1 1 \ ■) ; /■ > ■') THEIUmiLSElSOFSTilllOllinOL-B COMPRISES STANDARD WORKS In every department of instruction and of every grade. The teacher in want of a book fo ticular purpose or class, will always find the best of its kind in our catalogue. No other S( claims to be as complete as this. None is so extensive or so judiciously selected. Amonj volumes a high standard of merit is maintained, as it is our aim never to permit our impr poor or 1 country— frecdori almost ur Parke Nat the n volur series Davie Ac— callec the pi Same —For intere them XlfIont« —The a nun SteMc Che they Clark' iiovel Worir Ac- new Searii ando for B ARE J* Hergy Senger^ £ I Englis Comf finers exd's Sistor Engh Pen at — Smi Draw Jfaturt Chen: Chah Impor TON'S Ethic Tfie T^' ■'hesi A DESCRIPTIVE CATALOGUE of all these and many more may be obtained by c stamp to the Publishers, A- S. BAHNBS & COHPAHY, ' National Educational Publishers, 111 & 113 WILLIAM STREET. NEW YO jr ttuu 1 iticiice — noLBKOOK's wormai fliemoas — piOKTHJUNus xeauuei o ^boi»iant THE WORMAN SE RIES IN MODE RN LAN&UAGE. lOMPLETE Course in German By JAMES H. WORMAN, A.M. .MKJSrrARY GtERMLAl^ST G-R-A^INlMAIi, COMPLETK GERMAlSr G-KAMMAIt, COLLiEOIAXltl G-ERMIAT^ llJhZATi'E^Tl, KLEMEISTTARY GJ-EFUVtiV^ READErj, IIVIAIS^ COPV-UOOKS, GJ-ERnVlAlSr ECHO. TORY OF* G-ERMLA.N" IL.ITEP^A.TTJRE, OERlVIAlSr JLN-D ENGS^LISH LEXICON-. n i: OERSfAy GBAMMAJIS of Worman are widely preferred on ac- ■ their clear, explicit method (on tlio convei-uation plan), introclucing a eyptem ./y and comparison with the learuei-s' own language and oUiers commonly . of understanding the spoken language, and of correct pronun- 1 great tsuccess. .. » .,,..-.-......, .una of nouns and of irregular verbs are of great value to the The use of heavy type to indicate etymological changes, is new. The Vocabu- tfnonyiiiicai—VLlso a new feature. »V> /.' >f I .V'S GEItMAN TtFJADJEn contains progrespive Pelections vi ," : jf of the very bent German author;*, including thn-e couipli-te i>lay8, r : . iuiuhased in eeparate form for advanced ttudents who have com- ■ rler. 1'. 1 eminent authors. Notes after the text. References to all Gler- i: arnon use, and SLC ad^-iiyite Vocabulary; also. Exercises for . iutu thu Uciinan. MOWirfV'.v GETtHTAy^ ECTTO (Tk^tfi^hfl* Erhfi) Is entirely a new ' " ••■ " without translation, ;v. !i' language. By this -[)raK> II. j-<., PB0rBB6UK or BianiCS SfATBEMATine, COLUMBIA COLLRGK. A. S. BARNES & C M T A X Y, NEW YORK AND CHICAaO. 1875. IN MEMORrAM h/AI^^ DAVIES' MATHEMATICS. THE WEST POINT COURSE, And Only Thorough and Complete Mathematical Series. iisr Tia:R,EE f.a.i?,ts- . ; . I.; a-OMMOi^r school oouese. I>avies' Primary iArcthmetic- — The fundamental principles displayed in - , Obj«>c-t Lessone^ , IC^vi^'^ inte.U^ctTial' Arithmetic— Referring all operations to the unit 1 as the only'tangible liasls for logical development. Savies' Slements of Written Arithmetic— A practical introduction to the whole subject. Theory subordinated to Practice. Davies^ Practical Arithmetic**— The most successful combination of Theory and Practice, clear, exact, brief, and comprehensive. II. ACADEMIC COUESE. Davies' University Arithmetic*— Treating the subject exhaustively as a science^ in a logical series of connected propositions. Davies' Elementary Alg-ebra-*— A connecting link, conducting the pupil easUy from arithmetical processes to abstract analysis. Da vies' University Algebra ■*— For institutions desiring a more complete but not the fullest course in pure Algebra. Davies' Practical lYIathematics-— The science practically applied to the useful arts, as Drawing, Architecture, Surveying, Mechanics, etc. Davies' Xilementary CS-eometry .— The important principles in simple form, but with all the exactness of vigorous reasoning. Savies' Elements of Surveying.- Re-written in 1870. The simplest and most practical presentation for youths of 12 to 16. III. COLLEGIATE OOUESE. Davies' Bourdon's Algebra- *— Embracing Sturm's Theorem, and a most exhaustive and scholarly course. Davies' University Algebra-*— A shorter course than Bourdon, for Institu- tions have less time to give the subject. Davies' Legendre's Geometry-— Acknowledged tJie only satisfactory treatisa of its grade. 300,000 copies have been sold. Davies' Analytical Geometry and Calculus-— The shorter treatises, combined in one volume, are more available for American courses of study. Davies' Analytical Geometry- 1 The original compendiums, for those de- Davies' DifT- & Int- Calculus- ' siring to give full time to each branch. Davies' Descriptive Geometry-— With application to Spherical Trigonome- try, Spherical Projections, and Warped Surfaces. Davies' Shsdes, Shadows, and Perspective-— A succinct expcsition of the mathematical principles involved. Davies' Science of IKlathematics-- For teachers, embracing I. Gbammar of Akithtietic, ni. Logic and Utility of Mathematics, II. OuTxiNES OF Mathematics, IV. Mathematicax Dictionakt. * Keys may be obtained from the Publishers by Teachers only. Entered, according to Act of Congress, in the year 1859, ^y CHARLES DAVIES, In the Qerk's Office of the District Court of the United States for the Southern District of New York, N. E. A. PREFACE. Algebra naturally follows Arithmetic in a course of eoieii- tific studies. The language of figures, and the elementary combinations of numbers, are acquired at an early age. When the pupil passes to a new system, conducted by letters and signs, the change seems abrupt; and he often experiences much difficulty before perceiving that Algebra is but Arithmetic written in a different language. It is the design of this work to supply a connecting link between Arithmetic and Algebra ; to indicate the unity of the methods, and to conduct the pupil from the arithmetical processes to the more abstract methods of analysis, by easy and simple gradations. The work is also introductory to the University Algebra, and to the Algebra of M. Bourdon, which is justly considered, both in this country and in Europe, as the best text-book on the subject, which has yet appeared. In the Introduction, or Mental Exercises, the language of figures and letters are both employed. Each Lesson is 80 arranged as to introduce a single princi]>le, not known iii 92G515 IV PREFACE. before, and the whole is so combined as to prepare the pupil, by a thorough system of mental training, for those processes of reasoning which are peculiar to the algebraic analysis. It is about twenty years since the first publication of the Elementary Algebra. Within that time, great changes have taken place in the schools of the country. Tlie sys- tems of mathematical instruction have been improved, now methods have been developed, and these require correspond- ing modifications in the text-books. Those modifications have now been made, and this work will be permanent in its present form. Many changes have been made in the present edition, at the suggestion of teachers who have used the work, and favored me with their opinions, both of its defects and merits. I take this opportunity of thanking them for the valuable aid they have rendered me. The criticisms of those engaged in the daily business of teaching are invalu- able to an author; and I shall feel myself under special obligation to all who will be at the trouble to communicate to me, at any time, such changes, either in methods or lan- guage, as their experience may point out. It is only through the cordial co-operation of teachers and authors — by joint labors and mutual efforts — that the text-books of the country can be brought to any reasonable degree of perfection A Key to this volume has been prepared for the use of Teachers only. CONTENTS. CHAPTER I. DEFnnnoKS Aim kxtlanatoby biohs. PAOB.