the greater in- creased by 9 shall be to the less diminished by 4 as 3 : 1. 77. Divide a into two parts such that the greater in- creased by b shall be to the less diminished by c as m : n. 78. What two numbers are as 3:4, while if 8 be added to each the sums will be as 5 : 6 ? 79. Divide 127 into two parts, such that the difference between the greater and 130 shall be equal to five times the difference between the less and 63. 104 ELEMENTARY ALGEBRA. SECTION XIV. EQUATIONS OF THE FIRST DEGREE CONTAINING TWO UNKNOWN QUANTITIES. 109. Independent Equations are such as cannot be de- rived from one another, or reduced to the same form. Thus, a; + y= 10, I + 1=5, and 4a:+32^=:40— y are not independent equations, since any one of the three can be derived from any other one ; or they can all be reduced to the form x -\- y = 10. But a; -j- y = 10 and 4:X = 1/ are independent equations. 110. To find the value of several unknown quantities, there must be as many independent equations in which the unknown quantities occur as there are unknown quantities. From the equation x -}- i/ = 10 we cannot determine the value of either a: or y in known terms. If y is transposed, we have x= 10 — y; but since y is unknown, we have not determined the value of X. We may suppose y equal to any number whatever, and then x would equal the remainder obtained by subtracting y from 10. It is only required by the equation that the sum of two numbers shall equal 1 ; but there is an infinite number of pairs of numbers whose sum is equal to 10. But if we have also the equation 4x = y, vre may put this value of y In the first equation, X -{- y = 10, and obtain a: -j- 4 a: = 10, or a; = 2 ; then 4 a: = 8 = y, and we have the value of each of the unknown quantities. ELIMINATION. Ill, Elimination is the method of deriving from the given equations a new equation, or equations, containing one less unknown quantity. The unknown quantity thus excluded is said to be eliminated. EQUATIONS OF THE FIRST DEGREE. 105 There are three methods of elimination : — I. By substitution. II. By comparison. 111. By combination. CASE I. 112, Elimination by substitution. 1. Given |*^ + ^y = 2^l, to find a;and y. OPERATION. 4a: + 6y = 23 (1) 6a: + 4y= 22 (2) (3) 6x + 4(?i^)= 22 (4) 25X + 92 — 16aj = 110 (5) ?^ = 3 (7) x= 2 (6) 23 — 4 z y = — s — Transposing 4 x in (1) and dividing by 5, we have (8), which gives an expression for the value of y. Substituting this value of y in (2), we have (4), which contains but one unknown quantity ; i. e. y has been eliminated. Reducing (4) we obtain (6), or a: == 2. Substituting this value of x in (3), we obtain (7), or y = 3. Hence, RULE. Find an expression for the value of one of the unknown quantities in one of the equations, and substitute this value for the same unknown quantify in the other equation. Note. — After eliminating, the resulting equation is reduced by the rule in Art. 102. The value of the unknown quantity thus found must be substituted in one of the equations containing the two unknown quantities, and this reduced by the rule in Art. 102. Find the values of x and y in the following equations : — 2. Given 1^ + !'= in. Ans. j- = l«- 106 ELEMENTARY ALGEBRA. - + ^=12 3. Given ^ - - L. Ans. j^=^^• 5 + ^ - . J (5^ = 15. 4. Given 5. Given (2. -I- 3 = 0) ^^^^ ( (a;+y — 29 = 0) ^ ■J 2 "^ 3 36 >• . (2a7+ 32/ = 2 ) x= 1. ^ = 22. 6. Given ] 2~3-" ^L (3a: — 3^ = 16) CASE II. 113« Elimination by comparison. 1. Given {2rJy = 2?}' *^ ^^^ ^ ^^^ y- OPERATION. a: — 2y = 6 (1) 2a: — y = 27 (2) a: = 6 + 2s^ (3) ^_ 27 + y (^^ «l/ — " . 2 6 + 2y=^' + ^ (5) 12 + i!, = 21+y (6) P= 6 0) a;= 6 + 10 = 16 (8) Finding an expression for the value of x from both (1) and (2), we have (3) and (4). Placing these two values of x equal to each other (Art. 13, Ax. 8), we form (5), which contains but one unknown quantity. Reducing (5) we obtain (7), or* ?/ = 5. Sub- stituting this value of y in (3), we have (8), or a: = 16. Hence, EQUATIONS OF THE FIRST DEGREE. RULE. 107 Find an expression for ihe value of the same unknovm quantity from each equation, and put these expressions equal to each other. By this method of elimination find the values of x and y in the following equations : — {. Given -l^^""! — ^[.. 3. Given + y = 12 = 8 4. Given |3a: + 5y = 2| 5. Given (3(x-y)-9 = 0) 7. Given i6*-5y=17|. ( 2a; — v = 13> 8. Given ■ :-!=« Ana. ( V = 3. Ans ( V = = 24. Ans. Ans. (a: = 2. 108 ELEMENTARY ALGEBRA. CASE III. 114i Elimination by combination. 1. Given ■< ^ ^ ?- , to find x and y. OPERATIOS . Zx — 2y— 1 (1) 2a; — 3y = 3 (2) Qx — iy— 14 (3) 6 a; — 9.y= 9 (4) by= 5 (5) 2a; — 3 =3 0) y= 1 (6) a;=:3 (8) If we multiply (1) by 2, and (2) by 3, we have (3) and (4), fn which the coefficients of x are equal; subtracting (4) from (3), we have (5), which contains but one unknown quantity. Redu- cing (5), we have (6), or y = 1 ; substituting this value of y in (2), we obtain (7), which reduced gives (8), or a; = 3. 2. Given - -X y __ 2 4 13 "^ 2 6 12 - , to find X and y. • DPI SRATION. !-!= ' w .-1=12 (2) (3) 9-1= 6 (6) 'i-^^ (4) y = 12 (1) a: = 18 (5) If we multiply (1) by 2, we have (3), an equation in which y has the same coefficient as in (2) ; since the signs of y are different in (2) and (3), if we add these two equations together, we have (4), which contains but one unknown quantity. Reducing (4), we have (5), or a: = 18. Substituting this value of x in (1), we have (6), which reduced gives (7), or 3/ =-12. Hence, EQUATIONS OF THE FIRST DEGREE. RULE. 109 Multiply or divide the equations so that the coefficients of the quantity to he eliminated shall become equal ; then, if the signs of this quantity are alike in both, subtract one equa- tion from the other ; if unlike, add the two equations to- getlier. Note. — The least multiplier for each equation will be that which will make the coefficient of the quantity to be eliminated the least common multiple of the two coefficients of this quantity in the given equations.' It is always best to eliminate that quantity whose coefficients can most easily be made equal. By this method of elimination find the values of x and y in the following equations : — 3. Given |^- + 3y = 33> ^^ jx x=3. 4. 4. Given f 8a: + 6t/ (lOx — 3y }■ Ans 6. Given ■ 19 • 6 y __ 12 5 Ans. 1^ = 27. \y = 12. 6. Given fx-f y -^— 1 =0 1. Given 2 T X 110 ' ELEMENTARY ALGEBRA. 115* Find the values of x and y in the following Examples. Note. — Which of the three methods of elimination should be used depends upon the relations of the coefficients to each other. That one which will ehminate the quantity desired with the least work is the best. 1. Given P- + 32/ = 25|. \'!x + 2t/ = 2Si (5x— y= 0> 2. Given 3. Given 4. Given 2 x—2y 3 — y=6 (x—l , 17 ( 2x— .4y==17 Ans. y ' Ans Ans. = 3. 15. fx = ll. ly= 1. (. = 9. 5. Given 2a;-{-3y 1 a;-|-22/ + 3 Z "~ 3 ~ 2 ^__2.-2y^3 Ans, (a; = ll. (2^= 5. 6. Given Y. Given 7+3 -^ L 8 + 16 — "^^r —3 3— - ^^ ^ — y 3: + y ___ J. EQUATIONS OF THE FIRST DEGREE. Ill 8. Given 2x_, 5y 8^ T ' T 15 3 7 Ans 9. Given 10. Given 5x + 54 + 5 y = 102 ar + y I y^ '^y — ^ 4 •"3'" 2 11. Given \ 4 + 3 "= ^ [ • ( 4 a; — 3 V = 25 ; Ans ( v = = 10. 20. m \ ^-l-: = 48. 28. (.y = — 3. 12. Given ^ — .V _o Ans 1 — 13. Given 5 J Adb, 10. 20. 14. Given ^-±y = y^2 4y — 4 2 15. Given 112 16. Given ELEMENTARY ALGEBRA. 3 — ^ 3J IT. Given r ^+y _^ 1 "X" — ^~"3 18. Given ^ 2y 3a: a: + 3 4 7 — y 3 6 14 19. Given < 4a;— 7y 5 3rr + y X — 4 = y + 3 PROBLEMS PRODUCING EQUATIONS OF THE FIRST DEGREE CON- TAINING TWO UNKNOWN QUANTITIES. 116t Many of the problems given in Section XIII. con- tain two or more unknown quantities ; but in every case these are so related to each other that, if one becomes known, the others become known also ; and therefore the problems can be solved by the use of a single let- ter. But many problems, on account of the complicated conditions, cannot be performed by the use of a single letter. No problem can be solved unless the conditions given are suflScient to form as many independent equa- tions as there are unknown quantities. 1. A grocer sold to one man Y apples and 5 pears for 41 cents ; to another at the same rate 11 apples and 3 pears for 45 cents. What was the price of each ? EQUATIONS OF THE FIRST DEGREE. 113 SOLUTION. Let X = the price of an apple, and y =z " " "a pear. Then, by the conditions. 'Jx + 5y = 41 (1) and Hx-f- 3y= 45 (2; 55x + 15ff = 225 (3) 21a;+15y=123 (4) 2l + 5yr=41 (•?) 34x=102 (5) y= 4 (8) x= 3 (6) We multiply (2) by 6 and (1) by 8, and obtain (3) and. (4); subtracting (4) from (3) we have (5), which reduced gives (6), or X = 3. Substituting this value of x in (1), we have (7), which re- duced gives (8), or y = 4. 2. There is a fraction such that if 2 is added to the numerator the fraction will be equal to ^ ; but if 3 is added to the denominator the fraction will be equal to ^. What is the fraction ? SOLUTION. Let - = the fraction. y Then, by the conditions, '-±1 = \ (1) and^,=i (2) Zx = y-\-^ (3) 2x + 4=y (4) X — 4 = 3 (5) x=1 (6) 14 + 4 = 18=y a) ^ = f, (8) Clearing (1) and (2) of fractions, we obtain (3) and (4) ; sub- tracting (4) from (3), we obtain (5), which reduced gives (G), or a; = 7. Substituting this value of x in (4), we have (7), ot y= 18. Hence, - - ~ 114 ELEMENTARY ALGEBRA. 3. There are two numbers whose sum is 28, and one fourth of the first is 3 less than one fourth of the second. What are the numbers ? Ans. 8 and 20. 4. The ages of two persons, A and B, are such that 5 years ago B's age was three times A's ; but 15 years hence B's age will be double A's. What is the age of each ? Ans. A's, 25; B's, 65. 6. There are two numbers such that one third of the first added to one eighth of the second gives 39 ; and four times the first minus five times the second is zero. What are the numbers ? 6. Find a fraction such that if 6 is added to the nu- 'nerator its value will be ^, but if 3 be added to the de- nominator its value will be -^ ? Ans. /j. t. What are the two numbers whose difference is to their sum as 1:2, and whose sum is to their product as 4 : 3 ? SOLUTION. Let X = the greater and y = the less. Thenx — i/:x + y=l:2 (1) x-]-y : xy = 4.: S (2) 2x — 2y = x + y (3) Sx + Sy = 4:xy (4) x = Sy (5) 9y + Sy=l2f (6) x = 3 (1) l=:y (8) Having written (1) and (2) in accordance with the statement in the problem, we form from them (3) and (4) by Art. 106. Re- ducing (3), we obtain (5) ; substituting this value of x in (4), we have (6), which, though an equation of the second degree, can be at once reduced to an equation of the first degree by dividing each term by y ; performing this division and reducing, we obtain (8) or y = 1 ; substituting this value of y in (5) we obtain (7), or ar = 3. EQUATIONS OF THE FIRST DEGREE. 115 8. What are the two numbers whose difference is to their sum as 3 : 20, and three times the greater minus twice the less is 35 ? 9. There is a niimb^r consisting of two figures, which is seven times the sum of its figures ; and if 36 is sub- tracted from it, the order of the figures will be inverted. What is the number ? Ans. 81. 10. There is a number consisting of two figures, the first of which is the greater ; and if it is divided by the sum of its figures, the quotient is 6 ; and if the order of the figures is inverted, and the resulting number divided by the difference of its figures plus 4, the quotient will be 9. What is the number? Ans. 54. 11. As John and James were talking of their money, John said to James, " Give me 15 cents, and I shall have four times as much as you will have left." James skid to John, "Give me 7^ cents, and I shall have as much as you will have left/' How many cents did each have ? Ans. John, 45 cents ; James, 30 cents. 12. The height of two trees is such that one third of the height of the shorter added to three times that of the taller is 360 feet ; and if three times the height of the shorter is subtracted from four times that of the taller, and the remainder divided by 10, the quotient is 17. Re- quired the height of each tree. Ans. 90 and 110 feet. 13. A farmer who had $41 in his purse gave to each man among his laborers $2.50, to each boy SI, and had $15 left. If he had given each man S4 and then each boy $3 as long as his money lasted, 3 boys would have received nothing. How many men and how many boys did he hire ? 116 ELEMENTAKY ALGEBRA. 14. A man worked 10 days and his son 6, and they received $ 31 ; at another time he worked 9 days and Ms son 1, and they received $29.50. What were the wages of each ? m 15. A said to B, "Lend me one fourth of your money, and I can pay my debts." B replied, "Lend me $100 less than one half of yours, and I can pay mine." Now A owed $1200 and B $1900. IIow much money did each have in his possession ? Ans. A, $800 ; B, $1600. 16. If a is added to the difference of two quantities, the sum is b ; and if the greater is divided by the less, the quotient will be c. What are the quantities ? . he — ac , h — a Ans. ■ - and -• c — 1 c — 1 lY. A man owns two pieces of land. Tliree fourths of the area of the first piece minus two fifths of the area Df the second is 12 acres ; and five eighths of the area of the first is equal to four ninths of the area of the second. How many acres are there in each ? Ans. 1st, 64 acres ; 2d, 90 acres. 18. A and B begin business with different sums of money; A gains the first year $350, and B loses $500, and then A's stock is to B's as 9 : 10. If A had lost $500 and B gained $350, A's stock would have been to B's as 1:3. With what sum did each begin ? Ans. A, $1450; B, $2500. 19. If a certain rectangular field were 4 feet longer and 6 feet broader, it would contain 168 square feet more ; but if it were 6 feet longer and 4 feet broader, it would contain 160 square feet more. Required its length and breadth. EQUATIONS OF THE FIRST DEGREE. 117 20. A market-man bought eggs, some at 3 for Y cents and some at 2 for 5 cents, and paid for the whole $2.62 ; he afterward sold them at 36 cents a dozen, clearing $0.62. How many of each kind did he buy? 21. A and B can perform a piece of work together in 12 days. They work together 7 days, and then A fin- ishes the woRk alone in 15 days. How long would it take each to do the work ? Ans. A 36 and B 18 days. 22. "I was ten times as old as you 12 years ago,'' said a father to his son ; " but 3 years hence I shall be only two and one half times as old as you.'' What was the age of each ? 23. If 3 is added to the numerator of a certain frac- tion, its value will be f ; and if 4 is subtracted from the denominator, its value will be j-. What is the fraction? 24. A farmer sold to one man Y bushels of oats and 5 bushels of corn for $12.76, and to another, at the same rate, 5 bushels of oats and 7 bushels of corn for $13.40. What was the price of each ? 25. Find two quantities such that one third of the first minus one half the second shall equal one sixth of a ; and one fourth of the first plus one fifth of the second shall equal one half of a. . 34 a , 15 a Ans. — and -23^. 26. A person had a certain quantity of wine in two casks. In order to obtain an equal quantity in each, he poured from the first into the second as much .as the second already contained; then he poured from the sec- ond into the first as much as the first then contained ; and, lastly, he poured from the first into the second as much as the second still contained ; and then he had 16 gallons in each cask. How many gallons did each origi- nally contain? Ans. 1st, 22 ; 2d, 10 gallons. 118 ELEMENTARY ALGEBRA. SECTION XV. EQUATIONS OF THE FIRST DEGREE CONTAINING MORE THAN TWO UNKNOWN QUANTITIES. 117. The methods of elimination given for solving equa- tions containing two unknown quantities apply equally well to those containing more than two unknown quantities. ( ^+ y— ^= 4 1. Given -l^x -\- ^y -{- Az izx — 2y\-bz = A. = n k to = 5) find X, y, and z. OPERATION. a: + y — 2 = 4 (1) 2x + 3y + 4.z=l7 (2) 3x — 2r/+ 5z= 5 (3) 2x + 2y — 2z= 8 (4) 3x + 3y— 32; =12 (5,^ y + 62= 9 (6) by— 82= 7 (7{ 5y + 302=45 (8j x + 3 — 1=4 (13) y + 6 = 9 (11) 382 = 38 (9) x = 2 ' (14) y = S (12) z= 1(10) Multiplying equation (1) by 2 gives equation (4), which we sub- tract from (2), and obtain (6) ; multiplying (1) by 3 gives (5), and subtracting (5) from (3) gives (7). We have now obtained two equations, (6) and (7), containing but two unknown quantities. Mul- tiplying (6) by 5, we obtain (8), and subtracting (7) from (8), we obtain (9), which reduced gives 2=1. Substituting this value of z in (6), and reducing, we obtain y == 3. Substituting these values of y and z in (1), and reducing, we obtain x = 2. 2. Given - X + y =26' y +z=29 z -|- t^> = 66 w -\-u =Sl M + « = 4:6 ■ , to find u, w, X, y, and z. EQUATIONS OF THE FIRST DEGREE. 119 OPERATION. z + y«26 (1) y + « = 29 (2) *-Hw-56 (8) ir + « =« 81 (4) « + x»-46 (5| y + z = 28 «— z— 3 to+z=53 «— z = 28 X— z- 8 (6) i»+z=53 (7) « — z=28 (8) 2z = 18 (9) y = 17 (11) z = 12 (12) ir = 44 (13) u = 37 (14) z= 9(10) Here we subtract (1) fipom (2), and obtain (6) ; then (6) from (3), and obtain (7) ; then (7) from (4), and obtain (8) ; then (8) from (5), and obtain (9), which reduced gives (10), or x = 9. Sub- stituting this value of x in (1), (6), (7), and (8), and reducing, we obtain (11), (12), (13), and (14), or y = 17, z = 12, w = 44, and w = 37. Hence, for solving equations containing any number of unknown quantities, RULE. Fr(mi the given equations deduce equations one less in number, containing one less unknown quantify; and con- tinue thus to eliminate one unknown quantity after an- other , until one equation is obtained containing but one unknown quantity. Reduce this last equation so as to find the value of this unknown quantity ; then substitute this value in an equation containing this and but one other unknown quantity, and reducing the resulting equation, find the value of this second unknown quantity ; substitute again these values in an equation containing no more than these two and one otJier unknown quantity, and reduce as before ; and so con- tinue, till the value of each unknown quantity is found. Note. — The process can often be very much abridged by the exercise of judgment in selecting the quantity to be eliminated, the equations from which the other equations are to be deduced, the method of elimination which shall be used, and the simplest equa- tions in which to substitute the values of the quantities which have been found. 120 ELEMENTARY ALGEBRA. Find the values of the unknown quantities in the fol- lowing equations : — 3. Given X -{- f/ -\- z -{- w = 16" x-{-i/-\-w-\-u= 14 X-\-1/-{-Z-\- It z=l[>^ Note. — If these equations are added together and the sum di-^ vided by 4, we shall have x -\- y -{- z -{- w -\- u = 20 ; and if from this the given equations are successively subtracted, the values of the unknown quantities become known. r ^ g U = 3. Ans. 2 2y + ^ 'ZbJ Ans rx= 1. iz = n. 6. Given a;— y — 2;=: 1 X-{-2l/— 10z=: 1 2x — 4y-f- 3s= 1 rx =z 5. Ans. -|y r= 3. U = i. T. Given \^+y+\^=22 1 ,1,1 i^ + iy + 2^ 1 , 1 . , 1 24 10 Ans. E 20. 12. 32. EQUATIONS OF THE FIBST DEGREE. 121 8. Given ^ 1,1 5 x'^y— 6 y^2 12 1 , 1 ___ 3 Note. — The best method for this example is that used in Ex- ample 3, without clearing of fractions. 9. Given ( ^+ y+ ^= 6j fxz= -] 2a: + 3y + 4^ = 20 [■ . Ans. U = xz= 1. 2. 3. 10. Given 11. Given rx + iy = 3U ]y + ^z = ll[ iz +ix = 29) x + y = a 11 rx-^y = a^ /a? = ^ (a — ft -j- c) ■\y + ^ = f>y- Ans. }y = i{a + b—c) (a: + z = c) iz=^{b-\-c — a) M Ans. < f/z=- abx-\-abyz=:a-\- b raox-^a()y=za-\- b ^ Given \bcy-\-bcz:=b +c) 13. Given I x+ 21 =^y+28 9x = 2y \ 122 ELEMENTARY ALGEBBA. PROBLEMS PRODUCING EQUATIONS OF THE FIRST DEGREE CON- TAINING MORE THAN TWO UNKNOWN QUANTITIES. 118. 1. A merchant has three kinds of flour. He can sell 1 bbl. of the first, 2 of the second, and 3 of the third for $85; 2 of the first, 1 of the second, and ^ bbl. of the third for $45.50 ; and 1 of each kind for $41. What is the price per bbl. of each ? Ans. 1st, $ 12 ; 2d, $ 14 ; 3d, $ 15. 2. Three boj'^s. A, B, and C, divided a sum of money among themselves in such a manner that A and B re- ceived 18 cents, B and C 14 cents, and A and C 16. How much did each receive? Ans. A, 10 ; B, 8 ; C, 6 cents. 3. As three persons, A, B, and C, were talking of their ages, it was found that the sum of one half of A's age, one third of B's, and one fourth of C's was 33 ; that the sum of A's and B's was 13 more than C's age ; while the sum of B's and C's was 3 less than twice A'a age. What was the age of each? Ans. A's, 32 ; B's, 21 ; C's, 40. 4. As three drovers were talking of their sheep, says A to B, "If you will give me 10 of yours, and C one fourth of his, I shall have 6 more than C now has." Says B to C, *'If you will give me 25 of yours, and A one fifth of his, I shall have 8 more than both of you will have left." Saj^s C to A and B, "If one of you will give me 10, and the other 9, I shall have just as many as both of you will have left." How many did each have? 5. Divide 32 into four such parts that if the first part is increased by 3, the second diminished by 3, the third multiplied by 3, and the fourth divided by 3, the sum, difierence, product, and quotient shall all be equal. Ans. 3, 9, 2, and 18. EQUATIONS OF THE FIRST DEGREE. 123 6. If A and B can perform a piece of work together in 8|2j. days, B and C in 9^r^ days, and A and C in 8J days, in how many days can each do it alone? Ans. A in 15, B in 18, and C in 21 days. 7. Find three numbers such that one half of the fijst, one third of the second, and one fourth of the third shall together be 56 ; one third of the first, one fourth of the second, and one fifth of the third, 43 ; one fourth of the first, one fifth of the second, and one sixth of the third, 35. 8. The sum of the three figures of a certain number is 12 ; the sum of the last two figures is double the first ; and if 297 is added to the number, the order of its fig- ures will be inverted. What is the number? Ans. 417. 9. A man sold his horse, carriage, and harness for $450. For the horse he received $25 less than five times what he received for the harness ; and one third of what he received for the horse was equal to what he received for the harness plus one seventh of what he received for the carriage. What did he receive for each? Ans. Horse, $225; carriage, $175; harness, $50. 10. A man owned three horses, and a saddle which was worth $45. If the saddle is put on the first horse, the value of both will be $30 less than the value of the 'second ; if the saddle is put on the second horse, the value of both will be $55 less than the value of the third ; and if the saddle is put on the third horse, the value of both will be equal to twice the value of the second minus $ 10 more than one fifth of the value of the first. What is the value of each horse ? Ans. Ist, $100; 2d, $175; 3d, $275. 11. The sum of the numerators of two fractions is 7, and the sum of their denominators 16 ; moreover the sum of the numerator and denominator of the first is equal 124 ELEMENTARY ALGEBRA. to the denominator of the second ; and the denominator of the second, minus twice the numerator of the first, is equal to the numerator of the second. What are the fractions ? Ans. f and f . l/Q. A man bought a horse, a wagon, and a harness, for $ 180. The horse and harness cost three times as much as the wagon, and the wagon and harness one half as much as the horse. What was the cost of each ? 13. A gentleman gives $600 to be divided among three classes in such a way that each one of the best class is to receive $10, and the remainder to be divided equally among those of the other two classes. If the first class proves to be the best, each one of the other two classes will receive $5 ; if the second class proves to be the best. each one of the other two classes will receive $4f ; but if the third class proves to be the best, each one of the other two classes will receive $2, What is the number in each class? 14. A cistern has 3 pipes opening into it. If the first should be closed, the cistern would be filled in 20 min- utes ; if the second, in 25 minutes ; and if the third, in 80 minutes. How long would it take each pipe alone to fill the cistern, and how long would it take the three together ? Ans. 1st, 85f minutes ; 2d, 46-^7j- minutes ; 3d, 35^^^ minutes. The three together, IQ-^r minutes. 15. Three men. A, B, and C, had together $24. Now if A gives to B and C as much as they already have and then B gives to A and C as much as they have after the first distribution, and again C gives to A and B as much as they have after the second distribution, they will all have the same sum. How much did each have at first? Ans. A, $13 ; B, $1, and C, $4. EQUATIONS OF THE FIRST DEGREE. 125 SECTION XYI. POWERS AND ROOTS. 119. A Power of any quantity is the product obtained by taking that quantity any number of times as a factor; and the exponent shows how many times the quantity is taken (Art. 24). Thus, a=ia^ is the first power of a ; a a=Ma^ " . second power, or square, of a ; a a a:=: a^ " third power, or cube, of a ; a a a az:^ a^ " fourth power of a ; and so on. Though not strictly within the definition, the quantity itself is called the first power, and the first root of itself. 120. In order to explain the use of negative indices, we form, by the rules of division, the following series: — ' 8 4 3 2 1^1 111 ^' «' «' «' «' ^' a' ^' ^' a<' d'^ a\ a\ a», a^ a\ a^ a-\ a"*, a"*, a'^ a-\ We form the first series as follows: a* divided by a gives a*; a* by a, gives a*; a' by a, gives a'; a' by a, gives a; a by a, gives 1 ; 1 by a, gives -; - by a, gives ,; , by a, gives -j, and so on. The second series is formed in the same way from a' to a; but if wc follow the same rule of division from a toward the right as from rt* to a, viz. subtracting the index of the divvsor from that of the div- l^nd^ a divided by a, gives cfi\ cfi by a, gives a-^ ; read a, with the negative index one; a~* by a, gives o""; a~* by a, gives a~*\ and Bo on. From this we learn, Ist. That the power of every quantity is 1 ; 2d. That cr^, a"*, ar^, &c., are only different ways of •^ 111^ wrxtinq -, -,, — ,» (Jcc. ^ a or or 126 ELEMENTARY ALGEBRA. Any two quantities at equal distances on opposite sides of a°, or 1, are reciprocals of each other. 121. The rules given for the multiplication and divis- ion of powers of the same quantity (Arts. 50 and 54) apply equally well whether the exponents are positive or negative. For 1 a' a^ X ar^ = a^ X -, = ~, = ci^ a« -T- a-^ = a^ -f- - = a^ X ci' = c^ 2_^ a' a' a" The following examples in multiplication are to be done according to the rules for the multiplication of powers of the same quantity by each other, given in Art. 50 ; and those in division, by the rule for the division of powers of the same quantity by each other, given in Art. 5JL. 1. Multiply x' by x'^. Ans. x^. 2. Multiply a^ by a'^. 3. Multiply x^ by x'^. Ans. a;^ or 1. 4. Multiply y-'^ by y*. 5. Multiply a-^x^y^ by a~^x~^y'^. Ans. ar^x^y^, or ar^y''. 6. Multiply Ix-^y-^z by 3xV^*- 1. Multiply llx'^fz-^ by A^x-^y-^^^, 8. Multiply ^ll|-^' by 5 a^ b-"- c\ 9. Divide x^ by x~^. Ans. x^^. 10. Divide x^ by x''^. 11. Divide a:~" by x~^. Ans. x-*. EQUATIONS OP THE FIBST DEGREE. 127 12. Divide y-^ by f. 13. Divide y' by y*. Ans. y'^. 14. Divide a~*6c^ by a^h~* c'^. Ans. a~'^U*c*. 15. Divide 16x'y-*2 by 4a;-y-*2:«. 16. Divide 4ar»y-»z by 2aar-2y-«2*. IT. Divide *la^hx-^y^ by 10 a i*^ a:* y-«2«. 18. Divide 144<7«6c-^x*y-'2 by 16 «''6-^c-='ar^/. 122t It follows from the preceding article that a factor may he transferred from the numerator of a fraction to its denominator, or vice versa, provided the sign of the expo- nent of the factor is changed from -\- to — , or — to +. For ^ = a« X ^4 = a« X a:-* = a'x-* a a 1 v> _^ ^ -,= ^=a^-, = aX^ = a2/ y ""y ^ ""y * ar'^y * "^ "~x-^y — = 1 X - = — X (/ ^ X dx 1. Transfer the denominator of ^-^ — r to the numerator. bc'y-'^ Ans. cU Ir^ c'"^ x^ y . 2. Transfer the numerator of -j- , - , to the denommator. 3. Transfer the denominator of /, . to the numerator. a - 10(8a»r« (2y)» + 6(3z«) (2y)4 — (2y)5. Ans. 243x" — 810z»y-j-1080z*y>— 720x*y»-}-240x*y*— 32/. Note. — Any letters, as a and b, might be substituted for 3 a:" and 2 y, and the expansion of (a — by written out, and then the Talues of a and b substituted. 3. Expand (a' — Sby. Ans. a'—UaH + 54ca*b^—lOSaH^ + Slb*. 4. Expand {x^^y^y. 136 ELEMENTARY ALGEBRA. 5. Expand (2 a + iy. Ans. 8 a3+ 84 a'' + 294 a + 343. 6. Expand (2ac — xy. Ans. 16a*c^ — 32a3c3a; + 24a2c2a:2 — 8aca;«+a:4. 7. Expand (a^x--2yy. 8. Expand (^ + xY. 9. Expand (I- ly. ^^«-il + I + ¥ + 2^« + x*. Ans ^__i^ f i^_5a^ 5aa;* a:* ■ 32 48 "f" 36 54 »" "162 243 10. Expand — 1 V. 11. Expand (^^^_L.)'. 27 "^ ^^ 8 64 12. Expand (i + iy. 13. Expand /ac— iV" 14. Expand /^a: + -Y, 15. Expand (l — -V. 16. Expand /'2 a^ — IV. n. Expand (l+lY. 18. Expand (i-iy. EVOLUTION. 137 131* The Binomial Theorem can be applied to the ex- pansion of a polynomial. Thus, in a -\- b — c, a -\- b can be treated as a single term, and the quantity can be written (a -J- i) — c. In like manner, a -\- b -\- x — y can be written (a -]- b) -{- (x — y). In such cases it is easier to substitute a single letter for the enclosed terms, and after the expansion to substitute the proper values. 1 . Expand (a + ^ — &-|-8aft«-f 6» = (3a*-f3a&-l-i»)6. If therefore 15875 is divided by 3 a' + 3 a 6 -f &» it will give 6, tlie units of the root. But 6, and therefore 3 a 6 -j- &*, a part of the divisor, is unknown, and we must use 3a^ = 2700 as a trial divisor. 15875 -^ 2700, or 158 -^ 27 = 5, a number that cannot be too small but may be too great, because we have divided by 3 a* instead of the true divisor, 3 a^ -{- 3 a b -\- l^. Then b = 5, and 3a'-|- 3aft-f^= 2700 -f- 450 -f- 25 = 3175, the true divisor; and (3 a' 4- 3 a 6 -f- />») 6 = 31 75 X 5 = 15875, and therefore 5 is the unit's figure of the root, and 35 is the required root The work will appear as follows : — 144 ELEMENTARY ALGEBRA. OPERATION. 4 2 8 T 5 (35 Root. 21 Trial divisor, 3 a^ = 2 Y ' 3ab=z 450 b^=z 2 5 True divisor, B a^ + 3ab + b^ = 3 1 75 1 6 8 1 5 Dividend. 158T5 Hence, to extract the cube root of a number, RULE. Separate the number into periods of three figures each, by placing a dot over units, thousands, Sc. Find the greatest cube in the left-hand period, and place its root at the right. Subtract this cube from the left-hand period, and to the re- mainder annex the next period for a dividend. Square the root figure, annex two ciphers, and multiply this result by three for a trial divisor ; divide the dividend by the trial divisor, and place the quotient as the next figure of the root. Multiply this root figure by the part of the root previously obtained, annex one cipher and multiply this result by three ; add the last product and the square of the last root figure to the trial divisor, and the sum will be the true divisor. Multiply the true divisor by the last root figure, subtract the product from the dividend, and to the remainder annex the next period for a dividend. Find a new trial divisor, and proceed as before, until all the periods have been employed. Note 1. — The notes under the rule in square root (Art. 139) apply also to the extraction of the cube root, except that 00 must be annexed to the trial divisor when the root figure is 0, and after placing the first point over units the point must be placed over every third figure from this. Note 2. — As the trial divisor may be much less than the true EVOLUTION. 145 divisor, the quotient is frequently too great, and a less number must be placed in the root. 2. Find the cube root of 1819144T. OPERATION. l8t Trial Divisor, 8a* =1200] Sab =- 360 6« = 36 l8t True Divisor, 3a'-|-3a& + 6» 2d Trial Divisor, 18191447 (2 63 _8 10191 1st Dividend. 1596J 8c^ =202800 Sab = 2340 6^= 9 9576 2d True Divisor, 3a«-f3a6-f-6»=205149 61544 7 2d Div. 615447 We suppose at first that a represents the hundreds of the root and 6 the tens: proceeding as in Ex. 1, we have 26 in the root Then letting a represent the hundreds and tens together, i. e. 26 tens, and b the units, we have 3 a', the 2d trial divisor, = 202800 ; and therefore 6 = 3; and 3 a^ -\- S ab -\- b*, the 2d true divisor, = 205149; and 263 is the required root. Note. — Though the 1st trial divisor is contained more than 8 times in the dividend, yet the root figure is only 6. 3. Find the cube root of 68116.47. OPERATION, 6 8116.4 7 0(4 0.9 5+ 64 48 0.0 10 8.0 .8 1 4 7 1 6.4 7 49 8.8 1 50 18.4 30 6.13 50 .0025 50 24.56 7 5 4417.9 2 9 2 9 8.5410 00 251.228375 4 7.3126 25^ 146 ELEMENTARY ALGEBRA. 4. Find the cube root of 292420Y. Ans. 143. 5. Find the cube root of 8120601. Ans. 201. 6. Find the cube root of 36926037. 7. Find the cube root of 67911.312. 8. Find the cube root of 46417.8. 9. Find the cube root of .8. Ans. .928+. 10. Find the cube root of .17164. 11. Find the cube root of .0064. 12. Find the cube root of 25.0001Y. 13. Find the cube root of 2.7. Note. — As a fraction is involved by Involving both numerator and denominator (Art. 127), the cube root of a fraction is the cube root of the numerator divided by the cube root of the denominator. 14. What is the cube root 2 r ? Ans. f. 15. What is the cube root of //^ ? 