-i Algebra — Defin-tions — Explanation of the Signs 3S-41 Examples in writing Algebraic expressions 41 Interpretation of Algebraic language 42 CHAPTER II. FUNDAMENTAL OPERATIONS. Addition — Rule — Examples 43-60 Subtraction — Rule — Examples — Remarks 50-66 Multiplication — Monomials — Polynomials 66-6S Division — Monomials 63-68 Signification of the symbol a* 68-70 Division of Polynomials — Examples 71-76 CHAPTER III. USEFUL FORMULAS. FAOTOBINO, ETO. Formulas (1), (2), (3), (4), (5), and (6) 76-79 Factoring 79-81 Greatest Common Divisor 81-84 lioast Common Multiple . .^ B4-87 CHAPTER IV FRAOnONS. Transformation of Fractions , , , 89 To Reduce an Entire Quantity to a Fractional Form 90 V VI CONTENTS. To Reduce a Fraction to its Lowest Terms 90 To Reduce a Fraction to a Mixed Quantity ■. 92 To Reduce a Mixed Quantity to a Fraction 93 To Reduce Fractions to a Common Denominator » 94 Addition of Fractions 96 -9c Subtraction of Fractions 98- ?9 Multiplication of Fractions , 99- 10^ Division of Fractions 102- 1 05 CHAPTER y. EQUATIONS OF THE FIRST DEGREK. Definition of an Equation — Different Kinds 106-106 Transformation of Equations — First and Second 106-110 Solution of Equations — Rule 110-114 Problems involving Equations of the First Degree 115-130 Equations involving Two Unknown Quantities 130-131 Elimination — By Addition — By Subtraction — By Comparison. . 131-143 Problems involving Two Unknown Quantities , 143-148 Equations involring Three or more Unknown Quantities 148-159 CHAPTER VI. FOBMATION OF P0WEK8. Definition of Powers 160-161 Powers of Monomials 161-163 Powers of Fractions 163-165 Powers of Biuomials 165-167 Of the Terms— Exponents— Coefficients 167-170 Binomial Formula — Examples 170-172 CHAPTER VII. SQUARE BOOT. RADICALS OF THE SECOND DEGREE. Definition— Perfect Squares— Rule— Examples 173-179 Square Root of Fractious 179-181 Square Root of Monomials 181-183 Imperfect Squares, or Radicals 1 83-1 87 Addition of Radicals 187-189 Subtraction of Radicals 189-190 () N T K N T S . Vii rAGES. Multiplication of Radicals 190-191 Division of Kadicals 191-192 Square Root of Polynomials 193-197 CHAPTER VIII. ZQUATIONS or TUB SECOND DEORXX. Eiiuations of the Second Degree — Definition — Form 198-200 Incomplete Equations 200-209 Complete Equations— Rule 209-211 Four Forms 211-227 Four Properties 227-229 Formation of Equations of the Second Degree 229-231 Numerical Values of the Roots 281-236 Problems 236-240 Equations involving more than One Unknown Quantity 241-250 Problems 260-264 CHAPTER IX. ARrrnxETicAL and oeometrioal rRopoimoN. Ways in which Two Quantities may be Compared 256 Arithmetical Proportion and Progression 266-267 Last Term 267-260 Sum of the Extremes— Sum of Series 260-262 The Five Numbers — To find any number of Means 262-265 Geometrical Proportion 267 Various Kinds of Proportion 268-278 Geometrical Progression 278-280 Last Term- Sum of Scries 280-286 Progression having an Infinity Number of Terms 286-288 Tho live Numbers— To find One Mean 288-289 CHAPTER X. LooABrraus. Tlieory of Logarithms 290-295 SUGGESTIONS TO TEACHERS. 1. The Introduction is designed as a mental exercise. If thoroughly taught, it will train and prepare the mind of the pupil for those higher processes of reasoning, which it is the peculiar province of the algebraic analysis to develop. 2. The statement of each question should be made, and every step in the solution gone through with, without the aid of a slate or black-board ; though perhaps, in the begin- ning, some aid may be necessary to those unaccustomed to such exercises. 3. Great care must be taken to have every principle on which the statement depends, carefully analyzed ; and equal care is necessary to have every step in the solution distinctly explained. 4. The reasoning process is the logical connection of dis- tinct apprehensions, and the deduction of the consequences which follow from such a connection. Hence, the basis of all reasoning must lie in distinct elementary ideas. 5. Therefore, to teach one thing at a time — to teach that thing well — to explain its connections vnth other thmgs and the consequences which follow from such connections, would seem to embrace the whole art of instruction. viii ELEMENTAaY ALGEBRA, INTKODUCIION. MENTAL EXERCISES. LESSON L 1. John and Charles have the same number of apples; both together have twelve : how many has each ? Analysis. — Let x denote the number which John has; then, since they have an equal number, x will also denote the number which Charles has, and t^vice x, or 2ic, will denote the number which both have, which is 12. If twice x is equal to 12, x will be equal to 12 divided by 2, TE. — Wiien a- 18 written by itself, it is read one a5j ^anfl i^;tiie.sf?-iiic as, la* ; " * ' X or Ice, means once cc, or one a^ 2a;, " twice a;, or two a;, 3ic, " three times a*, or three a;, 4a;, " four times a;, or four x, &c., &c., &c. 2. What is a; + a; equal to ? " 3. What is a; + 2a; . equal to ? 4. What is a; + 2a; + x equal to ? 5. What is a; -f 5a; 4- x equal to ? 6. What is a; + 2a: + 3a; equal to ? 7. James and John together have twenty-four peaclies, and one has as many as the other ; how many has each ? Analysis. — Let x denote the number which James has ; then, since they have an equal number, x will also denote the number which John has, and twice x will denote the number which both have, which is 24. If twice x is equal to 24, X will be equal to 24 divided by 2, which is 12 ; therefore, each has 12 peaches. WRITTEN. Let X denote the number of peaches which James has ; then, 24 a; + a;, =r 2a; = 24; hence, a; — — =12. VERIFICATION. . A l^erification is the operation of proving that the num. ber found mil satisfy the conditions of the question. Thus, James' appiCS. John's apples. 12 4-12 = 24. Note. — Let the following questions be analyzed^ written^ and verified^ in exactly the sa^ne manner as the above. MENTAL EXERCISES. . 11 8. William and John together have 36 pears, and one has as many as the other : how many has each ? 9. What number added to itself will make 20? 10. James and John are of the same age, and the sum of their ages is 32 : what is the age of each? 11. Lucy and Ann are twins, and the sum of their ages Is 10: M'hat is the age of each? 12. What number is that Avhich added to itself will make 30? 13. What number is that which added to itself will make 50? 14. Each of two boys received an equal sum of money at Chiistmas, and together they received 60 cents: how much had each ? 15. What number added to itself -vvdll make 100? 16. John has as many pears as William; together they have 72 : how many has each ? 1 7. What number added to itself wHl give a sum equal to 46 ? 1 8. Lucy and Ann have each a rose bush with the same number of buds on each ; the buds on both number 46 : how many on each? LESSON n. 1. John and Charles together have 12 apples, and Charles has twipe as many as John : how many has each ? Analysis. — Let x denote the number of apples which John has ; then, since Charles has twice as many, 2x wijl denote his share, and a; 4- 2aj, or 3sc, will denote the number which they both have, which is 12.^ If 3a! is equal lo 12, X will be equal to 12 divided by 3, which is 4; therefore, John has 4 apples, and Charles, having tv\nce as many, has 8. 12 ^ »N'I RODUCTION. writt:^t. TiCt X denote the number of apples John has , then, 2x will denote the number of apples Charles has; and X -t 2x = 3x =z 12, the number both have; then, 12 X = -~ = 4, the number John has ; and, 2a; = 2 X 4 = 8, the number Charles has. , VERIFICATION. 