16. What is the cube root of f||? |-f| == |f|. Ans. f. Note. — If both terms of the fraction are not perfect cubes, and cannot be made so, reduce the fraction to a decimal, and then find the cube root of the decimal. A mixed number must be reduced to an improper fraction, or the fractional part to a decimal, be- fore its root can be found. 17. What is the cube root of ■^\? Ans. .899+. 18. What is the cube root of ^j? 19. What is the cube root of 3^ ? 20. What is the cube root of 117?? EVOLUTION. 14T EVOLUTION OP MONOMIALS. 1I3* As Evolution is the reverse of Involution, and since to involve a monomial (Art. 126) we multiply the exponent of each letter by the index of the required power, and prefix the required power of the numerical coefficient, Hence, to find the root of a monomial, RULE. Divide the exponent of each letter by the index of the re- quired root, and prefix the required root of the numerical cvefficient. Note 1. — The rule for the signs is given in Art. 136. As an even root of a positive quantity may be either positive or negative, we prefix to such a root the sign ± ; read, plus or minus. Note 2. — It follows from this rule that the root of the product of several factors is equal to the vroduct of the roots. Thus, V/"36 = v'l V^'O = 6. 1. Find the cube root of So:*/. Ans. 2xy^. 2. Find the square root of 4 x^. Ans. ±2x. . 3. Find the third root of — 125 a«ar. Ans. — 5 a^x^. 4. Find the fourth root of 81a"* b. Ans. ± 3 a-* b^. 5. Find the fifth root of 32 a^^ b\ Ans. 2 a* bK 6. Find the cube root of — 729 arV- Ans. — 9x1^. 7. Find the fourth root of 256 xV- 8. Find the cube root of — 512 a-H'. 9. Find the fifth root of 243 a:* y^. 148 ELEMENTARY ALGEBRA. Note. — As a fraction is involved by involving both numerator and denominator (Art. 127), a fraction must be evolved by evolving both numerator and denominator. 4 q2 2 a 10. Find the square root of r— r* Ans. ± ^r^,- Perform the operations indicated in the following ex- pressions : — 11. \/—129aH^c\ 12. (ida^x^f)^, 13. / 9 a? X* Y 36Z>V 14. it^a'^x" 15. {25Qa''x^'y^Y. ,4^10^,16\-J 16. /^Slan\ 11. \^a'"62"*c"»". SQUARE ROOT OF POLYNOMIALS. 144. In order to discover a method for extracting the square root of a polynomial, we will consider the rela- tion of a 4" ^ ^^ ^ts square, a^ ~\- 2 a b -{- h^. The first term of the square contains the square of the first term of the root ; therefore the square root of the first term of the square will be the first term of the root. The second term of the square contains twice the product of the two terms of the root ; therefore, if the second term of the square, 2 a b, is divided by twice the first term of the root, 2 a, we shall have the second term of the root b. Now, 2 a b -\- b^ = (2 a -\- b) b', therefore, if to the trial divisor 2 a we add b, when it has been found, and then EVOLUTION. 149 multiply the corrected divisor by h, the product will be equal to the remaining terras of the power after a^ has been subtracted. The process will appear as follows : — OPERATION. Having written a, the square a^-\-2ab-\-lt^{a-\-b root of a", in the root, we sub- a^ tract its square (a') from the 2a-|-i)2a64-^ g»ven polynomial, and have \ , \ VI 2ah 4- h" left. Dividing the ' first term ol this remamder, 2 a h, by 2 a, which is double the term of the root already found, we obtain ft, the second term of the root, which we add both to the root and to the divisor. If the product of this corrected divisor and the last term of the root is subtracted from 2 a ft -j- 6', nothing remains. 145* Since a polynomial can always be written and involved like a binomial, as shown in Art. 131, we can apply the process explained in the preceding Article to finding the root, when this root consists of any number of terms. 1. Findthe8quarerootofa2+2a5+ft=* — 2ac— 2ftc + (r». OPERATION. a^^2ab + P—2ac — 2bc + c^(a + b^c 2a + b)2ab + lP 2ab + if' 2a + 2b — c)—2ac — 2bc + c* -^2ac — 2bc-\-(^ Proceeding as before, we find the first two terms of the root a-\-b. Considering a -\- h zr 2^ single quantity, we divide the remainder — lac — 2ftc-)-c* by twice this root, and obtain — c, which we write both in the root and in the divisor. If this competed divisor is multiplied by — c, and the product subtracted from the dividend, nothing remains. 150 ELEMENTARY ALGEBRA. Hence, to extract the square root of a polynomial, RULE. Arrange the terms according to the powers of some letter. Find the square root of the first term, and write it as the first term of the root, and subtract its square from the given polynomial. Divide the remainder by double the root already found, and annex the result both to the root and to the divisor. Multiply the corrected divisor by this last term of the root, and subtract the product from the last remainder. Proceed as before with the remainder, if there is any. 2. Find the square root of 4 .<;^ — 4 a:^^ -[" y*. Ans. 2x — y^. 3. Find the square root of a^ -\-2ab-\-P-\-4:ac + 4 5 c + 4 c2. Ans. a + h + 2c. 4. Find the square root of 9 x^ — 12 x^ -{- 4=x^ -{- 6 ax^ — 4:ax-\-a^. Ans. Sx'^ — 2x-\-a. 5. Find the square root of4a^ -\- Sab — 4a -|~ ^^^ — 46+1 Ans. 2a + 2b—l. 6. Find the square root of 25 x^ — lOx^ -{- Gx"^ — x -\- {. Ans. 5^2 — oc -\- ^. 1. Find the square root of a?^ + 2 x^ — x^ — 2x^ -{- x^. 8. Find the square root of 4«^ — 4:ab -\- b^ — 4ac ^4.ad-[-2bc-\-2bd-\-c''-^2cd-[-d\ Ans. 2 a — b — c — d. 9. Find the square root of x^ — 4a?^ -[" ^^^ — ^^* J^bx'' — 2x-{- 1. 10. Find the square root of 4a4 -|- Sa^b — SaH^ ^I2ab' + 9b\ F.VOLUTION. 161 Note 1. — According to the principles of Art 136, the signs of the answers given above may all be changed, and still be correct Note 2. — No binomial can be a perfect square. For the square of a monomial is a monomial, and the scjuare of the polynomial with the least number of terms, that is, of a binomial, is a trinomial. Note 3. — A trinomial is a perfect square when two of its terms are perfect squares and the remaining term is equal to twice the product of their square roots. For, (a-|-6)» = a'-f 2a6 + 6» (a — 6/ = a' — 2a& + 6» Therefore the square root of a' ± 2 a 6 -[- i' is a ± 6. Hence, to obtain the square root of a trinomial which is a perfect square, Omitting the term that is equal to twice the product of the square roots of the other two^ connect the square roots of the other two by the tign of the term omitted. 1 1 . Find the square root of / + — * 4 A 4 Ans. 5-~ 12. Find the square root of x' -f- 2x + 1. Ans. X -\-\. 13. Find the square root of 4a:^ — 8xy -|- 4y^. 14. Find the square root of ^- — 2ab -\- ^h^. 15. Find the square root of 16y^ + 40yr* + 25 2*. Note. — By the rule for extracting the square root, any root whose inde.x is any power of 2 can be obtained by successive extractions of the square root Thus, the fourth root is the sqtiare root of the square root ; the eighth root is the square root of the square root of the square root; and so on. 16. Find the fourth root of a« — I2a«6 + 54a*6» — 108o^» + 8l6^ Ans. a" — Zb, 152 ELEMENTARY ALGEBRA. IT. Find the fourth root of -, + -1 + -^. 4- — 4- - Ans. -A X 18. Find the fourth root of x^ — 4.x' + 10a;« — - 16a:« J^lQx^—Ux^-\-\{)x^ — 4.x-\-l. Ans. t^^x+I. 146t To find any root of a polynomial. Since, according to the Binomial Theorem, when the terms of a power are arranged according to the power of some letter begin- ning with its highest power, the first term contains the first term of the root raised to the given power, therefore, if we take the re- quired root of the first term, we shall have the first term of the root. And since the second term of the power contains the second term of the root multiplied by the ne^jt inferior power of the first term of the root with a coefficient equal to the index of the root, therefore if we divide the second term of the power by the first term of the root raised to the next inferior power with a coefficient equal to the index of the root, we shall have the second term of the root. In accordance with these principles, to find any root of a polynomial we have the following RULE. Arrange the terms according to the powers of some letter. Find the required root of the first term, and write it as the first term of the root. Divide the second term of the polynomial by the first term of the root raised to the next inferior power and multiplied by the index of the root. Involve the whole of the root thus found to the given power, and subtract it from the polynomial. If there is any remainder, divide its first term by the di- visor first found, and the quotient will be the third term of the root. Proceed in this manner till the power obtained by involv- ing the root is equal to the given polynomial. EVOLUTION. 153 Note 1. — This rule verifies itself. For the root, whenever a new term is added to it, is involved to the given power, and whenever the root thus involved is equal to the given polynomial, it is evident that the required root is found. Note 2. — As powers and roots are correlative words, we have used the phrase given power, meaning the power whose index is equal to the index of the required root, and the phrase next inferior power meaning that power whose index is one less than the index of the required root. 1. Find the cube root of a« — 3 a'^ + 5 a* — 3 a — 1. OPERATION. Constant divisor, 3 a*) a' — 3 a^ -|- 5 a* — 3 a — 1 (a^ — a — 1 gg _ 3 flS -I- 3 q^ _ gg — 3 a*, Ist term of remainder. a«_.3a» + 6a« — 3a— 1 The first term of the root is a*, the cube root of a*, a* raised to the next inferior power, i. e. to the second power, with the co- efficient 3, the index of the root, gives 3 a*, which is the constant divisor. — 3 a*, the second term of the polynomial, divided by 3 o*, gives — a, the second terra of the root, (a* — a)' = a* — 3 a* 4" 3 a* — a* ; and subtracting this from the polynomial, we have — 3 a* as the first term of the remainder. — 3 a* divided by 3 a* gives — 1, the third term of the root, (a' — a — 1)'= the given poly- nomial, and therefore the correct root has been found. 2. Find the fourth root of 16 x* — 32 ar« y* + 24 ar* y* — 8ar/ + y". OPERATION. 4 X ( 2 x) • =* 3 2 a:") 1 6 a:* — 3 2 z" y» -f- 24 x« y* — 8 a; / -f y» ( 2 z — y» 16 a;* — 82a:»y»-|-24a:'y* — 8xy'-}-y' 3. Find the cube root of a» + 3 a" 6 + 3 a ft» + 6^ — 3 a^ c _>.6a6c — 36='c + 3ac« + 3ir»— c«. 4. Find the fourth root of 16 a* c* — 32 a» c» x + 24 a« (r* ar» — 8acx^ + x*. 7* 154 ELEMENTARY ALGEBRA. SECTION XVII. RADICALS. 147. A Radical is the indicated root of any quantity, as \/ X, d^ , \/2, 3^, &c. 148. In distinction from radicals, other quantities are called rational quantities. 149. The factor standing before the radical is the co- efficient of the radical. Thus, 2 is the coefficient of \/2 in the expression 2 \/ 2. 150. Similar Radicals are those which have the same quantity under the same radical sign. Thus, \/a, 2 \/a, and X h/a are similar radicals ; but 2 \/a and 2 \^b, or 2 x^ and 2 x* are dissimilar radicals. 151. A Surd is a quantity whose indicated root cannot be found. Thus, \/2 is a surd. The various operations in radicals are presented under the following cases. CASE I. 152. To reduce a radical to its simplest form. ^ Note. — A radical is in its simplest form when it contains no factor whose indicated root can be found. 1. Reduce \/^5a^b to its simplest form. OPERATION. ^15aH = A/25a^ X Sb=z^25a^xV^h = 5a\/Sb We first resolve 75 a' b into two factors, one of which, 25 a', is the greatest perfect square which it contains ; then, as the root of RADICALS. 155 the product is equal to the product of the roota (Art 143, Note 2)» we extract the square root of the perfect square 25 a", and annex to this root the factor remaining under the radical. Hence, RULE. Hesolve the quantity under the radical sign into two fac- tors, one of which is the greatest perfect power of the same name as the root. Extract the root of the perfect poicer, multiply it by the coefficient of the radical, if it has any, and annex to the result the other factor, with the radical sign between them. Reduce the following expressions to their simplest form : — 2. 3. ^/12a:. 9a« v/' 3c v/1 Ans. 2\/3x. Ans. 7 x' i^~x. 4. ^72a»6». Ans. 2a^96». 6. b^64:ab\ Ans. 10 6/^^4 a. 6. 1. SA^U1aH\ 25 \/ 56 a:. Ans. 21 a 6=^3. Ans. 60^7x. 8. 9. 4x/128r«y. -C/343x». 10. / 27a«c Y l2S2*y 11. 12. 1 1l€?C V 128x*y- Y 64 2 >v/16r^y» — 32 Vl6x«y«. — 32: rV Gx^y^Vl — 2ar'y« = 4a:y \/l — 2x2yS Ans. 156 ELEMENTARY ALGEBRA. 13. 4-^81a«c + 27a«. Ans. 12a4^3c + l. 14. (a + h) \/3a^ — Qab-\- Sb\ Ans. (a" — b^) ^"3. 15. 1 ^250x^f—12ox^y\ 16. (a: — y) (a^ar — a^y)*. 18. \/^^^16. -v/— 16 = \/16 V— 1=4:\/— 1, Ans. 19. 4/ —1250. 20. Vl9«' — 4^2 153. When a fraction is under the radical sign, it can be transformed so as to have only an integral quantity under the radical sign, by multiplying both terms of the fraction by that quantity which will make its denominator a perfect power of the same name as the root, and then re- moving a factor according to the Bule in Art. 152. 1. Reduce iy to its simplest form. OPERATION. /347. RADICALS. 16T le- Ana. 1^30. 10. .vi51 *' 1876 11. (a + 6)y/^. Ans. V^-6». CASE II. 151* To reduce a rational quantity to the form of a radical. 1. Reduce 3 a:* to the form of the cube root. OPERATION. Since 3 a:* is to be placed 3 2^ __ 3^ 27 a:* under the form of the cube root without changing its value, we cube it and then place the radical sign, ^, over it It is evident that ^ 21 z? = 3 a:". Hence, RULE. Involve (he quantity to the power denoted by the index of the root required, and ph/ce the corresponding radical sign over the power thus produced. 2. Reduce 4 a* 6 to the form of the square root. Ans. A^lQa^i^. 3. Reduce 2ab'^c~^ to the form of the fifth root. Ans. 4/32a'b^''c'^. 4. Reduce ^a'c^ to the form of the cube root. 9 168 ELEMENTARY ALGEBRA. 5. Reduce -z — r to the form of the fourth root. Bxyt 6. Reduce x — 2i/ to the form of the square root. Ans. \/ x^ — 41x1/ -\- 4: y^. 155% On the same principle the rational coefiScient of a radical can be placed under the radical sign, by involv- ing the coefficient to a power of the same name as the root indicated by the radical sign, multiplying it by the radical quantity, and placing the given radical sign over the product. 1. Place the coefficient of hi^ly under the radical sign. OPERATION. 6 A^ 2y = >^ 125 >ij/ 2^^ = ^ 250y In the following examples, place the coefficient under the radical sign. 2. 3>^4a;8y. Ans. >/x and b\fx. Ans. ^ s/x. It is evident that 3 times the ^ x and 6 times the ^~x make 8 times the y/a;. 3. Add V'S and \/ 50 together. OPERATION. In this case we make the radi- f^~~^ = 2 \/^ ^^^ ipa-rts similar by reducing them .-rx c /"H- *o t^®"* simplest form (Art. 152), ^ "^^ — — — and then add their coefficients as Sum = T \/ 2 in Example 2. Hence, RULE. Make the radical parts similar when they are not, and prefix the sum of the coefficients to the common radical. If the radical parts are not and cannot he made similar, con- nect the quantities with their proper signs. 4. Add 2\/60«a; and 3\/98aa;. Ans. 31\/2aic. 6. Add 4>^24ar8 and x^^l. Ans. 110:^^3. 6. Add \/2T and \/^363. Ans. 14\/'3. Y. Add -^512a:* and .^162y*. Ans. {4.x -{-Zy) 4^2. 8. Add ^1) and ^/\ VT=\/^\/5 = i\/5; V5 + i\/5 = f\/"5, Ana. 9. Add ^^ and ^1|^. Ans. ^ ^l2 . RADICALS. 161 10. Add Vl, 10^7f» and 6/v/20. Ana. 13 V^. 11. Add Vl^ and a/2^. CASE V. 158i To subtract one radical from another. 1. From \/T5 take \/'2T. OPERATION. We make the radical parts sim- .-=T r /-q ilar by reducing them to their sim- ~~ plest form (Art. 152). And 3 y/l taken from 5 y^ 3 evidently leaves 2 v' 3. Hence, V27 = 3^/3 2v^3 RULE. Make the radical parts similar when they are not, sub- tract the coefficient of the subtrahend from that of the min- uend, and prefix the difference to the common radical. If the radical parts are not and cannot be made similar, indi- cate the svbtracHon by connecting them with the proper sign. __ _ • 2. From ^^ take >^ 3. Ans. 2-^3. 3. From 9 s/ a^xy^ take 3 a \/ x^*. Au8. ^ayjs/x. 4. From T \/20x take 4\/45a:. Ans. 2\/5ar. 6. From /^ 500 take ^T08. Ans. 2 ^1^, 6. Prom 2^^^ take \/^. Ans. ttV^- 7. From >v/ | take \/^. Ans. ;^ V^O. 8. From 2^n6a:» take .^y891x». 9. From a 4/1^ take 7 *i/a^3^. 10. From >C^ 1174 take ^1892. 162 ELEMENTARY ALGEBRA. CASE VI. 159. To multiply radicals. 1. Multiply 3 V « by 5 f^l. OPERATION. 3 s/~a X 5 \/T = 3 X 5 X V « X V^ = 15 s/~ab As it makes no diflference in what order the factors are taken, we unite in one product the numerical coefficients ; and >^ ay^^h = \/~ab (Art. 143, Note 2). 2. Multiply 4\/2a6 by 5 /v^3«y. We reduce the radical OPERATION. . , . , parts to equivalent radicals 4v2a6= 4Y^ 8a o having a common index (Art. ply as ample. 