4 + 8 = 12, the number both have. 2. William and John together have 48 quills, and, Williara has twice as many as John : how many has each ? 3. What number is that which added to twice itselfj will give a number equal to 60 ? 4. Charles' marbles added to John's make 3 times as many as Charles has; together they have 51 : how many has each ? AxALTSis. — Since Charles' marbles added to John's make three times as many as Charles has, Charles must have one third, and John two thirds of the whole. Let X denote the number which Charles has ; then 2x will denote the number which John has, and x -\- 2cc, or 3x, will denote what they both have, which is 51. Then, if 3x is equal to 51, x will be equal to 51 divided by 3, which is 17. Therefore, Charles has 17 marbles, and John, having twice as many, has 34. WKITTEN. Let X denote the number of Charles' marbles; then, 2x will denote the number of John's marbles; and ' Sx = 51, the number of both; then, 51 a; — — = 17, Charles' marbles; and 17 X 2 = 34, Jolm's marbles. MENTAL BZESCI8E6. 13 5. What number added to twice itself will make 15 ? 6. AVhat number added to t\\nce itself will make 57 ? 7. What number added to twice itself will make 39? 8. What number added to twice itseff will give 90 ? 9. John walks a certain distance on Tuesday, twice 20* for on Wednesday, and in the two days he walks 27 miles: how far did he walk each day ? 10. Jane's bush has twice as many roses as Nancy's: and on both bushes there are 36 : how many on each ? 11. Samuel and James bought a ball for 48 cents ; Samuel paid twice as much as James : what did each pay ? 12. Divide 48 into two such parts that one shall be double the other. 13. Divide 60 into two such parts that one shall be double the other, 14. The sum of three equal numbers is 12 : what are the numbers ? Analysis. — Let x denote one of the numbers; then, since the numbers are equal, x \snll also denote each of the others, and x plus x plus x, or 3a! will denote their sum, which is 12. Then, if 3aj is equal to 12, x will be equal to 12 divided by 3, which is 4 : therefore, the numbers are 4, 4, and 4. WRriTEN. Let x denote one of the equal numbers ; then, x + X -h X = Sx = 12; and x = - = 4. VERIFICATION. 4 -f 4 + 4 = 12. 15. Tlie sum of three equal numbers is 24 : what are the numbers? 16. The sum of three equal numbers is 36 : what are the numbers? 14 I K T K O U U C T I O K . 17. The sum of three equal numbers is 54 : what are the numbers ? LESSON III. 1. What number is that which added to three times itself will make 48 r Analysis. — Let x denote the number; then, 3a! will denote three times the number, and x plus 3£c, or A^x, will denote the sum, which is 48. If Ax is equal to 48, X will be equal to 48 divided by 4, which is 12; there- fore, 12 is the required number. WEITTEN. Let X denote the number; then, 'dx — tln-ee times the number; and ic + 3a; = 4a; := 48, the sum : then, a; 1= — = 12, the required number. VEllIFICATION. 12 + 3 X 12 r= 12 + 38 =: 48. Note. — All similar questions are solved by the same form of analysis. 2. What number added to 4 times itself will give 40 ? ^ 3. What number added to 5 times itself will give 42 ? *! 4. What number added to 6 times itself will give 63? ^ 6. What number added to 7 times itself will give 88 ? 6. What number added to 8 times itself will give 81 ? 7. What number added to 9 times itself will give 100? 8. James and John together have 24 quills, and John has three times as many as James: how many has each? 9. William and Charles have 64 marbles, and Charles has 7 times as many as William : how many has each ? MENTAL KXERCI6E6. 15 10. James and John travel 96 miles, and James travels 1 1 times as far as Jolin : how far does each travel ? 11. The sum of the ages of a father and son is 84 years; and the father is 3 times as old as» the son : what is the aire of each? 12. There are two numbers of which the greater is 1 tmies the less, and their sum is 72 : what are the nmnbers? 13. The sum of four equal numbers is 64: what are the numbere ? 14. The sum of six equal numbers is 54; what are the numbers ? 15. James has 24 marbles ; he loses a certain number, and then gives away 7 times as many as he loses which takes all he has : how many did he give away ? Verify. 16. William has 36 cents, and divides them between his two brothers, James and Charles, giving one, eight times as many as the other : how many does he give to each ? 17. What is the sum of x and 3a;? Of a; and 7aj? Of X and 5a;? Of x and 12a;? LESSON rv. 1. If 1 apple costs lucent, what will a number of apples denoted by x cost:? Analysis. — Since one apple costs 1 cent, and since x denotes a?iy number of apples, the cost of x apples will be as many cents as there are apples : that is, x cents. 2. If 1 apple costs 2 cents, what will x apples cost? Analysis. — Since one apple costs 2 cents, and smce a denotes the number of a})ples, the cost will be twice as many cents as there are apples : that is 2a; cents. 3. If 1 apple costs 3 cents, what H-ill x apples cost ? 4. If 1 lemon costs 4 cents, wliat will x lemons cost? 16 INTRODUCTION. 5. K 1 orange costs 6 cents, what will x oranges cost ? 6. Charles bought a certain number of lemons at 2 cents apiece, and as many .oranges at 3 cents apiece, and paid in all 20 cents : how many did he buy of each ? Analysis. — ^Let x denote the number of lemons ; then, since he bought as many oranges as lemons, :t will also denote the number of oranges. Since the lemons were 2 cents apiece, 2x will denote the cost of the lemons ; and since the oranges were 3 cents apiece, 3a; will denote the cost of the oranges ; and 2a5 + 3a;, or 6aj, "vvill denote the cost of both, which is 20 cents. Now, since hx cents are equal to 20 cents, x will be equal to 20 cents divided by 5 cents, which is 4 : hence, he bought 4 of each. WEITTEN. Let X denote the number of lemons, or oranges ; then, 2a; — the cost of the lemons ; and 3a; = the cost of the oranges ; hence, 2a; -j- 3a; = 5a; — 20 cents = the cost of lemons and oranges; hence, X = = 4, the number of each. 5 cents VERIFICATION. 4 lemons at 2 cents each, give, 4x2= 8 cents. 4 oranges at 3 cents each, " 4 x 3 = 12 cents. Hence, they both cost, 8 cents +12 cents = 20 cents. v. A farmer bought a certain number of sheep at 4 dollars apiece, and an equal number of lambs at 1 dollar a])iece, and the whole cost 60 dollars: hew many did he buy of each ? 8. Charles bought a certain number of apples at 1 cent apiece, and an equal number of oranges at 4 cents apiece, and paid 50 cents in all : how many did he buy of each ? MENTAL EXERCISES. 17 9. James bought an equal number of apples, pears, and lemons ; he paid 1 cent apiece for the apples, 2 cents apiece for the pears, and 3 cents apiece for the lemons, and paid 12 cents in all : how many did he buy of each ? Verify. 10. A farmer bought an equal number of sheep, hogs, and calves, for which he paid 108 dollars; he paid 3 dollars apiece for the sheep, 5 dollars apiece for the hogs, and 4 dollars apiece for the calves : how many did he buy of each? 11. A farmer sold an equal number of ducks, geese, and turkeys, for which he received 90 shillings. The ducks brought Mm 3 shillings apiece, the geese 6, and the tui-keys 1 : how many did he sell of each sort ? 12. A tailor bought, for one hundred dollars, two pieces of cloth, each of which contained an equal number of yards. For one piece he paid 3 dollars a yard, and for the other 2 dollars a yard ; how many yards in each piece ? 