5^Sax= 6^ 9a^x^ (Art. 156), and then multi- Product = 20 ^W^W ^^y ^' ^^ *^' preceding ex- 3. Multiply ^a by \/ a. OPERATION. a^ X a2 zz= >^a'^ X a^ = \/a\ or a* Multiplying as in the preceding examples, we have ^ a*, or a^ ; but I = ^ -|- ^ ; i. e. the index of the product is the sum of the indices of the factors. From these examples we deduce the following RULE. I. Reduce the radical paints, if necessary, to equivalent radicals having a common index, and to the product of the radical parts placed under the common radical sign prefix the product of their coefficients. II. Boots of the same quantify are multiplied together by adding their fractional indices. RADICALS. 163 4. Multiply 3VT0by 4V'5. . Ans. 60 V 2. 6. Multiply 4 -C^ a ar* by a Vx. Ana. ^axA^/d^x. 6. Multiply oa/c by b^/c. Ads. a&c. 1. Multiply \/ x" by ^ x. Ans. x«. 8. Multiply ^fhyV^. Ans. 1*, or ^16807. 9. Multiply \^x by v^x. Ans. V^""^". 10. Multiply 2 /v/a + 6 by 6x^a + 6. Ans. 12 x-^ (« + *)*. 11. Multiply /k/x, *yx, and /^x together. Ans. x^*. 12. Multiply 3 a{/^ by 2 V 8. Ans. 6 >C/2. 13. Multiply a\/x-^y by h mJ xy. Ans. aJy. 14. Multiply (a + ft)i by (a — 5)^ Ans. (a^ — J^jl 16. Multiply i \/T by 3 \/ 1 16. Multiply 2 V'S by 4 V"8. CASjE VII. 160* To divide radicals. 1. Divide 60\/15x by 4\/6x. OPERATION. As division ia finding 60 \/T5x -i- 4 i^~hx= 15 V 3" * quotient which, multi- plied by the divisor, will produce the dividend, the coefficient of the quotient must be a number which, multiplied by 4, will give 60, the coefficient of the dividend, i. e. 15; and the radical part of the quotient must be a quantity which, multiplied by y^Sx, will give v^lSa:, i.e. v^3; the quotient required, therefore, is 15 v' 3. 164 ELEMENTARY ALGEBRA. 2. Divide 6 M^Ty by 2 .^ 2y. OPERATION. We reduce the radical parts to equivalent radicals having a com- mon index (Art. 155), and then divide as in the preceding example. 3. Divide \/a by /^ a. OPERATION. ^ ^ „i = ^a^^Zr^2 _ ^- or J Dividing as in the preceding examples, we have ^~a, or a^. But ^ = i — i ; i' e. the index of the quotient is the index of the divi- dend minus the index of the divisor. From these examples we deduce the following RULE. I. Reduce the radical parts, if necessary, to equivalent radicals having a common index, and to the quotient of the radical parts placed under the common radical sign prefix the quotient of their coeficients. II. Roots of the same quantity are divided by subtracting the fractional index of the divisor from that of the dividend. 4. Divide 16 a/ ax by S\/a^x. Ans. 2\/«~^. 6. Divide 4:A^d' — b"^ by 2 \/« — *• Ans. 2A/a -\- b. 6. Divide Q\/'21 hj Sa/S. Ans. 6. T. Divide *>/ x by \/x. Ans. v^"*""- 8. Divide \/a by Xl/b. Ans, i/r- 9. Divide 3 by \^~3. Ans. -v/'S. RADICALS. 165 10. Divide x by ^x. Ans. ^x^. 11. Divide 4a«\/x by 2cr^\/~y, Ans. ^^^Jy 12. Divide V 6" by ^5^ 13. Divide 4^1 by ^500a«x, Ans. 4. Find the cube root of x'^ ^ a^ b. Ans. 1/ -W"* 6. Find the fifth root of ar^\/^. Ans. \/x. 6. Find the fourth root of i VT- Ans. ^'^. 1. Find the cube root of 1 a,/S. Ans. ^TlT. 8. Find the square root of 12 /^b. POLYNOMIALS HAVING RADICAL TERMS. 163. It appears from the principles already established, that the laws which apply to calculations with quantities which have exponents, apply equally well whether the exponents are positive or negative, integral or fractional. The following examples, therefore, can be done by rules already given. 1. Add 4a — 3 \/y and 3 a + 2 \/y. Ans. la — Vy. 2. Add 3 X + >^ 135 and 1 x — ^ 1080. _ Ans. lOar — 3>^6. 168 ELEMENTARY ALGEBRA. 3. Add 2 V 28 — V 27 and 2 V 63 + \/i8. 4. Subtract 15 a; — \/50a from 13 a: — s/^a. Ans. 3\/^ — 2ar. 5. Subtract f^ as? — \/4 6 from ^J ax — \/16 6. Ans. A^ ax — Xfs/a — 2\/6. 6. Subtract .i^"32 — ^"242 from — 3 f^l — T \/"3. T. Multiply /v/a — \/^ by \/a — /\/a;. OPERATION. \/a — fn/ X a — ^ ah — /^ ax -\- /^ bx 8. Multiply xy -\- \^ ah by 4 — s/ ah, Ans. ^xy -\- (4 — a; 3/) /y/o^ — ah. 9. Multiply T + \/T0 by 6 — \/l0. Ans. 32 — \/T0. 10. Multiply \/a -j- V^ by \/« — V^- -^i^s. a — J. 11. Multiply V'5 — 4-^"3 by ^45 + /^"O. 12. Multiply ^ VI + T \/"3 by i Vl^ — T V 3. 13. YiWi^Q s/ ax -\- M^ ay -\-x-\- s/ xy \y^ s/ a-\' s/ X. OPERATION. \/a -j- \/a;) is/ ax -\- ^ ay -\- x -\- s/ xy (\/ar -f- \/y ^/ ax -f- a: V^ + V^ s/ ay +V^ RADICALS. Ig9 14. Divide V^ — V^ — V^ + \/^ by v^ — V^. Ans. s/~ot. — \/^. 16. Divide <^ x-\-^x-\-^y^-\-^y^ bya:+y* 16. Divide a: — y by V^ — Vy. Ans. V^ + V.y. 17. Divide4xy + 4>v/^--ary>v/^ — a*by4 — V"^. 18. Expand (V^ + \/~yy. Ans. x + 2 V^ + y. 19. Expand (a^ — IT^y. Ans. a — 2 4/?^+ 1. 20. Expand (^ — ^1)*. Ans. o^ — 4a*6i + 6a6 — 40*6' 4- ja 21. Expand (4 — V^)». Ans. 100 — 61^3. 22. Expand (a-i — ar-i)». Ans. a-l — 3 a-^ x-^ + 3 a-4a:-« — x-*. ?3. Expand (l-^ly. Ans. l_-4. + i. ?_ I i. 24. Expand (^|--^|y. Ans. f?-i4j^+:.y_l^ , ^ 25. Find the square root of a — 2 a^ i^ -j- i^. Ans. t^~a — /^~W, 26. Find the cube root of x* — 3 x^y^ + 3 ary^ — y. Ans. X — ys. 27. Find the fourth root of 16 a — 32 a^y* + ^4 ar^yl -8xV + y^. Ans. 2 a^~yf 170 ELEMENTARY ALGEBRA. SECTION XVIII. PURE EQUATIONS WHICH REQUIRE IN THEIR REDUCTION EITHER INVO- LUTION OR EVOLUTION. 164. A Pure Equation is one that coDtains but one power of the unknown quantity ; as, A^x -\- ac = h, 4 aj2 -|- 3 = T, or 14 a;" = a 5. 165. A Pure Quadratic Equation is one that contains only the second power of the unknown quantity ; as, 6x'^'-14:a=5lb, ay^=lScd, ov acz'^ 14. 166. Radical Equations, i. e. equations containing the unknown quantity under the radical sign, require Invo- lution in their reduction. 167. To reduce radical equations. 1. Reduce \/ a; — 3»=^. operation. V"^ — 3 = 8 Transposing, a/ x =^ 11 Squaring, x = 121 2. Reduce /^x — 4 + 7 = 10. operation. ^ 1^4 _|_ 7 == 10 Transposing, ^x — 4 = 3 /Cubing, as -^ 4 == 2t Transposing, *c =? 31 RADICAL EQUATIONS. * 171 3. Reduce ^^^'±^~^ y/a = v<.. OPERATIOM. Clearing of fractions, V/df2 + Va:= a Squaring", d^ + A^x= a" Transposing, ^x=z a^ — d^ Squaring, a: = (a^ — d'f Hence, to reduce radical equations, we deduce from these examples the following general RULE. Transpose the terms so that a radical part shall stand by itself; then involve each member of the equation to a power of the same name as the root ; if the unknown quantity is still under the radical sign, transpose and involve as before ; finally reduce as usual. 4. Reduce 4 + J + 3 \/lc = ^J-. Ans. x=l6. 5. Reduce - 4/ - = - . Ans. x = 2. 6. Reduce (>v/7+ 4)* = 2. Ans. x = 144. T. Reduce V 11 + arm \/x + 1. Ans. a; = 25. 8 Reduce ^x — 7 == a^ x -f 18 — VS. Ans. ar = 27. 9. Reduce ■ ^^ = -^ . Ans. x = X ex yj X 1 r>j 85 8z — 3 8x4-3 16. Reduce ,^ — - — r— = - — ^. 42 3x-}- 3 3x — 3 IT T> J a:«-f-x4-8 , x» 4- x — 8 „ 17. Reduce -^^-f- + -J—^- = 2. 174 ELEMENTARY ALGEBRA. 170i Equations containing radical quantities may re- quire in their reduction both Involution and Evolution; and in this case the rule in Art. 167, as well as that in Art. 169, must be applied. Which rule is first to be ap- plied depends upon whether the expression containing the unknown quantity is evolved or involved. 1. Reduce 17 — \/x« — 2 = 12. OPERATION. IT — Va:^ — 2 = 12 Transposing, &c., s/ x^ — 2 = 5 Squaring, a:^ — 2 = 25 Transposing and uniting, ^ = 27 Extracting the cube root, a: = 3 2. Reduce (V^c' — 4 + 3)^ = 125. Ans. a; = 2. 3. Reduce i/ — =z a^ x. Ans. x-=z ± - y 2a; .2 a-\-h 4. Reduce \f x -\- a =. \ X — a Ans. x= ± A^2a^-{-2ab + b\ 5. Reduce f- ^3 (.^ + ii)^^ 6. deduce Ai^2x^ + Sx^ + 24.x^+S2x=zx + 2. . 7. Reduce 4^9 {x' + 19) + 100 — ^ = ^^ • 171 1 Equations which contain two or more unknown quantities may require for their reduction involution, or evolution, or both. In these equations the elimination is effected by the same principles as in simple equations. (Arts. 112-114.) 3^ 1. Given -^5 4 "^^ [>-, to find x and y. 2x^ + , ^ — 14 ? y = 54) PURE EQUATIONS . ABOVE THE FIRST DEGREE. OPERATION. ^-H^* (1) 2x' + y= 54 (2) r ^-^« (3) r=iio (4) 15-?=14 C) a^= 25 (5) y= 4 (8) X =±5 (6) 176 Subtracting four times (1) from (2), we obtain (4), which re- duced gives (6), or X = ± 5 ; substituting this value of x in (1), we obtain (7), which reduced gives (8), or y = 4. Find the value of the unknown quantities in the follow- ing equations: — 2. Given \'^=''-^l. Ans.l^=±^- 3. Given |'"-'^='^|. Ans. \-=±^- Ma:«2 = 20 \ ra:== ± 1. 4. Given ^ 2 x ar = 10 >• . Ans. -j y = ± 3. (3y.z = 45) (2:= ±5. 6. Given K^-^^=n. Ans. |- = 2J- 6. Given |x« + y = 97 > (x — y = 1/ — 2x> 7. Given (x*--2y«=14 ) 176 ELEMENTARY ALGEBRA. PROBLEMS PRODUCING PURE EQUATIONS ABOVE THE FIRST DEGREE. 172. Though the numerical negative values obtained in solving the following Problems satisfy the equations formed in accordance v^ith the given conditions, they are practically inadmissible, and are therefore not given in the answers. 1. A gentleman being asked how many dollars he had in his purse, replied, " If you add 21 to the number and subtract 4 from the square root of the sum, the remainder will be 6.^' How many had he? SOLUTION. Let X = number of dollars. Then, ^/x + 2i—4.= 6 Transposing, /\/ x -{- 21 =. 10 Squaring, a; + 21 = 100 Transposing, x= 19, number of dollars. 2. Divide 20 into two parts whose cubes shall be in the proportion of 27 to 8. Ans. 12 and 8. 3. What two numbers are those whose sum is to the less as 8 : 3, and the sum of whose squares is 136? Ans. 10 and 6. 4. What number is that whose half multiplied by its third gives 54 ? 6. What number is that whose fourth and seventh multiplied together gives 46f ? Ans. 36. 6. There is a rectangular field containing 4 acres whose length is to its breadth as 8:5. What is its length and breadth ? PURE EQUATIONS ABOVE THE FIRST DEGREE. 177 V. There are two numbers whose sum is IT, and the less divided by the greater is to the greater divided by the less as 64 : 81. What are the numbers? Aus. 8 and 9. 8. The sum of the squares of two numbers is 65, and the difl'erence of their squares 33. What are the numbers ? 9. The sum of the squares of two quantities is a, and the difference of their squares b. What are the quantities ? Ans. ± VTTM^) a"d ± Vi la~^h). 10. A gentleman sold two fields which together con- tained 240 acres. For each he received as many dollars an acre as there were acres in the field, and what he received for the larger was to what he received for the smaller as 49 : 25. What are the contents of each ? Ans. Larger, 140; smaller, 100 acres. 11. What are the two quantities whose product js a and quotient hi . . — - , la Ans. ± V a ^ and ±1/7- 12. What two numbers are as m : n, the sum of whose squares is a? ^ mi/a , . nd'a Ans. ± -, ^ and ± -; ^ Y/(m«-}-n«) V(m'-}-n«) 13. What two numbers are as m : n, the difference of whose squares is a? , - ,- Ans. ± ^ and ± -, — 14. Several gentlemen made an excursion, each taking $484. Each had as many servants as there were gentle- men, and the number of dollars which each had was four times the number of all the servants. How many gen- tlemen were there? Ans. 11. 16. Find three numbers such that the product of the first and second is 12 ; of the second and third, 20 ; and the sum of the squares of the first and third, 34. 8» L 178 ELEMENTARY ALGEBRA. SECTION XIX. AFFECTED QUADRATIC EQUATIONS. 173f An Affected Quadratic Equation is one that con- tains both the first and second powers of the unknown quantity ; as, Sx^ — 4: x=: 16; or ax — bx^ = c. 174. Every affected quadratic equation can be reduced to the form x"^ -]- bx = c, in which h and c represent any quantities whatever, posi- tive or negative, integral or fractional. For all the terms containing a^ can be collected into one term whose coefficient we will represent by a ; all the terms containing x can be collected into one term whose coefficient we will represent by d; and all the other terms can be united, whose aggregate we will represent by e. Therefore every affected quadratic equation can be reduced to the form Dividing (1) by a, d e Letting - =b, and - = c, we have 175. The first member of the equation x^ -\- bx = c cannot be a perfect square. (Art. 145, Note 2.) But we know that the square of a binomial is the square of the first term plus or minus twice the product of the two terms plus the square of the last term; and if we can find the third term which will make x"^ -{- bx a perfect aa^-\-dx = e (1) 'a a (2) 3^-{-bx = c (3) EQUATIONS OP THE SECOND DEGREE. 179 square of a binomial, we can then reduce the equation x^ -\- bx = c. Since b x has in it as a factor the 8qi\are root of a:*, a:* can be the first term of the square of a binomial, and b x the second terra of the same square; and since the second term of the square is twice the product of the two terms of the binomial, the last term of the binomial must be the quotient arising from dividing the second term of the square by twice the square root of the first term of the square of the bi no- inial ; i. e. the last OPERATION. ^^ ^^ ^i^g ^j„^ x2-f-5ar = c (1) . , . bx b «" + ^^ + 4 = 4 + « (2) ^^^ therefore the J rn — third term of the x-\- -^=- ±. kI T '\~ ^ (^) square must be 2 Y * b\ . — to each mem- 4 ber, we have (2), an equation whose first member is a perfect square. Extracting the square root of each member of (2), and transposing, we obtain (4), or x ^ — 9 "*■ v/ T "^ ^' ^^'^^ ** * general expression for the value of x in any equation in the form of z* -|" ^ ^ = <^* Hence, as every affected quadratic equation can be re- duced to the form x^ -\- hx^=. c, in which b and c repre- sent any quantities whatever, positive or negative, integral or fractional, every affected quadratic equation can be re- duced by the following RULE. Beduce the equation to the form x^ -\- bx=z c, and add to each member the square of half the coefficient of x. Extract the square root of each member, and then reduce as in simple equations. 180 ELEMENTARY ALGEBRA. 1. Reduce Ta;2 — 28 a; + 14 = 238. OPERATION. *lf — 2Sx-{- 14 = 238 Transposing, T a;^ — 28 a: = 224 Dividing by Y, x"^ — 4a;= 32 Completing the square, x^ — 4 x -j- 4 = 36 Evolving, X — 2 = ±6 Transposing, a;:=:2±6 = 8, or — 4 Note. — Since in reducing the general equation x^ -{- bx = c vre find x = — ^±i/j -f-c, every affected quadratic equation must have two roots ; one obtained by considering the expression 2 ~\~ ^ positive, the other by considering this expression nega- / b^ tive. Whenever 4 / - -j- c = these two roots will be equal. o -D J ^ rr , 13 x^ , X 2. Reduce ^--- + -^ = - + -. operation. Clearing of fractions, 4a:^ — 2a: + 13 = lOa:^ -|- 5x Transposing, — Car* — 1x = — 13 Dividing by — 6, x^ -\ = — Completing the square, ^' + i) +^^ = ^, + ~ =~ Evolving, a; + ^ = ± ^ Transposing, ^ = — — ± j^ ^\, or— 2^ EQUATIONS OF THE SECOND DEGREE. 181 Note. — In completing the square, as the second term disappears when the root is extracted, we have written ( ) in place of it. 3. Reduce 3x^ — 25 + 6x = 80. Ans. x=z 5, or — t. 24 X 4. Reduce x = 3. Ans. x=z6, or — 4. X 5. Reduce2x + ^-±4=7. Ans. x = 2. ' X 1 Note. — In this example both roots are 2. x* 4- 4 6. Reduce 1x ^^ =zbx — 1. X — 4 Ans. X = 8, or — 1. t. Reduce 17 - ^-^-=^ == ^£^ + 10. Ans. X = 7, or — 27. 8. Reduce T , + 3 = ^* A^^s. x = 10, or — 1 J. x+5 9. Reduce j -^ = 4. ,/v T> J 16 100 — 9x « 10. Reduce —^ — = 3. x 4z" 176. Whenever an equation has been reduced to the form x^ -\- bx = c, its roots can be written at once; for this equation reduced (Art. 175) gives x = — - ± t /- + c. Hence, TJie roots of an equation reduced to the form x^ -\- bx = c are equal to one half the coefficient of x with the opposite sign, plus or minus the square root of the sum of the square of one half this coefficient and the second member of the equation. 182 ELEMENTARY ALGEBRA. In accordance with this, find the roots of x in the fol- lowing equations : — 1. Reduce a:^ -|- 8 a: = 65. a; = — 4 ± /s/16 + 65 = 6, or — 13, Ans. 2. Reduce a:2~ 10a; = — 24. x = b ± /v/25 — 24 = 6, or 4, Ans. 3. Reduce x'^ — Qx=z — ^, Ans. a; = 5, or 1. • 4. Reduce a:^ _|_ T a; = ITO. 7 /^9 ^ = — 2 ± \/ X + ^"^^ = ^^' ^^~ ^'^' ^"«- 6. Reduce a;^ + -x = -- ^ = -i±\/^ + ^ = i or -1, Ans. 1 9 6. Reduce x^ + -a: =-• Ans. a: = 1, or — 1^. Y. Reduce a;2 — -zx^^ — — • Ans. a; = -, or -• 5 25 5 5 8. Reduce - = | -j- 5^. Ans. a: = 7, or — 5^. SECOND METHOD OF COMPLETING THE SQUARE. 177. The method already given for completing the square can be used in all cases ; but it often leads to in- convenient fractions. The more difficult fractions are in- troduced by dividing the equation by the coefficient of x^y to reduce it to the form a;^-|-ia: = c. To present a method of completing the square without introducing these fractions, we will reduce equation (1) in Art. 174. EQUATIONS OF THE SECOND DEGREE. 