13. The sum of three numbers is 28 ; the second is tmce the first, and the third twice the second : Avhat are the numbers ? Verily. 14. The sum of three numbers is 64 ; the second is 3 times the fipt, and the third 4 tunes the second : what are the numbers ? LESSON V. 1. If 1 yard of cloth costs x dollars, what will 2 yards cost? Analysis. — ^Two yards of cloth will cost twice as much as one yard. Therefore, if 1 yard of cloth costs x dollars, 2 yards will cost twice x dollars, or 2x dollars. 2. If 1 yard of cloth costs x dollars, what will 3 yards cost? Why? 18 INTKODUCTION. 3. If 1 orange costs x cents, what will 1 oranges cost \ Why ? 8 oranges ? 4. Charles bought 3 lemons and 4 oranges, for which he paid 22 cents. He paid twice as much for an orange as for a lemon : what was the price of each ? An-alysis. — Let x denote the price of a lemon ; then, 2i8 will denote the price of an orange ; 3a5 wdll denote the cost of 3 lemons, and Sec the cost of 4 oranges ; hence, Zx plus 8a?, or 11 ic, will denote the cost of the lemons and oranges, which is 22 cents. If \\x is equal to 22 cents, x is equal to 22 cents divided by 11, which is 2 cents: therefore, the price of 1 lemon is 2 cents, and that of 1 orange 4 cents. AVKITTEN. Let X denote the price of 1 lemon ; then, 2a; = " 1 orange ; and, 3rr -f 8a; = 11a; r= 22 cts., the cost of lemons and oranges; 22 cts hence, x = — — — '- — 2 cts., the price of 1 lemon ; and, 2x2 = 4 cts., the price of 1 orange;. VERIFICATION. 3 X 2 =z 6 cents, cost of lemons, 4 X 4 = 16 cents, cost of oranges. 22 cents, total cost. 6. James bought 8 apples and 3 oranges, for which he paid 20 cents. He paid as much for 1 orange as for 4 applss-* what did he pay for one of each ? 6 A farmer bought 3 calves and 7 pigs, for which he paid 19 dollars. He paid four times as much for a calf as for a pig : what was the price of each ? *l. James bought an apple, a peach, and a pear, for which ho paid 6 cents. He paid twice as much for the peach as for MENTAL EXER0I6EB. 19 the apple, and three tunes as much for the pear as for the apple : what was the j)rice of each ? 8. Williaiu bought an ai)ple, a lemon, and an orange, for wliich lie paid 24 cents, lie paid twace as much for the lemon as for the apple, and 3 times as muck for the orange as for the apple : what was the j)rice of each ? 9. A fanuer sold 4 calves and 5 cows, for which he received 120 dollars. He received as much for 1 cow as for 4 calves: what was the price of each ? 10. Lucy bought 3 ])ears and 5 oranges, for which she piud 20 cents, giving t^^'ice as much for each orange as for jach pear: what was the price of each? 11. Ann bought 2 skeins of silk, 3 pieces of tape, and a penknife, for which she paid 80 cents. She paid the same for the silk as for the tape, and as much for the penknife as for both : what was the cost of each ? 12. James, John, and Charles are to divide 66 cents among^ them, so that John shall have twice as many as James, and Charles twice as many as John: what is the share of each ? 13. Put 54 aj)ple8 into three baskets, so that the second hall contain t^Wce as many as the first, and the third as 1 any as the first and second: how many will there be hi ich. 14. Divide 60 into four such parts that the second shaU be double the first, the third double the second, and the Iburth double the third : what are tlie numbers ? LESSON VL 1. If 2x + X is equal to Sx, what is 3x — x eqiuil to ? Written, 3x — x == 2x. 2. What is 4a; — CB equal to ? Written, 4in — jc — 3x. 20 IN TKOD UOTION. 3. Wliat is 8a3 minus 6a! equal to ? Written, 8a; — 6a; = 2x. 4. What is 12a; — 9a; equal to? Ans, Sa. 6. What is 15a; — 7a; equal to? 6. What is •I'Jx — 13a; equal to? Arts, 4a;. V. Two men, who are 30 miles apart, travel towards each other ; one at the rate of 2 miles an hour, and the other at the rate of 3 miles an hour : how long before they will meet? Analysis. — Let x denote the number of hours. Then, since the time, multiplied by the rate, will give the distance, 1x will denote the distance traveled by the first, and 3a! the distance traveled by the second. But the sum of the distances is 30 miles ; hence, 2a; 4- 3a; = 6a; = 30 miles; and if bx is equal to 30, x is equal to 30 divided by 6, which is 6 : ^ence, they will meet in 6 hours. WRITTEN. Let X denote the time in hours ; then, 2a; = the distance traveled by the 1st ; and 3a; = i* " 2d. By the conditions, 2a; 4- 3a; = 5a; = 30 miles, the distance apart ; hence, jc = — = 6 hours. o VERIFICATION. 2x6 z=z 12 miles, distance traveled by the first. 3x6 = 18 miles, distance traveled by the second 30 miles, whole distance. 8, Two persons are 10 miles apart, and are traveling in the same direction ; the first at the rate of 3 miles an hour, and the second at the rate of 5 miles : how long, before the second will overtake the first ? and, 5x = a and, 5x — 3x — 2x or, X = 10 2 MENTAL EXEB0ISK8. 21 Analysis. — Let x denote the time, in hours. Then, Sx will denote the distance traveled by the first in x hours; and 5x the distance traveled by the second. But when the second overtakes the first, he will have traveled 10 miles more than the first : hence, 5aj — 3a; = 2a; = 10; if 2a; is equal to 10, x is equal to 5 • hence, the second will overtake the first in 5 hours. WRITTEN. Let X denote the time, in hours: then, 3a; = the distance traveled by the 1st; 2d; = 1 hours ; = 5 hours. ^ VKRIFICATIQN. 8x5 = 15 miles, distance traveled by 1st. 6 X 5 = 25 mUes, " " 2d. 25 — 15 = 10 miles, distance apart. 9. A cistern, holding 100 hogsheads, is filled by two pipes ; one discharges 8 hogsheads a minute, and the other 12 : in what time will they fill the cistern? 10. A cistern, holding 120 hogsheads, is filled by 3 pipes; tlie first discharges 4 hogsheads in a minute, the second 7, and the third 1 : in what time will they fill the cistern ? 11. A cistern which holds 90 hogsheads, is filled by a pipe ^vhich discharges 10 hogsheads a minute ; but there is a waste pipe which loses 4 hogsheads a minute : how long will it take to fill the cistern ? 12. Two pieces of cloth contain each an equal number of yards ; the first cost 3 dollars a yard, and the second 5, and both pieces cost 96 dollars : how many yards in each ? 1 3. Two pieces of cloth contain each an equal number of yards ; the first cost 7 dollars a yard, and the second 6 ; the first 22 INTRODUCTION. cost 60 dollars more than the second : how many yards m each piece ? 14. John bought an equal number of oranges and lemons the oranges cost him 5 cents apiece, and the lemons 3 ; and he paid 56 cents for the whole: how many did he buy of each kind ? 15. Charles bought an equal number of oranges and lemons; the oranges cost him 5 cents apiece, and the lemons 3 ; he paid 14 cents more for the oranges than for the lemons : how many did he buy of each ? 16. Two men work the same number of days, the one receives 1 dollar a day, and the other two : at the end of the time they receive 54 dollars : how long did they work ? LESSON^ yii. ^ 1. John and Charles together have 25 cents, and Charles has 5 more than Jolm : how many has each ? Analysis. — Let x denote the number which John has ; then, a; + 5 will denote the number which Charles has, and a; -h 85 + 5, or 2a; 4- 5, will be equal to 25, the number they both have. Since 2x + 5 equals 25, 2x will be equal to 25 minus 5, or 20, and x will be equal to 20 divided by 2, or 10: therefore, John has 10 cents, and Charles 15. WRITTEN. Let X denote the niunber of John's cents ; then, ^ a; -h 5 =r " Charles' cents; and, jc -f a; -h 5 =. 25, the number they both have ; or, 205 -J- 5 ^ 25: ; and, 2a; = 25—5 == 20; hence, 20 X ■=. — =10, John's number; and, z 10 + 5 = 15, Charles' number. MENTAL BXKK0I6B6. VEBIPICATION. 10 Chwlea'. + 15 = 25, the sum. ChM-le*' 15 John's. - 10 = 5, the difference. 23 2. James and John have 30 marbles, and John has 4 more than James : how many has each,? 3. AVilHam bought GO oranges and lemons ; there were 20 more lemons than oranges: how many were there of each sort ? 4. A farmer has 20 more cows than calves ; in all he has 86 : how many of each sort? 5. Lucy has 28 pieces of money in her purse, composed of cents and dimes; the cents exceed the dimes in number by 16 : how many are there of each sort ? 0. Wliat number added to itself, and to 9, will make 29 ? Y. What number added to twice itself, and to 4, will make 25 ? 8. What number added to three tunes itself, and to 12, will make 60 ? 9. John has five times as many marbles as Charles, and *'liat they both have, added to 14, makes 44 : how many has tach? 10. There are three numbers, of which the second is t^vice the first, and the third twice the second, and when 9 is added to the sum, the result is 30 : what are the numbers? 11. Divide 17 into two such parts that the second shall be two more than double the first: what are the parts? 12. Divide 40 Jnto three such parts that the second shall be twice the first, and the third exceed six times the first by 4 : what are the parts? 13. Charles has twice as many cents as James, and John 24 INTKODUCTION. has twice as many as Charles ; if 7 be added to what they all have, the sum will be 28 : how many has each ? 14. Divide 15 into three such parts that the second shaU be 3 times the first, the third twice the second, and 5 over ; what are the numbers ? 15. An orchard contains three kinds of trees, apples, pears, and cherries; there are 4 times as many pears as apples, twice as many cherries as pears, and if 14 be added, th«^ number will be 40 ; how many are there of each ? LESSOR vm. 1. John after giving away 5 marbles, had 12 left: how many had he at first ? Analysis. — Let x denote the number ; then, x minus 5 will denote what he had left, which was equal to 12. Since X diminished by 5 is equal to 12, x will be equal to 12, increased by 5 ; that is, to 17 : therefore, he had 17 marbles. WRITTEN. Let X denote the number he had at first; then, 05 — 5 = 12, what he had left; and X = 12 + 5 = 17, what he first had. VERIFICATION. 17 — 5 = 12, what were left. 2. Charles lost 6 marbles and has 9 left : how many had he at first ? 3. \yilliam gave 15 cents to John, and had 9 left: how many had he at first ? 4. Ann plucked 8 buds from her rose bush, and there svere 19 left : how many were there at first ? MENTAL EXERCISES. 25 6. William took 27 cents from his purse, and there were lU left: how many were there at first? 6. The sum of two numbers is 14, and their difference ig 2: what are the numbers ? Analysis. — ^The diff*erence of two numbers, added to the lees, will gi^•c the greater. Let x denote the less number; then, a; 4- 2, will denote the greater, and jc + aj + 2, will denote their sum, which is 14. Then, 2a; + 2 equals 14; and 2a; equals 14 minus 2, or 12: hence, x equals 12 divided by 2, or 6 : hence, the numbers are G and 8. VEIUFICATION. 6 + 8 = 14, their sum; and S — 6 = 2, their difference. 7. Tlie sum of two numbers is 18, and their difference 6 : what are the numbers ? 8. James and John have 26 marbles, and James has 4 more than John : how many has each ? 9. Jane and Lucy have 16 books, and Lucy has 8 more than Jane : how many has each ? 10. William bought an equal number of oranges and lemons ; Charles took 5 lemons, after which William had but 25 of both sorts : how many did he buy of each ? 11. Mary has an equal number of roses on each of two bushes ; if she takes 4 from one bush, there will remain 24 en both : how many on each at first ? i2. The sum of two numbers is 20, and their difference b 6 : what are the numbers ? Analysis. — ^If x denotes the greater number, a; — - 6 will denote the less, and a; + a; — 6 will be equal to 20 ; hence, 2x equals 20 + 6, or 26, and x equals 26 divided by 2, eqiuiJF 13; hence the numbers are 13 and 7. 2 iJO I N T R O D D C r I O N . WRITTEN. Let X denote the greater ; then, aj — 6 = the less ; and a; 4- a — 6 = 20, their sum ; hence, 22; = 20 + 6 = 26 ; or, 26 X = —- = 13 ; and 13 — ~ 7. VERIFICATION. 13 + 7 = 20 ; and, 13 — 7 = 6. 13. The sum of the ages of a father and son is 60 ydflrs, and their difference is just half that number : what are the'r ages? 14. The sum of two numbers is 23, and the larger lacks 1 of being 7 times the smaller : what are the numbers ? 15. The sum of two numbers is 50 ; the larger is equal to 10 times the less, minus 5 : what are the numbers ? 16. John has a certain number of oranges, and Charles has four times as many, less seven ; together they have 53 : how many has each ? 17. An orchard contains a certain number of apple trees, and three times as many cherry trees, less 6 ; the whole num- ber is 30 : how many of each sort ? LESSON IX. 1. If a; denotes any number, and 1 be added to it, what will denote the sum ? A?is. ic -f- 1. 2. If 2 be added to jb, what will denote the sum ? If 3 be added, what ? If 4 be added ? roduct will be equal to Charles' number; how many has each ? ''^ ^ 22. A bosket is filled with apples, lemons, and oranges, in all 26 ; the number of lemons exceed the apples by 2, and the number of oranges is double that of the lemons : how many are there of each ? 5 " LESSON XL 1. John hiis a certain number of apples, the half of which IS e(iual tn 10: how many has he? Analv.^is. — Let x denote the number of apples; then, X divided by 2 is equal to 10; if one half of x is equal to 10, twice one-half of a?, or a^ is equal to twice 10, which is -•>; hence, x is equal to 20. Note. — A similar analysis is applicable to any one of thi* fractional units. Let each question be solved according to the analysis. 2. John has a certain number of oranges, and one-tliird of his nuMiber is 15: how many h.as he? 3. If one-fifth of a number is 6, what is the number? 4. If one-twelfth of a number is 9, what is the number?.^ 5. What number added to one-half of itself will give a sum equal to 1 2 ? Analysis. — Denote tin? number by x; then, x plus one half of a; equals 12. But jb plus one-half of a equals three halves of x: hence, three halves of x equal 12. If throe |,iKv>^ nf X equal 12, one-half of x equals one-third of 12, 32 INTRODUCTION. or 4. If one-half of x equals 4, x equals twice 4, or 8, hence, x equals 8. WRITTEN. Let X denote the number; then, 1 3 03 + Tiic = ^^^ = 12 ; then, -a = 4, or JB = 8. VERIFICATION. 8-f-?rrr8 + 4 = 12. 6. What number added to one-third of itself will give a sum equal to 12? 7. What number added to one-fourth of itself will give a sum equal to 20 ? 8. What number added to a fifth of itself will make 24 ? 9. What number diminislied by one-half of itself will leave 4 ? Why ? 10. What number diminished by one-third of itself wiD leave 6 ? 11. James gave one-seventh of his marbles to William, and tlien has 24 left : how many had he at first ? 12. What number added to two-thirds of itself will give a siim equal to 20 ? 13. What number diminished by three-fourths of itself will leave 9 ? 14. What number added to five-sevenths of itself will make 24 ? 15. What number diminished by seven-eighths of itself w ill leave 4 ? 16. AVhat number added to eight-ninths of itself will make 34? ELEMENTARY ALGEBRA. CIIAPrER I. DEFINITIONS AND EXPLANATORY SIGNS. 