183 1. Reduce ax^ -\- dx = e. OPERATION. ax' + dx = e (1) a^a^-^-adx^ae (2) a'^x^ + adx + j=j + ae (3) ax X Multiplying (1) by a, the coefficient of a^, we obtain (2), in which the first term must be a perfect square. Since adx, the second terra, has in it as a factor the square root of a* r", a* 2* can be the first tenn of the square of a binomial, and adx the second term ; and since the second term of the square is twice the product of the two terms of the binomial, the last term of the binomial must be the second term of the square divided by twice the square root of the first term of the square of the binomial, or „ — = - ; and therefore the 2 ax 2 d* term required to complete the square is — , which is the square of d* one half of the coefficient of x in (1). Adding - to both members of (2), we obtain (3), whose first member is the square of a binomial. Extracting the square root of (3) and reducing, we obtain (5), or l{-Ht+-} Hence, to reduce an aflfected quadratic equation, we have this second RULE. Reduce the equation to the form ax^ -\- dx = e; then mul- tiply tlie equation by the coefficient of ar*, and add to each member the square of half the coefficient of x. Extract the square root of each member, and then reduce as in simple equations. 184 ELE]VIENTARY ALGEBRA. Note 1. — This method does not introduce fractions into the equa- tion when the numerical part of the coefficient of x is even. When the coefficient of x* is unity, this method becomes the same as the first method. Note 2. — If the coefficient of 3? is already a perfect square the square can be completed without multiplying the equation, by add- ing to both members the square of the quotient arising from dividing the second term by twice the square root of the frst. This method also becomes the same as the first method when the coefficient of 3^ is unity. Note 3. — As an even root of a negative quantity is impossible or imaginary, the sign of the first term, if it is not positive, must be made so by changing the signs of all the terms of the equation. 2. Reduce 3 a:^ + 8 a; = 28. OPERATION. 3x2_|_8a: = 28 Completing the square, 9 a;^ + ( ) -f 16 = 16 + 84 = 100 Extracting square root, 3a;-[-4=± 10 Whence, 3a; = — 4 ±10 = 6, or — 14 And x = 2, or — 4f 3. Reduce 25ar» — 10 a; = 195. operation. 25a;2_-i0a:=195 Completing sq. by Note 2, 25a;2— () + 1 = 1 + 195 = 196 Extracting square root, 5x — 1 = ± 14 Whence, 5a:=l ± 14 = 15, or— 13 And x= 3, or — 2f 4. Reduce 5 a;^ — 20 a; = — 15. Ans. a; = 3, or 1. 5. Reduce Ta:2 — 8 a; =12f. Ans. a; = f ± f V 26. 6. Reduce Y a;'' — 4:ax=-—-' Ans. a; =: — , or — -• Y. Reducer- ^"7 =x — 3. Ans. x= 10, or 31. 14 — z S •* EQUATIONS OF THE SECOND DEGREE. 185 8. Reduce ar^ + l^ = — i^ — 4. ' 4 4 9. Reduce — = g — = — ^ THIRD METHOD OF COMPLETING THE SQUARE. 178i The method of the preceding Article introduces fractions whenever the numerical coefficient of x is not even. To present a method of completing the square without introducing any fraction, we will again reduce equation (1) in Art. 174. 1. Reduce ax^ -{- dx = e. OPERATION. aar^-j- dx = e (1) ' a a (2) ^■+"a'*'"*"4a«""4a«"+"a (3) a»a:» + 4arfx + rf2==ef- + 4ae W 2ax + d= ± Vrf^ + 4«e (5) _ — d± v/d»4-4atf (6) Dividing (1) by a, the coefficient of 2*, we have (2); then com- pleting the square according to the Rule in Art. 1 75, we have (3) ; and if we multiply (3) hy 4 a\ it will give (4), an equation free from fractions (unless o, d, or c in (1) are themselves fractions), and one whose first member is the square of a binomial. To pro- duce this equation directly from (1), we have only to multiply (1) by 4a; i. e. by four times the coefficient of z*, and add to both members rf* ; i. e. the square of the coefficient of x. Reducing we have (S), which is a general expression for the value of x in any equation in the form o{ ax^ -{- dx = e. 186 ELEMENTARY ALGEBRA. Hence, to reduce an affected quadratic equation, we have this third RULE. JReduce the equation to the form ax^ -\- dx=z e; then mul- tiply the equation by four times the coefficient of x^ and add to each member the square of the coefficient of x. Extract the square root of each member, and then reduce as in simple equations. Note. — The third Note under the Rule in Art 177 is applica- ble in all cases. 2. Reduce 5x'^ — 1x = 24:, OPERATION. 5 a:2 — 7 r = 24 Multiplying by 5 X 4 and adding 7^ to each member, 100.r — { ) + 49 = 49 + 480 = 529 Extracting the square root, lOx — 7 = ±23 Transposing, 10 x = 7 ± 23 = 30, or — 16 Whence, a: = 3, or — 1.6 Note. — The multiplication of the coefficient of x^ need only be expressed. Its coefficient after evolving is double its original coef- ficient. 3. Reduce ^^"^"^^^ = 293. OPERATION. 44 x^ — 15 a: = 293 7 Clearing of fractions, 44 a;^ — 15 a: = 2051 Completing square, 176 X 44ar2 — { ) + 225 = 225 + 360976 = 361201 Evolving, 88 a: — 15 = ± 601 Transposing, 88 a: = 15 ± 601 = 616, or — 586 293 Whence, a: = 7, or -— 44 4. Reduce 1 x^ — 15 a; = — 2. Ans. a: = 2, or |. EQUATIONS OF THE SECOND DEGREE. 187 6. Reduce —5 ^ — rir = ^ — 3. XT O X -f- V 6. Reduce — 1—^ + - == 5. 7. Reduce — ^ + ? = 3. o „j 4X-I-4 3x— 3 8. Reduce ' -z X 2x — 1 1 3 9. Reduce 7 — 2x ' 2x-f-4 Ans. x = 1, or- -28. Ans. X = 3, or — ■u- Ans. X = = 2, or - -i lOx-flO 3x 13 10* 10. Reduce ^x^ — a^ = x — b. h , lAcf — V Ans. a: = -±y/-3^^. 11. Reduce 5 — 3x-^ = llOar-*. Note. — Multiply by x*. 12. Reduce s/l^ + V^* = 6 \/^. Note. — Divide by y^x! 179. The rules which have been given for the solution of affected quadratic equations apply equally well to any equation containing but two powers of the unknown quan- tity whenever the index of one power is exactly twice that of the other. By the same reasoning as in Art. 1T4, it can be shown that all such equations can be reduced to the fonn aa:^" -|" ^^ = ^> or It will be seen that the first member is composed of two terms so related that they may be the first two terms of a binomial square, and we can supply the third by one of the rules already given for completing the square. OPERATION. x^ — 2x^ = 4.8 (1) r6_.2x«+ 1 = 1 + 48 = 49 (2) x^—l=±1 (3) x^ = S,or — 6 (4) a: = 2, or ^— 6 (5) 188 ELEMENTARY ALGEBRA. 1. Reduce ;r^ — 2a:^==48. Since the squu*e root of x^ isa^, it is evident that the second term contains as one of its factors the square root of the first term ; i. e. the first member of the equation is composed of two terms so related that they may be the first two terms of the square of a binomial. Completing the square, we have (2) ; extract- ing the square root of each member of (2), we obtain (3) ; transpos- ing we have (4), and extracting the cube root of (4) we have x = 2, or ^^^ 2. Reduce Sx^ — 4.x^ = 160. OPERATION. 3a;l_4^l— 160 (1) 36;k^— + 16 = 16+ 1920 = 1936 (2) 6a;^ — 4=±44 (3) 6 a:^ = 48, or — 40 (4) x^ = 8, or — 5f- (5) . x^'= 2, or ^ — -s^o (6) a:= 16, or (— Y)^ (7) In this equation the index of the higher power is exactly twice that of the lower. Completing the square we have (2) ; extract- ing the square root of each member of (2), we have (3) ; trans- posing, we have (4), which divided by 6 gives (5); extracting the cube root of (5), we have (6), which involved to the fourth power gives (7). 3. Reduce ^ + ^ = A. Ans. x = ^J, or — ^J. it 4 o 2i 4. Reduce -^a;^ + f/^x=:l. Ans. a; = j, or — 8. EQUATIONS OF THE SECOND DEGREE. 189 5. Reduce a: — § >/ x = 44^. Ans. x = 49, or 40|. 6. Reduce x* — x^ = 0. Ans. a; = 1, or 0. 7. Reduce 3ar* — 2ar»+3 = 228. Ans. X =z ± S, or ± 5 \/ — ^. 8. Reduce 3 x^- — 2 x" = 8. Ans. x = aC^2, or <^ — J. 9. Reduce ^A^-±J = t±ll . Ans. x = 64, or 4. 10. Reduce x* — a x^ = i. Ans.x = ±y/(|±^^ + J). 180. A polynomial may take the place of the unknown quantity in an affected quadratic equation. In this case the equation can be reduced by considering the polyno- mial as a single quantity. 1. Reduce (x — 4)^ — 2 (x — 4) = 8. OPERATION. (x — 4)2 — 2 (x — 4) = 8 (1) (x-4)2-() + l = l-f 8 = 9 (2) X — 4--l=±3 (3) X = 6 ± 3 = 8, or 2 (4) Considering x — 4 as a single term, and completing the square, we have (2) ; extracting the square root, transposing, &c., we have (4), or X =« 8, or 2. Note. — We might put (x — 4)=y; then (x — 4)* = 3^, and the equation becomes y* — 2 y = 8. After finding the value of y in this equation, x — 4 must be substituted for y. 2. Reduce V5 + x + /^5 + x = 6. Ans. x= 11, or 7ft, 3. Reduce 4 + 2x — x3 + j^>v/44- 2x — x» = i. Ans. X = 1 ± ^\/T9, or 3, or — 1. 190 ELEMENTARY ALGEBRA. 4. Reduce x + 7 — T s/ x -f- Y = 8 — hs/x + t. Ans. a; = 9, or — 3. 6. Reduce ix — 6)^ — Z s/ x — 5 = . Ans. itr = 9, or 5 + ^25. 6. Reduce 7?-\-Zx-\- s/ x^ _[_ 3 a: -f- 6 = 14. Note. — Add 6 to both members. Ans. a; = 2, or — 5, or — f ±1 V^. 1, Reduce 4 + aj2 — 2a: — 2 V6 — 2a: + x^=l. Ans. a: = 3, or — 1, or 1 ± 2 \/ — 1. 60 8. Reduce Vx^ + a: + 6 = , — 4. Ans. a: = 5, or — 6, or — \±.\ \/377. 181. Of the methods given for completing the square, the first is the best when the coefiScient of the less power of the unknown quantity is even, and the coefficient of the higher power is unity, or when these become so by reduc- tion ; the second method is better than the third when- ever the coefficient of the less power of the unknown quan- tity is even. When the equation cannot be reduced by the first method without introducing fractions, if the co- efficient of the higher power of the unknown quantity is a perfect square, and the coefficient of the less power is divisible without remainder by twice the square root of the coefficient of the higher power, the method given in Note 2, Art. 177, is the best. Let each of the follow- ing equations be reduced by the method best adapted to it. 1. Reduce 4x2— 14 _ 3^:2 __ i2a; — 1. Ans. a: = 1, or — 13. 2. Reduce 36ar^ + 24 a; =1020. Ans. a: = 6, or — 5|. EQUATIONS OF THE SECOND DEGREE. 191 3. Reduce x — i + ? = 21. Ans. x = 21, or f . „ , 9, — x x — 2 2a:— 11 4. Reduce —2 6~ = x-S * 5. Reduce g "~ 5 "" To ~ ^^* 6. Reduce — ^ — = ~^ — ^* Ans. a; = 3(c — d), or 3rf. 7. Reduce |±^ = ^ + 2§. 8. Reduce — = 9 — X — 4 ^ 9. Reduce - — y = ^ 10. Reduce ^ + ^ = ?• Ans. x == 1 ± Vl— «"• 11. Reduce 3x + 3 = 13 + -• 3a: — 3 « , 3x — 6 12. Reduce 5 X — — ^==2xH ^ 13. Reduce ^-1^ + 1 = ^- U. Reduce 2 >v/^ — ^x — *l = 5. Ans. X = 16, or 7^. 15. Reduce 2/s/«^^+3V2^= -^j^=- Ans. X = 9 a, or — a. 15 16. Reduce 4>\/^ — \/2x+ 1 = ,^^ ■ Ans. X == 4, or — 2f . 17. Reduce 5 V25 — x = 6 \/26 — x -f x — 13. Ans. X = 16, or 9. 192 ELEMENTARY ALGEBRA. 18. Reduce ^ii = V 4~+ V 2^H^^'. Ans. X = 12, or 4. 19. Reduce 6 + 4x-i — 12 a:-^ = 100 ar-^. 20. Reduce i ^~^' + 6 .^x — y = y • 21. Reduce 3 a;* — 24 a;^ — 80 = 304. 22. Reduce ^ -J- 10 = 1 + 4:r». Ans. a: = ± 4, or ± 2 V— 2. Ans. a: =: \/"9, or .^. 23. Reduce 5 x* — 3 a:^ + ?1 = 27, ' 4 Ans. a;=^ ± iv 6, or ± 3/v/— ^V- 24. Reduce 2 x^ — 5 a:^ + 4 = 2. 25. Reduce 6 a;^ + 1184 = 5a;i 26. Reduce \/^ + 3 — ^x -f 3 = 2. Ans. X = 13, or — 2. 27. Reduce x^ — x/ar^ -j- a; — 5 = 25 — a:. Note. — By transposing — x and subtracting 5 from each mem. ber, make the expression without the radical in the first member like that under the radical; then complete the square, &c. 28. Reduce a:^ — 2ar + 3V2a:2 — 6ar~ll=x + 33. Ans. a; = 6, or — 3, or f ± ^ \/^T3. 29. Reduce 21 a:* + 11 a:^ _ 69 =:= 321 — (11 ar* + 5 a;^). Ans. x= ± ^ VT3, or ± ^ \/ — lb. 30. Reduce 7— J—— = ^ 4- (2 a: — 4/ 8 ' (2a: — 4)* 31. Reduce (a:2— 4a:)2z=12x — 3a:2. 32. Reduce a; + (a;2 — ar)2 = a:2 -I- 6112. EQUATIONS OF THE SECOND DEGREE. 193 PROBLEMS PRODUCING AFFECTED QUADRATIC EQUATIONS WITH BUT ONE UNKNOWN QUANTITY. 182. Though the numerical negative values obtained in solving the following Problems satisfy the equations formed in accordance with the given conditions, they are prac- tically inadmissible, and are therefore not given in the answers. 1. Divide 40 into two parts such that the sum of their squares shall be 1042. SOLUTION. Let X = one part ; then 40 — x = other part Then, x« -}- (40 — x)' = 1042 Expanding, a^ _|- 1600 — 80 a: -j- x» = 1042 Transposing and uniting, a:" — 40a: = — 279 Whence, x = 20 ± 1 1 = 81, or 9 And, 40 — X = 9, or 31 2. Divide 20 into two parts such that their product will be 99J. Ans. ^ and 10^. 3. The ages of two brothers are such that the age of the elder plus the square root of the age of the younger is 22 years, and the sum of their ages is 34 years. What is the age of each? • Ans. Elder, 18; younger, 16. Note. — The other answers found by reducing the equation, viz. 25 and 9, satisfy the conditions of the equation only upon consid- ering y' 9 = — 3. To make the problem correspond to these an- swers, the word "plug** must be changed to "minus." 4. A merchant had two pieces of cloth measuring to- gether 96 yards. The square of the number of yards in the 194 ELEMENTARY ALGEBRA. longer is equal to one hundred times the number of yards in the shorter. How many yards are there in each piece ? Ans. 60 and 36. 5. Find two numbers whose difference is 3, and the sum of whose squares is 117. Ans. 9 and 6. 6. A merchant having sold a piece of cloth that cost him S42, found that if the price for which he sold it were multiplied by his loss, the product would be equal to the cube of the loss. What was his loss ? Note. — If the word " loss " were changed to gain; the other an- swer, — 7, or as it would then become, -}- 7> would be correct. Ans. $6. T. Find two numbers whose difference is 5, and prod- uct 176. Ans. 11 and 16. 8. There is a square piece of land whose perimeter in rods is 96 less than the number of square rods in the field. What is the length of one side ? Ans. 12 rods. 9. Find two numbers whose sum is 8, and the sum of whose cubes is 152. 10. A man bought a number of sheep for $240, and sold them again for $6.75 apiece, gaining by the bargain as much as 5 sheep cost him. How many sheep did he buy ? Ans. 40. 11. Find two numbers whose difference is 4, and the sum of whose fourth powers is 1312. Note. — Let x — 2 and x -|- 2 be the numbers. Ans. 2 and 6. 12. A man sold a horse for $312.50, and gained one tenth as much per cent as the horse cost him. How much did the horse cost him? Ans. $250. 13. The difference of two numbers is 5, and the less minus the square root of the greater is 7. What are the numbers? Ans. 11 and 16. EQUATIONS OF THE SECOND DEGREE. 195 14. A and B started together for a place 300 miles dis- tant. A arrived at the place 7 hours and 30 minutes be- fore B, who travelled 2 miles less per hour than A. How many miles did each travel per hour? Ans. A, 10 ; B, 8 miles. 15. A gentleman distributed among some boys $15; if he had commenced by giving each 10 cents more, 5 of the boys would have received nothing. How many boys were there? Ans. 30. 16. Find two numbers whose sum is a, and product h. Ans. "f-^ and ^-^^ 17. A merchant bought a piece of cloth for $45, and sold it for 15 cents more per yard than he paid. Though he gave away 5 yards, he gained $4.50 on the piece. How many yards did he buy, and at what price per yard ? Ans. 60 yards, at 75 cents per j'^ard. 18. A certain number consists of two figures whose sum is 12; and the product of the two figures plus 16 is equal to the number expressed by the figures in inverse order. What is the number? Ans. 84. 19. From a cask containing 60 gallons of pure wine a man drew enough to fill a small keg, and then put into the cask the same quantity of water. Afterward he drew from the cask enough to fill the same keg, and then there were 41 j gallons of pure wine in the cask. How much did the keg hold ? Ans. 10 gallons. 20. There is a rectangular piece of land 75 rods long and 65 rods wide, and just within the boundaries there is a ditch of uniform breadth running entirely round the land. The land within the ditch contains 29 acres and 96 square rods. What is the width of the ditch ? Ans. .5 of a rod. 196 ELEMENTARY ALGEBRA. SECTION XX. QUADRATIC EQUATIONS CONTAINING TWO UNKNOWN QUANTITIES. 183. The Degree of any equation is shown by the sum of the indices of the unknown quantities in that term in which this sum is the greatest. Thus, ^xy — 2a: = 7 is an equation of the second degree, hx'y'-^- xif^a'c " '' " fourth 5^4 _ 14^ _ ^2^.3 .. Ci i^ fifth Note. — Before deciding what degree an equation is, it must be cleared of fractions, if the unknown quantities appear both in the denominators and in the numerators or integral terms; and also from negative and fractional exponents. 184. A Homogeneous Equation is one in which the sum of the exponents of the unknown quantities in each term containing unknown quantities is the same. Thus, ^x^ — ^xy -\- y"^ z=:i\^ or x^-\- 3:r^2 + 3ar2y + / = 2T or x^ — ^.x^yAr Gx^/ — 4a;/ + ^^ = 256 is a homogeneous equation. 185. Two quantities enter Symmetrically into an equa- tion when, whatever their values, they can exchange places without destroying the equation. Thus, x^ — 2xy + / — 25 or a:»+3ar2y + 3:r/4- /= 8 or x' + 2xy-\-y''^ 2x + 2y = 24 QUADRATIC EQUATIONS. 