1. Quantity is anything which can be increased, diminished, and measured; as number, distance, weight, time, &c. To measure a thing, is to find how many times it con- tains some other thing of the same kind, Uiken as a stand- ard. The assumed standard is called the unit of measure. 2. Mathematics is the science which treats of the measurement, properties, and relations of quantities. In pure mathematics, there are but eight kinds of quantity, and consequently but eiglit kinds of llNrrs, viz.: Units of Number ; Units of Currency ; Units of Length ; Units of Surface; Units of Volume ; Units of Weight; Units of Time ; and Units oi Angular Measure. 3. Algebra is a branch of Mathematics in which the quantities considered are represented by letters, and the operations to be performed are indicated by signs. 1. What 19 quantity ? What is the opcratioD of measuring a thing? Whit is the assumed standard called ? 2. What is Mathematics ? How many kinds of quantity are there ic .he pure mathematics? Name the unit« of thoe^ quantities. 8. What is Algebra? !♦ 34 ELEMENTARY ALGEBRA. 4. The quantities employed in Algebra are of two kinds, Known and Unknown : Known Quantities are those whose values are given ; and Unknown Quantities are those whose values are rC' quired. Known Quantities are generally represented by the lead ing letters of the alphabet, as, «, ^, c, &c. Unknown Quantities are generally represented by tho final letters of the alphabet ; as, x^ y, 2, &c. When an unknown quantity becomes known, it is often denoted by the same letter with one or more accents ; as, ik', x'\ x". These symbols are read: x prime j x second; X thirds dbc. 5. The Sign of Addiiton, +, is called plus. When placed between two quantities, it mdicates that the second is to be added to the fii'st. Thus, a -f ^, is read, a plus 6, and indicates that b is to be added to a. If no sign is wi-itten, the sign -f is understood. The sign -f , is sometimes called the positive sign, and the quantities before Avhich it is written are called j^ositive quan- tities^ or additive quantities. 6. The Sign of Subtraction, — , is called mimis. When placed between two quantities, it indicates that the second is to be subtracted from the first. Thus, the expression, 4. How many kinds of quantities are employed in Algebra? How are they distinguished ? What are known quantities ? What are unknown quantities? By what are the known quantities represented? By what are the unknown quantities represented ? When an unknown quantity becomes known, how is it often denoted? 5. What is the sign of addition called? When placed between two quantities, what does it indicate ? 6. What is the sign of subtraction called ? When placed between two •■juautitifts, what does it indicate? DEFINITION OK TERMS. 35 c — dy road c minus f7, iiidicaltsthat c? is to be subtracted from c. If a stands for 6, and d for 4, then a — d \s equal to 6 — 4 , which is equal to 2. Tlie sign — , is sometimes called the negative sign, and the qu:intiti*is before which it is written ai*e called negative quaur FilieSy or aubtractive quantities. T. Tlie Sign qp Multipucation, x , is read, mnltipUul bi/y or into. When placed between two quantities, it indi- cates that the first is to be multiplied by the second. Thus, a X b indicates that a is to be nmltiplied by b. If o stand?? for 7, and b for 5, then, a X b \a equal to 7 x 5, which is equal to 35. The multiplication of quantities is also indicated by simply writing the letters, one after the other ; and sometimes, by placmg a point between them ; thus, a X b signifies the same thing as ab^ or as a.b. a X b X c signifies the same thmg as abc^ or as a.b.c. 8. A Factou is any one of the multipliers of a product. Factors are of two kinds, numeral and literal. Thus, in the expression, 5aic, there are four factors : the numeral factor, 5, and the three literal fiictors, a, ^, and c. 9, The Sign of Division, -f-, is read, divided by. When written between two quantities, it indicates that the first is to be divided by the second. 7. How is the sign of multiplication road ? "When phiced between two quantities, what does it indicate? In how many ways may m;iltip)ieatiou be indicated? 8. V«hat is a factor? IIow many kinds of factors are there T Bow many factors arc there in Zabc ? 9. Uuw is the sign of division read? When wntten between two quan- tiJicfl, what does it indicate? IIow many ways are there of indicating division ? 86 ELEMENTARY ALGEBRA. There are three signs used to denote division. Thus, a -T- 5 denotes that a is to be divided by b, 7 denotes that a is to be divided by h, a I b denotes that a is to be divided by b. lO. The Sign of Equality, =, is read, equal to. Wlien M rittcn between two quantities, it indicates that they are equal to each other. Thus, tlie expression, a + 6 =r c, in- dicates that the sum of a and ^ is equal to c. If a stands for 3, and b for 5, c will be equal to 8. ai. The Sign of Inequality, > <, is read, greater thaii^ or less than. When placed between two quantities, it indicates that they are unequal, the greater one being placed at the opening of the sign. Thus, the expression, a > ^, indicates that a is greater than b\ and the expres- sion, c <^ d^ indicates that c is less than d. 12. The sign . * . means, therefore^ or consequently. 13. A Coefficient is a number written before a quan. tity, to show how many times it is taken. Thus, a-\-a-\-a-\-a-\-a = 5a, In which 5 is the coefficient of a. A coefficient may be denoted either by a number, or a letter. Thus, 5x indicates that x is taken 5 times, and ax 10. What is the sign of equality ? When placed between two quanti- ties, what does it indicate ? n. How is the sign of inequality read? Which quantity is placed on ^ the side of the opening ? 12. What does .*. indicate? 18. What is a coefficient? How many times is a taKen in 5a. By what may a coefficient be denoted ? If no coefficient is written, what coefficient is understood ? In 5ar, how many limes is ax taken? Ho-v niauy tiraea is r taken ? DEFINITION OF TERMS. 37 indicates that x is taken a times. If no coefficient is ^\Tit- ten, the coefficient 1 is understood. Thus, a is the same as la. 14. Ax ExroxENT is a number wriiten at tlic right and Rl»ove a quantity, to indicate how many times it is taken as a factor. Thus, a X a is written a\ a X a X a (C o^ a X a X a X a (( a\ &0., Ac, m which 2, 3, and 4, are exponents. The expreshions aro read, a square, a cube or a third, a fourth ; and if we have o**, in which a enters m times as a factor, it is read, a to the mth, on simply a, mth. The exj^onent 1 is goneraUy omitted. Thus, a* is the same as a, each denoting that a enters but once as a factor. 15. A Power is a product which arises from the multi- plication of equal factors. Thus, a X a = a^ is the square, or second power of a. a X a X a = a^ IS the cube, or third power of a. axax«xa = a* is the fourth power of a. a X a X ... . = a" is the mth power of a. 16. A Root of a quantity is one of the equal factors, riie radical sign, \/ , when placed over a quantity^ indi- cates that a root of that quantity is to be extracted. Tlie root is indicated by a number written over the radical sign, 14. What is an exponent? In a', how many times is a taken as a fao- or? When no exponent is written, what is understood? 16. What is a power of a quantity? What is the third power of 2» Of 4 ? Of 6 ? 16. What is the root of a quantity? What indicates a root? What indicates the kind of root? Wh.it is the index of tho pquare root? 0( the ctibc root ? Of the mth root ? 38 ELEMENTARY ALGEBRA. called an index. When the mdex is 2, it is generally omit- ted. Tims, \/a^ or y/a, indicates tlie square root of a. ya indicates tbe cuhe root of a. ^/a indicates the fourth root of a. \/a mdicates the fnih root of a. ] ? . An Atx^.ebraic Expression is a quantity written in algebraic language. Thus, „ j is the algebraic expression of ttree tunes ( tbe number denoted by a ; - 2 ^ '^s the algebraic expression of five times I the square of a ; is the algebraic expression of seven times la^b^ -j the the cube of a multiplied by the square of b ; is the algebraic expression of the differ- Sa — 5h -l ence between three times a and five times b ; r is the algebraic expression of twice the „ „ „ , ,,„ square of a. diminished by three times 2a^ — Sab + 4h^ ^ ^/ i 1 i- ^ j . i ^ J the product oi a by t>, augmented by I four times the square of b. 18 A Term is an algebraic expression of a sm gle quan- tity. Thus, 3a, 2ab, — 5cfb\ are terms. 19. The Degree of a term is the nmnber of its literal factors. Thus, „ j is a term of the first degree, because it oontains but ( one literal factor. 17. What is an algebraic expression 1 8. What is a term ? 1 9. What is tlio dogree of a term ? What detenniriop the (Vgrpe of a term ? Trf'ij DEFINITION OP TERMS. 39 • j is of tlie second degree, because it contains two lite- ( ral factors, is of the fourth degree, because it contains four literal factors. Tlie degree of a term is determined by the sum of the exponents of all its letters. 20. A MoKOiHAL is a single term, unconnected with any other by the signs + or — ; tlms, 3a^ Slt^a, are monomials. a I. A PoLYNOiTiAL is a collection of terms connected ()}• the signs -H or — ; as, 3a — 5, or, 2a^ — db + 4h^. 22. A Binomial is a poljTiomial of two temis ; as, a -f 6, 3a2 _ c^ 6ab — c^. 23. A Trinomial is a polynomial of three terms ; as, abc — a^ + c\ ab — (jh — f. 21. Homogeneous Terms are those which contain the p;»iiK' number of literal factors. Thus, the terms, abc^ — «\ 4- c^, are homogeneous ; as are the terms, ab^ — gh. 25. A Polynomial is tiomogeneous, when all its terms !'' homogeneous. Tlius, the polpiomial, abc — a^ + c^, is iJiogeneous; but the polynomial, ab — gh — f is not ho- iii<»geneou8. 26. Snrn.AR Terms are those which contain the same Ht.'ial factors affected \^'iih the same exponents. Thus, *lah 4- 3rt/> — 2a^ - 1». What 19 a monomial ? 21. What is a polynomial? 'Zl. What is a binomial? .:;. What is a trinomial? . \. What are homojjpneous terms ? •J."). When i« a polTnomial homogeneous ? L'-i. What nrp «:niil:'.r tpnn'J ? 40 ELEMENTAKY A L O E B E A. . are similar terms ; and so also are, but the terms of the first polynomial and of the last, are not similar. ' 27. The Yinculttm, , the Bar \ , the Parens thesis^ ( ) , and the Brackets^ [ ] , are each used to con- nect several quantities, which are to be operated upon in the Fame manner. Thus, each of the expressions, a X 'a^i^^c X Jc, f 5 (a + ^> + c) X aj, and [a + 5 4- c] x j», indicates, that the sum of a, J, and c, is to be multiplied by JB. 28. The RECirROCAL of a quantity is 1, divided by that quantity; thus. are the reciprocals of . 7 ^ a, a -{• . -• c 29. The Numerical Yalue of an algebraic expression, is the result obtained by assigning a numerical value to each letter, and then performing the operations mdicated. Thus, the numerical value of the expression, ah -\- he + cL when, a = i, J — 2, c = 3, and d =: 4, is 1x2 + 2x3 + 4 = 12; by performing the indicated operations. 27. For what is the vincular used? Point out the other ways in which this may be done ? 28. What is the reciprocal of a quantity? 29. What is the numerical value cf an algebraical exprepsion? 1 ALOBBRAIC EXPRESSIONS 4l EXAMPLES IX WETTING ALGEBSAIC EXPEESSIONS. 1. Write a added to b. Ans. a -f ft« 2. Write b subtracted from a. Ans. a — b. Write the following : 8. Six times the square of a, minus twice the square of b. 4. Six times a multiplied by b, diminished by 5 times c cube multiplied by d. 6. Nine times a, multiplied by c plus d, diminished by 8 times b multiplied by d cube. 6. Five times a minus b, plus 6 times a cube into b cube. I. Eight times a cube into d fourth, into c fourth, plus 9 times c cube into d fifth, minus C times a into b^ into c square. 8. Fourteen times a plus J, multiplied by a minus 5, plus 6 times a, into c plus f/. 9. Six times a, into c plus t7, minus 5 times ft, into a plus c, minus 4 times a cube ft square. J 1 0. Write a, multiplied by c plus d, plus / minus ^. II. Write a divided by ft + c. Three ways. 12. Write a — ft divided by a + ft. 13. Write a polynomial of three tenns; of four terms; of five, of sb:. ^ 14. Write a homogeneous binomial of the first degree; of the second ; of the tliird ; 4th ; 6th ; 6th. 15. Write a homogeneous trinomial of the first degree; with its second and third terms negative; of the second degree; of the 3rd; of the 4th. 16. Write in the same column, on the slate, or black-board, a monomial, a binomial, a trinomial, a polj-nomial of four terms, of five terms, of six tenns and of seven tenns, and aJJ of the sanie degree. 42 ELEMENTARY ALGEBRA. INTKEPRETATION OF ALGEBRAIC LANGUAGE. u Find the numerial values of the folloTving expressions, when, a = 1, ^ — 2, c = 3, c? =. 4. 1. ab + he. A?is, 8. ' 2. a + be -{- d. A?is. 11. 3. ad -\- b — c. Ans. '6. 4. ab •\- be — d. Ans. 4. 5. (a + b) e^^- d. , • - Ans. 23. 6. (a + b) {d — h.) ^ Ans. 6. 7. (a^.+ «c7,) c -\- d. Ans. 22. 8. {ab -h o) (ac? — a). Ans. 15. 9. ^a2^2 _\2(a + c? -1- 1). ..4/i6\ 0. 10. ^ ^ X (« 4- <^) ^;z5. 10. tt^":}. 52 _|_ c2 a3 _^ J3 -|_ c^ - f/ . 11. r X r • Ans. 32. 7 2 a54 _ c _ ^3 J^2 - h -\- d^ 12. x^= -r-^- — ^^ A?is. 4. 6 33 Fmd the numerical values of the folio ^^-ulg expressions, when, a = 4, b = Sy c = 2, and d = 1.. 13. ^ - 7: + c — d. Ans. 2. ^ , ^/''d) a — d\ U. 5f ^ 3~/' ^^15. 15. 15. [(rt2Z> + l)cn -f- (a^b + cZ). ^715. 1. 16. 4(abc - -^l X (30c3 — aPd^). A?is. 11088. ,^ « + ft + c , abed , 4^2+ ft^ - Sab Qab ab Add the follownng : (3.) (4.) a Sab a lab \2ab 4- 2a \5ab (5.) lac 5ac I2ac (6.) -f ^abc Sabc + labc • ..u. What ia addition ? 31. What is the rule for addition when the terms are Biniilarand hftve M ELEMENTARY ALGEBRA. .0') (8.) (9.) (10.) — Zabq — Sad — 2adf — 9abd — 2abc — 2ad — Qadf — I5abd — 5abc — had — '^adf — 2\abd Hence, when the terms are similar and have like signs : RULE. Add the coefficients^ and to their sum prefix the common sign y to this, annex tJite common literal part, EXAMPLES. (11.) (12.) (13.) 9ah + ax 8ac^ — db^ \ba¥c^ — I2abc^ Sab + Sax lac^ — 8b^ 12ab^c* — 15abc^ \2ab + 4aa; Sac"^ — db'^ ab^c* — abc^ When the terms are similar and have unlike signs, 32. The signs, + and — , stand in direct opposition to each other. If a merchant writes + before his gains and — before ?iis losses, at the end of the year the sum of the plus numbers will denote the gains, and the sum of the minus numbers the losses. If the gains exceed the losses, the difference, which is called the algebraic sum,, will be plus ; but if the losses exceed the gains, the algebraic sum will be minus. 1. A merchant in trade gained $1500 in the first quarter of tlie year, $3000 in the second quarter, but lost $3000 in the third quarter, and $800 in the fourth : what was the re- sult of the year's business? 1st quarter, +1500 3d quarter, —3000 2d " 3000 4th " — 800 -h 4500 — 3800 + 4500 — 3800 = + YOO, or $700 gain. 32. What is the rule when the terms are s'milar and havt; unlike sigiia ? ADDITION. 45 2. A merchant in trade gained llOOO in the first quarter, and $2000 the second (jiiarter ; in tlie tliird quarter he lost llaOO, and in the fourth quarter $1800.