197 186. Quadratic equations containing two unknown quan- tities can generally be solved by the rules already given, if they come under one of the three following cases: — I. When one of the equations is simple and the other quadratic. II. When the unknown quantities enter symmetrically into each equation. 111. When each equation is quadratic and homogeneous. CASE I. 187. When one of the equations is simple and the other quadratic. 1. Given 1^^+^^,"= ^- I, to find X and y. OPERATION. 2x-f2y = 22 (1) 3x'-{-if=in (2) 3/=ll— x(3) 3z»-f-121 — 22a:-|-3:»=lll (4) 42^— 22x = — 10 (5) 4«a:«—()-fll«= 121— 40 = 81 (6) 4x= 11 ±9 = 20, or 2 (7) y=6,orl0^(9) a:=5, or^ (8) From (1) we obtain (3), or y = 11 — x. Substituting this value of y in (2), we obtain (4), an affected quadratic equation, which reduced gives (8) ; and substituting these values of x in (3), we obtain (9). In this Case the values of the unknown -quantities can generally be found by substituting in the quadratio eqvaHon the value of one unknown quantity found by reducing the siviple equation. 198 ELEMENTARY ALGEBRA. 2. Given | ^^ " ^^ | , to find a: and y. xy = 2^ (1) OPERATION, X — y = 3 (2) x' — 2xy + 7f= 9 (3) 4:xy ==112 (^) x^ + 2xy + y'=l2l (S) x + y =±n (6) 2x = 14, or — 8 a) 2y = 8, or — 14 (8) X = 1, or — 4 (9) y = 4, or — 7 (10) Subtracting four times (1) from the square of (2), we obtain (5) ; extracting the square root of each member of (5), we obtain (6); adding (2) to (6), we obtain (7) ; subtracting (2) from (6), we ob- tain (8) ; and reducing (7) and (8), we obtain (9) and (10). Note. — Though Example 2 can be solved by the same method as Example 1, the method given is preferable. By this method find the values of x and y in the fol- lowing equations : — 3. Given |^ "^ = H. Ans. j ^ = ^• 4. Given |^+^=13|. Ads. j^ = ^^or6. 6. Given < ^ ^ >■ • \5x + y = 29> 6. Given j ^y-24> iSx — 2yz=lQi Zxy = 45 ^y = 15 7?- -2xy+y« = 4 ^ — y = ±2 QUADRATIC EQUATIONS. 199 CASE II. 188. When tlie unknown quantities enter symmetrically into each equation. 1. Given | ^ + ^ = ^ J | , to find x and y. (x'-j-y*= 152) OPEIIATION. x+y = % (I) 2:»+y=152 (2) a:^ + 2x^ + 3/^ = 64 (3) ar«- xy + y^=19 (4) (5) (6) a) (8) 2x = 10, or 6 (9) 2y = 6, or 10 (10) X = 5, or 3 (11) y = 3, or 5 (12) Squaring (1), we obtain (3); dividing (2) by (1), we obtain (4); subtracting (4) from (3), we obtain (5), from which we obtain (6) ; subtracting (6) from (4), we obtain (7) ; extracting the square root of each member of (7), we obtain (8) ; adding (8) to (1), wc ob- tain (9); subtracting (8) from (1), we obtain (10); and reducing (9) and (10), we obtain (11) and (12). Note 1. — It must not be inferred that x and y are equal to each other in these equations ; for when x = 5, y = 3 ; and when X = 3, y = 5. In all the equations under this Case the values of the two unknown quantities are interchangeable. Note 2. — Although 3* -^ r^ = \b2 is not a quadratic equation, yet as we can combine the two given equations in such a manner as to produce at once a quadratic equation, we introduce it here. 200 ELEMENTARY ALGEBRA. 2. Given i„.„ ^^ r»to find x and y. U2-fy2_2a; — 2y = 3> ^ OPERATION. a:y=6 (1) a:2_|_^_2a;_2y= 3 (2) 2a:y =12 (3) (^ + 2//-2(a: + 2/) = 15 (4) (3; + yy- () + 1 = 16 (5) x + 3/=l±4 = 5,or — 3 (6) a: = 3,or2,or ""^^/~^\ 7) y=2,or3,or^:iiHyj:iL^(8) Adding twice (1) to (2), we obtain (4) ; completing tlie square in (4), we obtain (5) ; extracting the square root of each member of (5), and transposing, we obtain (6) ; and combining (1) and (6) as the sum and product are combined in the preceding example, we obtain (7) and (8). In Case II. the process varies as the given equations vary. In general the equations are reduced by a proper combination of the sum of the squares, or the square of the sum or of the difference, with multiples of the product of the two unknoiun quantities; and finally, of the sum, with the difference of the two unknown quantities. Note 3. — When the unknown quantities enter into each equation symmetrically in all respects except their signs, the equations can be reduced by this same method ; e. g. ar — ?/ = 7, and x^ — t/' = 511. In such equations the values of the unknown quantities are not inter- changeable. Note 4. — The signs ± ^ standing before any quantity taken in- dependently are equivalent to each other ; but when one of two quan- tities is equal to ± a while the other is equal to ^ &, the meaning is that the first is equal to -f- a, when the second is equal to — b; and the first to — a, when the second is equal to -|- ^« QUADRATIC EQUATIONS. 201 By this method find the values of x and y in the follow- ing equations : — 3. Given i2-+2y= IH . Ans. j- = 4'Or3. l3x»+3/ = 273) (y = 3,or4. (^^y— 8) .^„ (x = 9,or— 1. 4. Given ^ . •^, w«« f • -^^s* "^ , o (ar» — / = 728) Cy= 1, or — 9. 6. Given K + ^^t ^'^ «tl ' Note. — Divide the second equation by the first. 6. Given |^-^^ + y = U. CASE III. 189. When each equation is quadratic and homogeneous. 1. Given pa:y+ y^ = 5) to find a: and y. KZt? — ary = 10 ) OPERATION. 2xy+y» = 5 (1) Sa:'* — xy=10 (2) Let x=zvy 2vy^J^f = b (3) 3t;2y2 — vy2=io (4) 5 10 a) 2i;-^-l ~ Zv'—v . 15t;» — 5v=20v+ 10 (8) 3r^ — 6t; = 2 (9) r = 2, or — ^ (10) y=±l,or±Vl5 (12) x = ry= ±2, or qp i\/T5 (13) 202 ELEMENTARY ALGEBRA. Substituting vy for x in (1) and (2), we obtain (3) and (4) ; from (3) and (4) we obtain (5) and (6) ; putting these two values of y^ equal to each other, we obtain (7), which reduced gives (10) ; sub- stituting this value of v in (5), we obtain (11), which reduced gives (12) ; and substituting hx x = vy the values of v and y from (10) and (12), we obtain (13). Examples under Case III. can generally be reduced best by substituting for one of the unknown quantities ih^ product of the other by some unknown quantity, and then finding the value of this third unknown quantity. When the value of this third quantity becomes known, the values of the given unknown quantities can be readily found by substitution. Note. — Whenever, as in the example above, the square root is taken twice, each unknown quantity has four values ; but these values must be taken in the same order, i. e. in the example above, when y = -j-l, x = -f~2; when y = — l,a: = — 2; when y = -[- y^Ts, X •«= — \ y/Ts ; and when y = — y/ 15, a; = -]- ^ V^ 15. By this method find the values of x and y in the follow- ing equations : — 2. Given j ^^ - ^^ = 1^ a:= ± T, or ± 4\/— -J. y= ±5, or dbllV— ~i. Ans. < 3. Given \-'' + ^-y = ^n. '•1 Ans. ^-=±3,or±9^/-i. = ± 2,.or TS-i/— i- ixy — ^y = 3 — x^ ) Ans j^=±2,or±24V-V^. \y=± l,or qillV— T^T- 5. Given j'' ": '^^ = '/ ^^^l • ( x2 — 3=y2 + 2 > QUADRATIC EQUATIONS. 203 190« Find the values of x and y in the following Examples. Note. — Some of the examples given below belong at the same time to two Cases. Thus in Example 1 both the equations are symmetrical, and both are (juadratic and homogeneous, and there- fore it belongs both to Case II. and Case III. Example 3 belongs both to Case I. and Case II. 1. Given I , -y = 20) ^^^ Cx=±6,or±4. a; = 7, or 3. y = 3, or 7. ar» + ^-^ = 41 ) (y = ± 4, or ± 5. 2. Given I ^y= 6 ) U^+7x?/ = 55— y2) 3. Given i^+y= "^ I. Ans. j^ = 4^^^8- (y = 3, or 4. 4. Given | ^^=12 > (a:» + x = 32— y— y^> 5. Given \ ^T^'^^^l- Ans. | (sxy — 7 = 56) 6. Given \ ^-y= H. (a^y''-\-2xyz= 1295) Note. — Considering xy a single quantity, find its value in the second equation. 7. Given i^y--f=m. Note. — Subtract from the second equation three times the first, and extract the cube root of each member of the resulting equation. 8. Given j2x=y + 2x/= 168) -Ans. 1^ = 1'°' !• - (« =3, or4. 204 9. Given 10. Given 11. Given 12. Given 13. Given ELEMENTARY ALGEBRA. \ a:y=10 j na;2 — 2a:y=:88) Ans. |^ = = ±5. ±2. Ans. { a:=±4, or± 66 a^ y= ±3, or q: Its V^. FJrg-- { 2a: + 2y = 30~y2. 4xy = 60 > (6a:^-— 23/^= 10 ) Ans. fa-rr: 1000, or 8. 14. Given | 3a:^:^18) (0:^ — 2/^ = 65) Ans. ■! 625, or 1 a:= ± 3, or ± 2/^— 1, 3^=: ± 2, or ± 3^"^=^. 16. Given J^ (o:-^/) == 3 (V^^ + Vy) ) { a;y = 36 Ans. - ±\/- -23- -11 Tv/- 2 -23- -11 , or 9, or 4. , or 4, or 9. 16. Given IT. Given I x + y = Al ) Jo:*— /= n (x — 2^ =19) QUADRATIC EQUATIONS. 205 18. Given -{^ ^ ^ :f l . i X + y = 65 > 19. Given i^-'-^^^i^^ 20. Given i-* + 2x^y + 2xy^ + / = 95| (x«— x'f/— xy^ + y»= 5> 21. Given ■[ ^y= H. U* + y* = 272) 22. Given j^ +.y = H- U* + y* = 626) PROBLEMS PRODUCING QUADRATIC EQUATIONS CONTAINING TWO UNKNOWN QUANTITIES. 191. Though the numerical negative values obtained in solving the following Problems satisfy the equations formed in accordance with the given conditions, they are practically inadmissible, and, except in Example 4, are not given in the answers. 1. The sum of the squares of two numbers plus the sum of the two numbers is 98 ; and the product of the two numbers is 42. What are the numbers ? Ans, 7 and 6. 2. If a certain number is divided by the product of its figures the quotient will be 3 ; and if 18 is added to the number, the order of the figures will be inverted. What is the number "^ Ans. 24. 3. A certain number consists of two figures whose X roduct is 21 ; and if 22 is subtracted from the number. 206 ELEMENTARY ALGEBRA. and the sum of the squares of its figures added to the remainder, the order of the figures will be inverted. What is the number? Ans. 37. 4. Find two numbers such that their sum, their prod- uct, and the difierence of their squares shall be equal to one another. Ans. f ± ^ V5 and ^ ± ^ V5. 6. There are two pieces of cloth of difterent lengths; and the sum of the squares of the number of yards in each is 145 ; and one half the product of their lengths plus the square of the length of the shorter is 100. What is the length of each ? Ans. Shorter, 8 ; longer, 9 yards. 6. Find two numbers such that the greater shall be to the less as the less is to 2f, and the difference of their squares shall be 33. 7. The area of a rectangular field is 1575 square rods ; and if the length and breadth were each lessened 5 rods, its area would be 1200 square rods. What are the length and breadth ? 8. Find two numbers such that their sum shall be to 6 as 9 is to the greater, and the sum of their squares shall be 45. Ans. 9 Vl and 3 VT, or 6 and 3. 9. The fore wheels of a carriage make 2 revolutions more than the hind wheels in going 90 yards ; but if the circumference of each wheel is increased 3 feet, the car- riage must pass over 132 yards in order that the fore wheels may make 2 revolutions more than the hind wheels. What is the circumference of each wheel ? Ans. Fore wheels, 13^ feet ; hind wheels, 15 feet. 10. Find two numbers such that five times the square of the greater plus three times their product shall be 104, and three times the square of the less minus their prod- uct shall be 4. KATio AXD r::o?onTio-M. 207 SECTION XXI. RATIO AND PROPORTION. 102. Ratio is the relation of one quantity to another of the same kind ; or, it is the quotient which arises from dividing one quantity by another of the same kind. Ratio is indicated by writing the two quantities after one another with two dots between, or by expressing the division in the form of a fraction. Thus, the ratio of a to b is written, a : h, or ^ ; read, a is to b, or a divided by b, 193. The Terms of a ratio are the quantities compared, whether simple or compound. The first term of a ratio is called the antecedent, and the other the consequent; and the two terms together are called a covplet. 194. An Inverse, or Reciprocal Ratio, of any two quan- tities is the ratio of their reciprocals. Thus, the direct ratio of a to ft is a : b, i. e. r; and the inverse ratio of a to ft is 1 1 . 1 1 ft , - : Tf 1. e. - -T- 7 = -» or ft : a. a b aba 195. Proportion is an equality of ratios. Four quan- tities are proportional when the ratio of the first to the second is equal to the ratio of the third to the fourth. The equality of two ratios is indicated by the sign of equality (==) or by four dots (: :). Thus, a : b = c : d, or a : b: : c : d, or -r=z-,; read, a to ft h a equals c to d, or a is to ft as c is to d, or a divided by ft equals c divided by d. 208 ELEMENTARY ALGEBRA. 196i In a proportion the antecedents and consequents of the two ratios are respectively the antecedents and con- sequents of the proportion. The first and fourth terms are called the extremes, and the second and third the means. 197. When three quantities are in proportion, e. g. a : b = b : c, the second is called a mean proportional be- tween the other two ; and the third, a third proportional to the first and second. 198. A proportion is transformed by Alternation when antecedent is compared with antecedent, and consequent with consequent. 199. A proportion is transformed by Inversion when the antecedents are made consequents, and the conse- quents antecedents. 200. A proportion is transformed by Composition when in each couplet the sum of the antecedent and consequent is compared with the antecedent or with the consequent. 201. A proportion is transformed by Division when in each couplet the diff'erence of the antecedent and conse- quent is compared with the antecedent or with the con- sequent. THEOREM I. 202. In a proportion the product of the extremes is equal to the product of the means. Let a : b = c : d i. e. Clearing of fractions, a c h~d RATIO AND PROPORTION. 209 THEOREM II. 203t If the product of Ivjo quantities is equal to the prod- uct of two others, the factors of either product may he made the extremes^ and the factors of the other the means of a proportion. Let ad=:bc Dividing by hd, h^^d i. e. a : b = c : d THEOREM III. 204. If four quantities are in proportion, they will he in proportion by alternation. Let a : b = c : d By Theorem I. ad=bc By Theorem II. a -. c=^b : d THEOREM IV. 205. If four quantities are in proportion, they mil he in proportion by inversion. Let a : b = c : d By Theorem I. ad= be By Theorem 11. b : a = d : c THEOREM V. 206. If three quantities are in proportion, the product of tJie extremes is equal to the square of tJie mean. Let a : b = b : c By Theorem I. ac = If^ THEOREM VI. 207. If four quantities are in proportion, they will he in proportion hy composition. a b — c 'd t + ^ = a ' b ~ c + rf d a + b: b = c + d: d 210 ELEMENTARY ALGEBRA. Let a \h:=.c \ d \. e. Adding 1 to each member; or i. e. THEOREM VII. 208. If four quantities are in proportion, they will he in proportion by division. Let a : b =z c : d i. e. Subtracting I from each member or i. e. a THEOREM VIII. 209. Two ratios respectively equal to a third are equal to each other. Let a : b=:m '. n and c : d=m : n . ^ a m . c m 1. e. - = - and ,== - on d n Hence (Art. 13, Ax. 8), |= f i. e. a '. b = c : d THEOREM IX. 210. If four quantities are in proportion, the sum and difference of the terms of each couplet will he in proportion. a c b~~d l-^-l-^ a — b c — d b ~ d -b: b = c — d: d RATIO AND PROPORTION. 211 Let a : h:=c : d By Theorem VI. a -\- b : b = c -{- d : d {I) and by Theorem VII. a^—b . b = c — d : d (2) From (1 ), by Theorem III. a-{'b:c-\-d=b:d From (2), by Theorem III. a — b:c — dz=b:d By Theorem VIII. a + b:c + d=a — b : c — d Hence, by Theorem III. a -\- b : a — b = c -\- d : c — d TUEOREM X. 211* Equimultiples of two quantities have tJw same ratio as tfie quantities themselves. For by Art. 83, ? = ^ •^ mo i. e. a : b = ma : mb Cor. It follows that either couplet of a proportion may be multiplied or divided by any quantity, aiid the result- ing quantities will be in proportion. And since by Theo- rem III. if a : b = ma : mb, a : ma == b : mb, or ma : a = mb : b, it follows that both consequents, or both ante- cedents, may be multiplied or divided by any quantity, and the resulting quantities will be in proportion. THEOREM XI 212, If four quantities are in proportion, like powers or like roots of these quantities will he in proportion. Let a : b = c : d a c b=d Hence, ^=^ i. e. t^ : l^ = c" : d^ Since n may be either integral or fractional, the theorem is proved. 212 ELEMENTARY ALGEBRA. THEOREM XII. 213. If any number of quantities are proportional, any antecedent is to its consequent as the sum of all the antece- dents is to the sum of all the consequents. Let a : b = c : d=ze :f Now ab=zab (1) and by Theorem I. ad=zbc (2) and also af^=:be (3) Adding(l), (2), (3), a(H-^ + /) = b {a -\- c -^ e) Hence, by Theorem II. a : b =z a-\- c -\-e •.b-\- d -\-f THEOREM XIII. 214. If there are two sets of quantities in proportion, their products, or quotients, term by term, wilt be in proportion. Let a : b = c : d and e:f=g :k By Theorem I. ad = bc (1) and eh=fy (2) Multiplying (1) by (2), adeh = b cfg ; (3) Dividing (1) by (2), ad be (4) From (3), by Theorem IL ae : bf=zcg : dh and from (4), a b c d PROBLEMS IN PROPORTION. 215. By means of the principles just demonstrated, a proportion may often be very much simplified before making the product of the means equal to the product of the extremes ; and a proportion which could not oth- erwise be reduced by the ordinary rules of Algebra may often be so simplified as to produce a simple equation. RATIO AND PROPORTION. 213 1. The cube of the smaller of two numbers multiplied by four times the greater is 96 ; and the sum of their cubes is to the difference of their cubes as 210 : 114. What are the numbers ? SOLUTION. Let X = the greater and y = the less. Then 4x/ = 96 (1) x'+y :ar» — / = 210 : 114 (2) From (2), by Theo. X^ Cor. ar' + y' : a:* — y* = 35 : 19 By Theorem IX. 2a:» : 2y» = 64 : 16 By Theorem X., Cor. a:« : / = 27 : 8 By Theorem XI. a: : y = 3 : 2 By Theorem I. 2x = 3y (3) From (1) and (3) we find x = 3 and y = 2. 2. The product of two numbers is 78 ; and the differ- ence of their cubes is to the cube of their difference as 283 : 49. What are the numbers ? SOLUTIOX I^t X = the greater and y = the less. Thenxy=78 (1) I* — 3/»: a:» — 3x«y + 3a:y» — y» = 283 : 49 (2) From (2), by division, Sx^y — 3 x^ : (x — y)' = 234 : 49 Dividing 1st couplet by a: — y, 3xy :(x — y)' = 234 : 49 Dividing antecedents by 3, xy : (x — y)' = 78 : 49 Substituting the value of zy, 78 : (x — y)* = 78 : 49 Dividing antecedents by 78, 1 : (x — y)' = 1 : 49 Extracting the square root, 1 : x — y = 1 : 7 Whence, x — y = 7 (3) From (1) and (3) we find x = 13 and y = 6. 3. The sum of the cubes of two numbers is to the cube of their sum as 13 : 25 ; and 4 is a mean proportional be- tween them. What are the numbers? 214 ELEiMENTARY ALGEBRA. 4. The difference of two numbers is 10 ; and their prod- uct is to the sum of their squares as 6 : 3*7. AVhat are the numbers ? SOLUTION. Let X = the greater and y = the less. Then a;— 2/= 10 (1) xy\ t? -\- f = ^ -. ^1 {^^ From (2), by Theorem X., Cor. 2xy\ o? -{- f = \1 -.^1 By Theorem IX. o? -\- Ixy -{- y^ \ x^ — 'Ixy -\- f = 4.^ -. lb By Theorem XL x ■\- y \ x — y = 7:5 By Theorem IX. ' 2a::2^/=12:2 By Theorem X., Cor. x\y =6:1 By Theorem I. a; = 6 y (3) From (1) and (3) we find a; = 12 and y = 2. 5. The product of two numbers is 136 ; and the dif- ference of their squares is to the square of their differ- ence as 25 : 9. What are the numbers ? Ans. 8 and IT. 6. As two boys were talking of their ages, they dis- covered that the product of the numbers representing their ages in years was 320, and the sum of the cubes of these same numbers was to the cube of their sum as T : 27. What was the age of each? Ans. Younger, 16; elder, 20 years. Y. As two companies of soldiers were returning from the war, it was found that the number in the first multi- plied by that in the second was 486, and the sum of the squares of their numbers was to the square of the sum as 13 : 25. How many soldiers were there in each company? Ans. In 1st. 27; in 2d, 18. 8. The difference of two numbers is to the less as 100 is to the greater ; and the same difference is to the greater as 4 is to the less. What are the numbers ? J^OTE. — Multiply the two proportions together. (Theorem XIII.) PROGRESSION. 215 SECTION XXII. PROGRESSION. 216. A Progression is a series in which the terms in- crease or decrease according to some fixed law. 217. The Terms of a series are the several quantities, whether simple or compound, that form the series. Tlie first and last terms are called the extremes, and the others the means. ARITHMETICAL PROGRESSION. 218. An Arithmetical Progression is a series in which each term, except the first, is derived from the preced- ing by the addition of a constant quantity called the com' mon difference. 219. When the common difference is positive, the series is called an ascending series, or an ascending progression ; when the common difference is negative, a descending se- ries. Thus, a, a -\- d, a -\- 2d, a -\- Zd, &c. is an ascending arithmetical series in which the common difference is d] and a, a — d, a — 2d, a — Zd, &c. is a descending arithmetical series in which the common difference is — d. 220. In Arithmetical Progression there are five elements, any three of which being given, the other two can be found : — 1. The first term. 2. The last terra. 216 ELEMENTARY ALGEBRA. 3. The commou difference. 4. The number of terms. 5. The sum of all the terms. 221. Twenty cases may arise in Arithmetical Progres- sion. In discussing this subject we shall let a = the first terra, I = the last term, d = the common difference,- n = the number of terms, S=. the sum of all the terms. CASE I. 222. The first term, common difference, and number of terms given, to find the last term. In this Case a, d, and n are given, and / is required. The suc- cessive terms of the series are a, a -\- d, a -\- 2d, a -{- Sd, o -j- 4 rf, &c. ; that is, the coefficient of d in each term is one less than the number of that term, counting from the left ; therefore the last or nth term in the series is a-\- (n—1) d •or Z = a -J- (n — 1) d in which the series is ascending or descending according as d is posi- tive or negative. Hence, RULE. To the first term add tJie product formed by multiplying the common difference by the number of terms less one. 1. Given « = 4, d =z 2, and w = 9, to find /. l=:a-\-(n — I) d = 4: + (9 — 1) 2 = 20, Ans. 2. Given a = t, d=3, and n = 19, to find /. Ans. /=61. PROGRESSION. 217 3. Given a = 29, rf = — 2, and n = 14, to find I Ans. / = 3. 4. Given a = 40, d=z 10, and n = 100, to find /. 5. Given a = 1, d=z ^, and n =. 17, to find /. 6. Given a = |^, rf = — ^^, and n = 13, to find /. 7. Given a = .01, d = ^ .001, and n = 10, to find /. CASE II. 223. The extremes and the number of terms given, to find the sum of the series. In this Case a, /, and n are given, and S is required. Now 5 = o + (o + d) + (a + 2d) + (a + 3d) + + Z or, inrerting the seriee, 5= i + ( i — d) + {l — 2d) -\- (l — Sd) + -fa Adding these together, 2 5= (a + + (a +0 + (o + + (a + + + (a + And since (a -j- /) is to be taken as many times as there are terms, hence 2 5 = n (a -f- or S = - (a -\- I). Hence, RULE. Find one half (he product of the sum of the extremes and the number of term^. Note. — If in place of the last term the common difference is given, the last term must first be found by the Rule in Case I. 1. Given a = 3, /= 141, and n = 26, to find S. ^= ^ (a + = IT (3 + 141) = 1872, Ans. 2. Given a = |, / = 25, and n = 63, to find S. Ans. ^=793f. 3. Given a = 4, are ascendine: series : -^64,-18,— 6,— 2, &c.) ^ 64, 32, 16, 8, &c.> , ,. ' ' ' ' V are descending series. — 8, — 16, — 32, — 64, &c. > ^ If the ratio is negative, the terms of the progression are alternately positive and negative. Thus, if the ratio is — 2 and the first term 3, the series will be 3, — 6, + 12, — 24, + 48, &c. ; but if the first term is — 3, — 3, +6, —12, +24, —48', &c. The positive terms of these two series constitute an as- cending progression whose ratio is the square of the given ratio ; and the negative terms a descending progression having the same ratio. 231* In Geometrical Progression there are five elements, any three of which being given, the other two can be found. These elements are the same as in Arithmetical Progres- sion, except that in place of the common difference we have the ratio. 10* o 226 ELEMENTARY ALGEBRA. 232t Twenty cases may arise in Geometrical Progres- sion. In discussing these cases we shall preserve the same notation as in Arithmetical Progression, except that instead of d = the common difference we shall use r = the ratio. CASE I. 233. The first term, ratio, and number of terms given, to find the last term. In this Case a, r, and n are given, and I required. The successive terms of the series are a, ar, ar^^ ar^, a?-*, &c. That is, each term is the product of the first term and that power of the ratio which is one less than the number of that term count- ing from the left; therefore the last or nth term in the series is or l=sar^-^. Hence, RULE. Multiply the first term by that power of the ratio whose index is one less than the number of terms. 1. Given a = Y, r = 3, and w = 5, to find /. /=ar"-^ = 1 X 3* =567, Ans. 2. Given a = 3, r = 2, and w = 9, to find I. Ans. /=768. 3. Given a = 64, r == J, and n = 10, to find /. Ans. I = ^. 4. Given a = — T, r = — 4, and w = 3, to find /. Ans. Z = — 112. 5. Given a = — ^, r = ^, and w = 5, to find /. Ans. 1==. — y¥s- 6. Given a = 5, r = — \, and n = 10, to find I. 7. Given a == — ^, r=^, and w = 8, to find /. 8. Given a = — 10, r = — 2, and n = 6, to find /. PROGRESSION. 227 CASE II. 234. The extremes, and the ratio given, to find the sum of the series. In this Case ct, /, and r are given, and S is required. Now S == a -{- ar -{- ar^ -\- at^ -{- -f- / (1) Multiplying (1) by r, rS=ar-\- aH -f ar* -f -|-^+^'* (2) Subtracting (1) from (2), rS — S= Ir — a Whence, v/^7 + y=r21 (3) 2x^2y — m (4) x- \- y = 3 (5) 2\/^=18 (6) xy = ^\ (1) Dividing (2) by (1), we obtain (3); adding (3) to (1), we ob- tain (4), which reduced gives (5) ; subtracting (3) from (1), we obtain (6), which reduced gives (7). Combining (5) and (7) as the sum and product are combined in Example 1, Art. 188, we obtain x = 27 and y = 3. Ans. 3, 9, 21. 3. Of four numbers in geometrical progression the dif- ference between the fourth and second is 60 ; and the sum of the extremes is to the sum of the means as 13 : 4. What are the numbers ? SOLUTION. Let X, xy, xy^, and xy^ represent the series. Then xf 64a: xy=:60 (1) x/ + x:xy' + xy=l3:i (2) if-!/ + l :.y=:13:4 (3) 4:f — iy + 4,=z 13y (i) 4a: = 60 (7) if—11y^ — i (5) x=l (8) y = 4 (6) PROGRESSION. 235 Dividing the first couplet of (2) by xy -\- x,yre obtain (3) ; from (3) we form (4), which reduced gives (G), or y = 4. Substituting this value of y in (1) and reducing, we obtain (8), or ar== 1. Ans. 1, 4, 16, 64. 4. Of four numbers in geometrical progression the sum of the first two is 10 and of the last two 160. What are the numbers ? Ans. 2, 8, 32, 128. 5. A man paid a debt of S310 at three payments. The several amounts paid formed a geometrical series, and the last payment exceeded the first by $240. Wliat were the several payments? Ans. $10, $50, $250. 6. In the series x, \/xi/, and y what is the ratio? A„s./f. 7. In the series -, x, y, and - what is the ratio? y' ' ^' X 8. There are four numbers in geometrical progression whose continued product is 64 ; and the sum of the series is to the sum of the means as 5 : 2. What are the num- bers ? Ans. 1, 2, 4, 8. 9. There are five numbers in geometrical progression ; the sum of the first four is 156, and the sum of the last four 780. What are the numbers ? 10. There are three numbers in geometrical progression whose sum is 126; and the sum of the extremes is to the mean as 17 : 4. What are the numbers? 11. The sum of the squares of three numbers in geo- metrical progression is 2275 ; and the sum of the ex- tremes is 35 more than the mean. What are the numbers ? 12. Of four numbers in geometrical progression the sum of the first and third is 52 ; and the difference of the means is to the difference of the extremes as 5 : 31. What are the numbers ? 236 ELEMENTARY ALGEBRA. SECTION XXIV. MISCELLANEOUS EXAMPLES. 1. From 6ac— 5ab-{-c^ take Sac— [Sab — (c — c^) + 1c}. Ans. Sac — 2ab + 2c^-^6c. 2. Reduce x^ y^ — ( — xi/'^-{-x^ ) xi/ — x"^ f — {3/^ — y {^y — ^^) } ) t^ ^^s simplest form. Ans. 2x^y^ -\- x^. 3. Reduce {a — h -\- cf — fa (c — a — h) -^ [b {a + h -f- c) — c (a — b — ^) I ) to its simplest form. Ans. 2(a'-\-b'-{-c^). 4. Reduce (x -\- a) a -{- 1/ — { & + ^) {'^ + ^) — y (x -\- a — 1) — {^ -\- y) {b — ct)} to its simplest form. Ans. a^ — b"^. 5. Reduce (a^ — b^)c — {a — b) [a {b^ c) — b {a — c)\ to its simplest form. Ans. 0. 6. Reduce {a -\- b) x — {b — c) c — ^{b — x)b—{b — c) {b -\- c)\ — ax to -its simplest form. Ans. 2b x — be. 1. Multiply a^ + 2aH — SaP by — {—SaH-{- a^ b^). 8. Multiply a"^ -|- 6 a^ _f- 9 by a^ — 6 a^ + 9. 9. Multiply a -\- b — c by n — b -\- c. 10. Divide 2% a- — 6 a^ — Ga^ — 4a^ — 96 a + 264 by 3a2 — 4a + 11. 11. Divide 1 — 18x2+810:^ by 1 -\-Qx + ^x\ 12 Divide ^ a^ -\- I —Aa^ — Q a hy 1 -|- 2 «2 — 3 a. 13. Divide 9 ^^ — 7 x^/ -\-2y^ by 3 x^ + 2 ^^^ — /. MISCELLANEOUS EXAMPLES. 287 14. Divide 23 a — 30 — T a» + 6 a< by 3 a — 2 a^ — 5. 15. Find the prime factors of a* — b*. 16. Find the prime factors of 4m'n'^ — 49 m^n^". 17. Find the prime factors of x^ — 2xi/'\-y^, 18. Find the prime factors of x^ — y". 19. Find the greatest common divisor of 6 a:* — lOx^y + ISy* and 4x« + 8 x'^y + 8 xy^ _|_ 4y8. Ans. x + y. 20. Find the greatest common divisor of 8 a 6* + 24 a 6* + 16 a 6 and 7 i« + 7 i« + 7 6* — 7 6^ Ans. ^ + b. 21. Find the greatest common divisor of 6 a:^ -|- ^ ^y — Bt/^ and 12x2 _^ 22 ary + 6y^ 22. Find the greatest common divisor of 4x -|- 4ar'' — 40 and 3a:r*y — 48 y. Ans. x — 2. 23. Reduce 7 — i,x / « . » ^ . mv to its lowest terms. (a — 6) (a* 4- 2 a 6 -f- ^) 24. Reduce , ,. to its lowest terms. a' — IT 25. Reduce ., . .^ ^ ~ \, ; — ir to its lowest terms. 26. Find the least common denominator and reduce z r— i ,~i « — -^ 15 to a single fraction. 1 — a 1 + a 1-f-o 1 — <^ n Ans. — 1 +a« 27. Find the least common denominator and reduce ■r-^ — 4 — r— i — i to a single fraction. 28. Find the least common denominator and reduce 4a»+3a6 _ 48a'6 ^ • 1 r ^• 4a«-3a6-^- 16a^-9a«y *^ ^ «^°^^^ ^'^^*^^'^- 29. Reduce to one fraction with the least possible de- . . a y — a*-|-a6 3ft — a , c nominator j- j—f U — . .bed cd ' bd 238 ELEMENTARY ALGEBRA. 30. Reduce to one fraction with the least possible de- a + & h 4- c , a -\- c nominator yr ^ r — 7 X7 h\ 1 (b — c){c-'a) {a — c){a — h) ' (6 — a)(c — 6) Ans. j-j r-7 r-7 77 = 0. (h — c) (c — a) (a — h) 4- 2 a: /2 — Zx (\Q — x)x\^ . , „ ^. 31. Find the least common denominator and reduce 3j-2a: Y . . Ans. z~. 2 -\- X 32. Reduce to one fraction with the least possible de- . ^ l+rc Ax \—x . 2x-\-Q7? nommator ^j^, - ^^^^ - j^^j^,- Ans. -^^--^, • 33. Reduce a — c ~ ^^ ^ to its simplest form. 34. Reduce ^~^ to its simplest form. 35. Reduce — —- \- x and — —, 2y each to a X — 2?/ ' X -\-2y ^ single fraction and find their product. Ans. , „ » 36. Subtract ~^— from 2^ — a; a;-h?/ 37. Subtract 3a:-[-^ from a; — &« 88. Multiply (-i^,y by ^-3! 39. Divide — — by f , , and multiply the result by a'. 40. Divide ^--^^--^, by -^— ^. 41. Divide T by (— ri; + z r)' m -f- n '' \a-\- a — 0/ iUSCELLANEOUS EXAMPLES. 239 Je^ rk- -J 4(a» — aft) , Sab 42. Divide -^, — f— ,;,- by -5 rj- b{a -\- by "^ a} — b* 43. Divide --^ i — by — ■ j — , and give a — X ' a 4- X '' a — x a -+- x ° 0*4- z" the answer in its lowest terms. Aus. —^ — • 2ax 44. Reduce r-r = 1 • What is the value of a a -{- b a — o X, if a = — 2 and 6 = 3? 3x — 1 45. Reduce x 1 + 46. 'Reduce(a-{-x){b — x) — a{b — c)-~^-^ = 0. What is the value of a:, if a = 2, b = — 3, and c = — 1 . 47. Reduce -^^^ = — ~ What is the value of b a ar, if a = 2, 6 = — 1, and c = 3 ? 48. Reduce a — J-±^ = 0. 1 — x ab 49. Find the value of x in the equation x = ^^r| ^^rj in its simplest form. a — b a-t-b 60. A man spends $2. He then borrows as much money as he has left, and again spends S2. Then borrowing again as much money as he has left, he again spends $2, and then has nothing left. How much money did he have at first? 61. If 5 is subtracted from a certain number, two thirds of the remainder will be 40. What is the number ? 62. Having a certain sum of money in my pocket, I lost c dollars, and then spent one ath part of what re- mained and had loft one />th part of what I had at first. What was the original sum ? What does the answer be- come if a = 3, 6 = 9, and c = 5 ? 240 ELEMENTARY ALGEBRA. 63. If I buy a certain number of pounds of beef at $0.25 a pound, I shall have $0.25 left; but if I buy the same number of pounds of lard at $0.15 a pound, I shall have $1.25 left. How much money have I? 54. Divide 84 into three parts so that one third of the first, one fourth of the second, and one fifth of the third shall be equal. 65. In a certain orchard 25 more than one fourth of the trees are apple trees, 2 less than one fifth are pear trees, and the rest, one sixth of the whole, are peach trees. How many trees are there in the orchard ? 56. A merchant spent each year for three years one third of the stock which he had at the beginning of the year ; during the first year he gained $ 600, the second $500, and the third $400. At the end of the three years he had but two thirds of his original stock. What was his original stock ? 57. From a cask of wine out of which a third part had leaked, 84 liters were drawn, and then the cask was half full. What is the capacity of the cask ? 58. A gentleman has two horses and a chaise. The chaise is worth a dollars more than the first horse and h dollars more than the second. Three fifths of the value of the first horse subtracted from the value of the chaise is the same as seven thirds of the value of the second horse subtracted from twice the value of the chaise. What is the value of the chaise and of each horse ? What are the answers if a = — 50 and J = 50 ? 59. A had twice as much money as B. A gained $30 and B lost $40. Then A gave B three tenths as much as B had left, and had left himself 20 per cent more than he had at first. How much did each have at first? MISCELLANEOUS EXAMPLES. 241 60. A number of men had done one third of a piece of work in 9 days, when 18 men were added and the work completed in 12 days. What was the original number of men? 61. A boatman can row down the middle of a river 14 miles in 2 hours and 20 minutes ; but though he keeps near the shore where the current is one half as swift as in the middle, it takes him 4 hours and 40 minutes to row back. What is the velocity of the water in the middle of the river ? Ans. 2 miles an hour. 62. A had three fifths as much money as B. A paid away $80 more than one third of his, and B $50 less than four ninths of his, when A had left one third as much as B. What sura had each at first? 63. A farmer hired a man and his son for 20 days, agreeing to pay the man $3.50 a day and the son $1.25 for every day the son worked ; but if the son was idle, the farmer was to receive $0.50 a day for the son's board. For the 20 days' labor the man received $67. How many days did the son work ? 64. I purchased a square piece of land and a lot of three- inch pickets to fence it. I found that if I placed the pick- ets 3 inches apart, I should have 50 pickets left ; but if I placed the pickets 2^ inches apart, I must purchase 60 more. How much land and how many pickets did I pur- chase ? Ans. 18906J square feet and 1150 pickets. 65. A criminal having escaped from prison travelled 10 hours before his escape was discovered. He was then pursued and gained upon 3 miles an hour. When his pursuers had been on the way 8 hours, they met an ex- pressman going at the same rate as themselves, who had met the criminal 2 hours and 24 minutes before. In what time from the commencement of the pursuit will the crimi- nal be overtaken ? Ans. 20 hours. 11 F 242 ELEMENTARY ALGEBRA. 66. In February, 1868, a man being asked the time, answered that the number of hours before the close of the month was exactly one sixth of 10 less than the number that had passed in the month. What was the exact time ? Ans. February 25th, 10 o'clock, p. m. 6*?. A and B owned adjoining lots of land whose areas were as 3 : 4. A sold to B 100 hectares of his, and after- ward purchased of B two fifths of B's entire lot ; and then the original ratio of their quantities of land had been re- versed. How much land did each own at first ? Ans. A, 300 ; B, 400 hectares. 68. A laborer Was hired for TO days ; for each day he wrought he was to receive $2.25, and for each day he was idle he was to forfeit $0.75. At the end of the time he received $118.50. How many days did he work? 69. A sum of money was divided equally among a num- ber of persons by giving to the first $100 and one sixth of the remainder, then to the second $200 and one sixth of the remainder, then to the third $300 and one sixth of the remainder ; and so on. What was the sum divided and wjiat the number of persons ? 70. A besieged garrison had a quantity of bread which would last 9 days if each man received two hectograms a day. At the end of the first day 800 men were lost in a sally, and it was found that each man could receive 2f hectograms a day for the remainder of the time. What was the original number of men ? 71. Find a fraction such that if 1 is added to the de- nominator its value will be ^ ; but if the denominator is divided by 3 and the numerator diminished by 3, its value will be f . 72. If 7 years are added to A's age, he will be twice as old as B ; but if 9 years are subtracted from B's age, he will be one third as old as A. What is the age of each ? MISCELLANEOUS EXAMPLES. 213 Y3. A, B, and C compare their fortunes. A says to B, " Give me S700 of your money, and I shall have twice as much as you retain." B says to C, "Give me $1400, and I shall have three times as much as you retain." C says to A, "Give me S420, and I shall have five times as much as you retain." How much has each? 74. An artillery regiment had 39 soldiers to every 5 guns, and 4 over, and the whole number of soldiers and oflScers was six times the number of guns and officers. But after a battle in which the disabled were one half of those left fit for duty, there lacked 4 of being 22 men to every 4 guns. How many guns, how many officers, and how many soldiers were there ? Ans. 120 guns, 44 officers, 940 soldiers. 75. A car containing 5 more cows than oxen was started from Springfield to Boston. The freight for 4 oxen was $2 more than the freight for 5 cows, and the freight for the whole would have amounted to $30 ; but at the end of half the journey 2 more oxen and 3 more cows were taken into the car, in consequence of which the freight of the whole was increased in the proportion of 6 to 6. What was the original number of cows and oxen, and what was the freight for each ? . j 9 cows and 4 oxen. I Freight for a cow, $2 ; for an ox, $3. 76. Two sums of money amounting together to $1600 were put at interest, the less sum at 2 per cent more than the other. If the interest of the greater sum had been in- creased 1 per cent, and the less diminished 1 per cent, the interest of the whole would have been increased one fif- teenth ; but if the interest of the greater had been increased 1 per cent while the interest of the other remained the same, the interest of the whole would have been increased one tenth. What were the sums, and the rates of interest? Ans. $1200 at 7 per cent ; $400 at 9 per cent. 244 ELEMENTARY ALGEBRA. TT. A and B can perform a piece of work together in lYf days. They work together 10 days, and. then B fin- ishes the work alone in 16| days. How long would it take each to do the work ? T8. The Emancipation Proclamation of President Lincoln was promulgated on the 1st day of January in a year rep- resented by a number that has the following properties: the second (hundred's) figure is equal to the sum of the third and fourth minus the first ; or to twice the sum of the jfirst and fourth ; the third is a third part of tlie sum of the four; and if 1818 is added to the number, the order of the figures will be inverted. What was the year ? 79. A and B can do a piece of work in a days ; A and C in b days ; B and C in c days. In how many days can each do it ? 80. A can do a piece of work in a days, B in ^ days, and C in c days. In how many days can A and B together do it? B and C together? A and C together? All three together ? 81. A market-man bought some eggs for $0.28 a dozen, and sold some of them at 3 for 8 cents and some at 5 for 12 cents, receiving for the whole $6.24, and clearing $0,64. How many did he sell at each rate ? 82. One cask contains 56 liters of wine and 40 of water, and another 96 of wine and 16 of water. How many liters taken from each cask will make a mixture containing 52 liters of wine and 24 of watei- ? 83. A and B are travelling on roads which cross each other. When B is at the point of crossing, A has 120 me- ter's to go before he arrives at this point, and in 4 minutes they are equally distant from this point ; and in 32 minutes more they are again equally distant from it. What is the rate of each? Ans. A's, 100; B's, 80 meters a minute. MISCELLANEOUS EXAJIPLES. 245 84. Multiply tT by a:". 85. Multiply x' by x"'. 86. Divide y^* by y-^ 87. IMvide a""* by 0*+*. 88. Transfer the denominator of — ^5 to the numerator. 89. Free -zr-f-^ ^^^^^ negative exponents. 90. Expand (— 2a«)*. 91. Expand (a^i)"*. 92. Expand (— 3a:-V)*- 93. Expand (x'" X a:")'. 94. Expand {x — >v/y)'. 96. Expand (a» — 2 6)». 96. Expand {1x—ff. 97. Expand (2x — 3)*. 98. Expand (3 a — 2 6)". 99. Find five terms of (x — y)*^. 100. Expand (2 — x — y)«. 101. Expand (3 ■— a — 6 + cf. 102. Find ^C^"^. 103. Find y/^;;r^- 104. Find V— 16a:«. 105. Find the square root of | — (2a + 2x — 4j i/^ + a«__4a + 2ax + 4 — 4a: + a:«. 106. Reduce and Vif. 109. From a^IM take a^JO. 110. Multiply i^f by i>?^J. 111. Divide —Sa/TO by V'5. 112. Divide a^'6 by / ■ » = ^^ — 130. 131. 132. Given 1 . l'^*, ^ !^ 1 . to find z and y. 133. Given \'\'^^~\l},to&ndx and y. 134. Given } IT^y "^^^V, to find x and y. C 5xy=T5) 248 ELEMENTARY ALGEBRA. 135. Given i ^ "^^ 7 ^! "^ !!H > to S^d x and y. l2x^i/ — 2xf= 140) ^ 136. Given j^x^y - 2:.^^ = 875 ) ^^ ^^^ ^ ^^^ 13Y. Given j^''^' "" ^^^, == ^H , to find x and y. ( a;2 4- y2 __ 25 i ^ 138. Given j ^ ^' 7 f ^^ = ^^^ I , to find a: and y. (3/_j-5a;2/z= 132) ^ 139. Given 1^' + 2^' + ^ + ^ == ^^l , to find a; andy. C X -\- 1/ — xy= 0) Ans. 1^=2, orH-3±V2T). (y = 2, or i(— 3TV21). 140. Given 1^^ + ^ = "^U , to find x and y. (a:* + 3^* = 1921) ^ 141. A drover sold a number of sheep that cost him $297 for $1 each, gaining $3 more than 36 sheep cost him. How many sheep did he sell ? 142. A merchant sold a piece of cloth for $75, gaining as much per cent as the piece cost him. What did it cost him? 143. A drover bought 12 oxen and 20 cows for $920, buying one ox more for $160 than cows for $6Q. What did he pay a head for each? 144. A started from C towards D and travelled 4 miles an hour. After A had been on the road 6^ hours, B started from D towards C, and travelled every hour one fourteenth of the whole distance, and after he had been on the road as many hours as he travelled miles an hour, he met A. What was the distance from C to D ? MISCELLANEOUS EXAMPLES. 249 145. A person bought a number of horses for S1404. If there had been 3 less, each would have cost him $39 more. What was the number of horses and the cost of each ? 146. Find a number of four figures which increase from left to right by a common difference 2, while the product of these figures is 384. Ans. 2468. 147. A rectangular garden 24 rods in length and 16 in breadth is surrounded by a walk of uniform breadth which contains 3996 square feet. What is the breadth of the walk? Ans. 3 feet. 148. A square field containing 144 ares has just within its borders a ditch of uniform breadth running entirely round the field and covering 381.44 centares of the area. What is the breadth of the ditch ? Ans. 0.8 meter. 149. A and B hired a pasture into which A put 5 horses, and B as many as cost him $5.50 a week. If B had put in 4 more horses, he ought to have paid $6 a week. What was the price of the pasture a week? Ans. $8. 150. A father dying left $3294 to be divided equally among his children. Had there been 3 children less, each would have received $ 183 more. How many children were there ? 151. A merchant bought a quantity of tea for $66. If he had invested the same sum in coffee at a price $0.77 less a pound, he would have received 140 pounds more. How many pounds of tea did he buy ? 152. Find two quantities such that their sum, product, and the sum of thoir squares shall be equal to one an- other. Ans. i (3 ± \/'^^^) and i {S T \/^^^). 163. Find two numbers such that their product shall be 6, and the sum of their squares 13. U* 250 ELEMENTARY ALGEBRA. 154. A and B talking of their ages find that the square of A's age plus twice the product of the ages of both is 3864 ; and four times this product, minus the square of B's age, is 3575. What is the age of each? Ans. A's, 42 ; B's, 25. 155. Find two numbers such that five times the square of the less minus the square of the greater shall be 20 ; and five times their product minus twice the square of the greater shall be 25. . 156. A and B purchased a wood-lot containing 600 acres, each agreeing to pay $11500. Before paying for the lot, A offered to pay $20 an acre more than B, if B would consent to a division and give A his choice of situ- ation. How many acres should each receive, and at what price an acre ? Ans. A, 250 acres at $ TO an acre ; B, 350 at $50. 157. A merchant bought two pieces of cloth for $175. For the first piece he paid as many dollars a yard as there were yards in both pieces ; for the second, as many dol- lars a yard as there were yards in the first more than in the second ; and the first piece cost six times as much as the second. What was the number of yards in each piece? ^ Ans. In 1st, 10 yards ; in 2d, 5. 158. Two sums of money amounting to $14300 were lent at such a rate of interest that the income from each was the same. But if the first part had been at the same rate as the second, the income from it would have been $532.90 ; and if the second part had been at the same rate as the first, the income from it would have been $490. What was the rate of interest of each? Ans. First, 7 per cent ; second, 7fV P^r cent. 159. Divide 29 into two such parts that their product will be to the sum of their squares as 198 : 445. MISCELLAKEOUS EXAMPLES. 251 160. What is the length and breadth of a rectangular field whose perimeter is 10 rods greater than a square field whose side is 50 rods, while its area is 250 square rods less than the area of the square field ? Ans. Length, To rods ; breadth, 30. 161. A rectangular piece of land was sold for $5 for every rod in its perimeter. If the same area had been in the form of a square, and sold 'in the same way, it would have brought S90 less; and a square field of the same perimeter would have contained 272^ square rods more. What were the length and breadth of the field ? Ans. Length, 49 ; breadth, 16 rods. 162. A starts from Springfield to Boston at the same time that B starts from Boston to Springfield. When they met, A had travelled 30 miles more than B, having gone as far in IJ days as B had during the whole time; and at the same rate as before B would reach Springfield in 6f days. How far fi-om Boston did they meet? Ans. 42 miles. 163. The product of two numbers is 90 ; and the dif- ference of their cubes is to the cube of their difference as 13 : 3. What are the numbers? 164. A and B start together from the same place and travel in the same direction. A travels the first day 25 kilometers, the second 22, and so on, travelling each day 3 kilometers less than on the preceding day, while B travels 14^ kilometers each day. In what time will the two be together again ? Ans. 8 days. 165. A starts from a certain point and travels 5 miles the first day, 7 the second, and so on, travelling each day 2 miles more than on the preceding day. B starts from the same point 3 days later and follows A at the rate of 20 miles a day. If they keep on in the same line, when will they be together ? Ans. 8 or t days after B Btarts. 252 ELEMENTARY ALGEBRA. 166. A gentleman offered his daughter on the day of her marriage $1000; or $1 on that day, $2 on the next, $3 on the next, and so on, for 60 days. The lady chose the first offer. How much did she gain, or lose, by her choice ? 167. The arithmetical mean of two numbers exceeds the geometrical mean by 2 ; and their product divided by their sum is 3^. What are the numbers ? 168. A father divided $130 among his four children in arithmetical progression. If he had given the eldest $25 more and the youngest but one $5 less, their shares would have been in geometrical progression. What was the share of each ? 169. The sum of the squares plus the product of two numbers is 133 ; and twice the arithmetical mean plus the geometrical mean is 19. What are the numbers ? 110. The sum of three numbers in geometrical progres- sion is in ; and the difference of the second and third minus the difference of the first and second is 36. What are the numbers ? 171. There are four numbers in geometrical progression, and the sum of the second and fourth is 60 ; and the sura of the extremes is to the sum of the means as 7 : 3. What are the numbers? Cambridge : Electrotyped and Printed by Welch, Bigelow, & Ca d 4 THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL PINE OP 25 CENTS W,LL INCREASE TO SO CENTS ON th/fo^r™ oCeROUE. ™ *'°° °'' ™^ SEVENT^^":™ ■ __sfvt4;i~ "" FEB 97 1941 __i^!C^L^i|3_ .r 1\ I^j^vig^^*^ ^^Rl^':--- ■■'^i^'l^^^ --ilV^ ^^^R-.^^ "^V; ,-^^ •'r>" / B^SiK^fi?#vy^ 1 ^^^W'"''f' c^:rrr""^^ ^^B '^... ''..\-0'^''^^^A^ Biwiiil siftitefl ■}-:.':'■■': ■J:t'4^.^-'-i'^ ^^■■^■sf':";;::::.. LD 21-100m-8,'34 m»wy'^^mm m^^j H''-:^;^''. i- ', ^^v^ ■^^^•■. ^^i'.'-.Z -i'.^''--- '•! S^:5«f'-t 7^^.r ;i^^r': •? . ^^t^ ^''IH r,' 91i39v' 5^ B^ THE UNIVERSITY OF CALIFORNIA UBRARY