-: what was the result of the year's business ? iPt quarter, + 1000 3d quarter — 1500 2d *' -t- 2000 4th " -- 1800 H- 3000 — 3300 + 3000 — 3300 = — 300, or $300 loss. 3. A merchant in the first half-year gained a dollars and lost b dollars ; in the second half-year he lost a dollais and gained b dollars : what is the result of the year's business ? Ist half-year, -\- a — 6 2d " - a + b - Result, Hence, the algebraic sum of a positive and negative quan- tity is their arithmetical diffcre7ice^ with the sign of tlie greater jyrefixed. Add the following : ^ah Aach"^ — Aa'^b'^c^ dab — Sacb^ + 6aV)^c^ - Gab aclP- — la^h^c^ bab — dacb'^ Hence, when the terms are similar and have unlike signs : I. Write the similar terms in the same column : n. Add the coefficients of the additive terms ^ and also the coefficients of the subtractive terms : UL Take the difference of these sums, prefix the sign of the greater, and then annex the literal part, EXAMPLES. 1. What is the sum of 2a2^>3 _ 5^2^3 4-702^3 ^ oa^ft^ - Wa^l^'t 46 KLKMENTAliY ALGEBRA Having written the similar teinns in the same cohimn, we find the sum of the positive coeffi- cients to be 15, and the sum of the negative coefficients to be — 16 : their difference is ■— 1 ; hence, the sum is — a^h^. — 5r/2*3 - am. 2. "Wliat is the sum of 'a/h + ^a^h — ^a^b + 4«^^> - Qo^b — a^bt 3. What is the sum of 1 1a?bc^ — 4a^bc^ H- Qa^bc'' - 8a^c^ -f 1 1 a^bc^ ? Ans. I la^bc", 4. What is the sum of 4a2j _ 8^2^ - ^a'b -f 11«2^,? 5. What is the sum of labc^ — abc^ — nabc? + ^ahc^ -f Q>abcn 6. What is the sum of 9c^3- 5W>3— 8ac2-h 20c63 4- Gac^ - 24c^1» Ans, X og Ans, 1d?-h, Ans, — 2«*V> Ans. \'^abc\ To add any Algebraic Quantities, 33. 1. What is the sum of 3a, 55, and — 2c? Write the quantities, thus, 3a + 55 — 2c; which denotes their sum, as there are no similar terms, 2. Let it be required to find the sum of the quantities, 3a2 __ 8a j -\- /h^ lab — 552 ba^ — bah, — 46^ 38. What is tlie rule for the addition of any lilgc'brain quantities? ADDITION. 47 From the preceding examples, we have, for tlie addition of algebraic quantities, the following RULE. L Write the quantities to be added, placing similar terms ill the same colum?i, and giving to each its proper sign: TL Add up each column separately and then annex tht dissimilar terms with their proper signs, EXAMPLES. 1. Add together the polynomials, 3a? — 2/>« - Aab, 5a^ - b^ + 2ab, and dab - 3c^ - 2^. Tlie terra Sa^ being similar to 5a\ we yn-iie 8a^ for the result of the reduction of these two terms, at the sjime time slightly crossing them, as in the first terra. 5/r- + lib - V- 4- d^(h — ^ ^ — 3^ 8a2 -I- ab - bb^ - Z(^ Passing then to the term — Aab, which is similar to 4- 2ab and -h dab, the three reduce to + ab, which if. phu'cd after 8a?, and the tenns crossed like the first terra. Passing then to the terms involving 6?, we find their euni to be — bh^, after which we write — 3c?. The marks are dra\vn across the terms, that none of them may be overlooked and omitted. (2.) (3.> (4.) labc + Ocrz 8aa; -\ db 12o — 6c — dabc — 3aa; bax — 96 — 3a- — 9c Aakc + 6aa; 13aa; — 6A. 9a — 16& Note. — If a = 5, 6 = 4, c = 2, a; = 1, what are the numerical values of the several sums above found ? 48 ELEMENTARY ALGEBSA. (5.) (6.) (7.) 9« -f / Gax — Sac Saf + (7 -f «i — Qa + (/ — lax — Qac ag — ^af — m — 2a — f ax 4- 1 7ac ah — ag -\- Zg -(8.) (9.) ' — 3a: — 3a^ — 5c — 7cc2 — l^aca; + Wd'h'^G^ hx — 9ab — 9e — 4x^ + kacx — 20a^b^c^ ^ (10.) ^'^ (11.) =7 — =i:t-t^ -i- 22h — 3c — , 7/ + Sg 19ah^ + 3a'S^ — Sax^ — 3A -f 8c — 2/ — 9^ -f 5a; - llah^ - 9a^b* + 9 ag^^ (12.)^^ (13.) 7a; — 9?/ + 52 + 3 — <7 8a + ^ — a; — 3y — 8 — ^ 2a — 6 + c — a;+y — 3s + l + 7^ — 3a +5 +2c^ — 2a; 4- 63/ + 32 — 1 — ^ — 65 — 3c + 3<^ U. Add'ltogether — 5 -|- 3c — c7 — 115c + 6/ — 5<7, 35 — 2c — 3d — e ^ 27/, 5c — 86? + 3/ — 1g, — 7^— 6c -f l7(?-f 96 - -cN/-i- 11^, — 35 — 5d—2e-{-6f—9g-\-h. Ans. — 85 — 109e + 37/ - 10^ + A. 15. Add together the polynomials 7a25 — 3c?5c — 852<3 — 9c3 + cd\ Sabc— ^a?b + Sc^ — 452c + cd\ and 4a25 — 8c3 4- 952c — 36?^ Ans. 6a^b 4- 5a5c — 35^^ — 14c3 + 2cd^ — 3d^. 16. What IS the sum of, ^a^bc + 65a; — 4a/, — 3a25c -05^7 + 14a/, - a/+ 95a; + 2a25c, + 6a/— 85a;+ 6a25c? Ans. lOa'^bc + 5a; + 15a/. 17. "What is the sum of a^n^ + Za^m + 5, — Qa'^11^ — 6a^m - 5, + 95 — 9a^m — ba^n? ? Ans. — lOahi^ — ^2ahn + 95. (22.) (23.) Hn -j- b) 5{a- - c^ 3(a -1- />) - 4(a2 - c^) 2{a. -f- b) - 1(^2 _ c2) ADDITION. ^ 18. Wliat is the Bum of ia^b'^c ~ \6a*x — 9 SUBTRACTION. 51 of the subtrahend, and adding. Whence, we may write, 15a; — {_ 9a;) = 24a;. 3. From lOoa; take a — b. The difference^ or remainder^ is such a quantity, as added to a — ^, will give the minuend, lOaa;: what is tliat qaair tity? If you change the signs of both ttnns of the subtrahend, and add, you liavc, lOax — a -h b. Is this the true remainder ? Certainly. For, if you add the remainder to the subtrahend, a — b^ you obtain the minuend, lOax. It is plain, that if you change the signs of all the terms of the subtrahend, and then add them to the mmuend, and to this result add the given subtrahend, the last sura can be no other than the given minuend ; hence, the Jirst result is the true difference, or remainder (Art. 36). Uence, for the subtraction of algebraic quantities, we have the following BULE. L Write t/te terms of the subtrahend under those of the miyiuend^ placiyig similar terms in the same column : IL Coficeive the sigyis of all the terms of the subtrahend to be clhanged from ■\- to —^ or from — ^o -j-, and t/veti proceed as in Audition, EXAMPLES OF M0X0MIAIJ8. (1.) (2.) (3.) From Sab. 6aaj 9abo take 2ab Sax lahc. Rem. ab Sax 2«//x? 62 ELEMENTARY ALGEBRA. From taks Rem. I' a + JB^ take a^ + i^ take y3 - 6a'b^. Ans. Sab"^ — a^b\ Ans. x'^y — y'^x. Ans. Sx'^y^ — xy. Ans. SaH/'^x — xyz. Ans. Ua'^b^. Ans. 34a^y^. Ans. — 40a^b^. Ans. x^y*i Ans. — 42(Px^y. Ans. — 97a^x^ Ans. a -I- a;2 4- y''-. a? — b^. Ans 2a^ + 2b^ IQa^x^y take ~ Ida'^x^y. Ans. + Sa'^x^y. take a^ 4- a^^. A?is. — 2x^. SUBTRACTION. 53 OBNEBAL EXAMPLES. (1.) flj (!•) n FromQac — 5ab+ c* *'s^ 6ac — 5tf/> + c* take (3^c ->? 3tf^ 4:: 7c^ ^ sM — ^>rrc — 3a b — 1c Rem. 3ac — 8a^ + c^ — Vc. ^^ _^ (2.) From Gax — a + 3b^ take 9aa; ^x -V b^ Rem. — 3aa? — a + x + 2b\ (4.) From 5a3 — 4a=i+ 2b^c take -f 2a3if3a2^^ 86^c - • ' 10 - 8a5 + (3.) c2- y<3 Qf/x - 3a:2 + 55 - 2/a; -i-3 -ha 5?/a; — na:2 4- 3 -t- 66 - -a. (5.) 4( lb — cd+. ;a2 5a^> - 4. -d - 5^2. 6. From a + 8 take c — 5. -47i5. a — c + 13 7. From Ca^ — ]5 take Oa^ v^ 30. ^7i«. — da^ - 45 8. From Gary — Sa^^a take -4- Ixy -f «V. -^1«5. I3xy — lifc^ 9. From a -\- c take — a — c. Ansl 2a -\- 2c 10. From 4(a + 6) take 2(rt + ^). ^;i5. 2(a + ^) 11. From 3(a + a) take (a + a.')- -^1'^^- 2(a + x) 12. From 9(0* — x^) take - 2(a2 _ x^). Ans. Il(a2 - x^) 13. From Qa} — loi^ take — Za^ + 9Z^'^. ^1/15. 9rt» - 24^2 14. From Sa" — 2^»'* take r^*" — 2/>\ yl;?.9. 2a'". 15. From ^chii^ — \ tr.ke 4 — 7c'^m^ J«5. IGc^/i* — 8. 10. From 6am -f y take 3am — x Am. Sam -f a; -f y 17. From 3aaj take 3aa; — y. Ans. -\ y. ^ 54 ELEMENTAKY ALGEBRA. -V- -t . _ 18. From — If + Sm — 8a; take — 6/ — 5m — 2a; 4- Sd 4- 8, Ans. —/-{- Bm — Gx — Sd — S. 19. From — a _ 55 + '?c 4- f^ take ib ~ c + 2d -{- 24 ^n5. — a — 96 + 8c — (/ - 2k. 20. From — 3a + 5 — 8c + ^e — 5/ + 3A - Ta; — 1 ;^./ take Ic -\- 2a — 9c + Se — 1x -\^ If — y — dl -f k. Ans. — 5a + ^» + c — e — ]2/ + 3A — 12y -f- 3/. jJl 21. From 2a; — 4a — 2^ + 5 take 8 — 56 + a 4 6a;. Ans. — 4a; — 5a + 36 — 3. 22. From Za -\- h -^ c — d — 10 take c -\- 2a — d. Ans. a 4- 6 — 10. 23. From 3a + 6 4- o -- c? — 10 take 6 — 19 4- 3a. Ans. c — fZ 4- 9. 24. From a^ 4- 36^6 4- aJ^ _ ahc take 6^ 4- aJ^ -f abe. • ^n5. a3 4- 362c — b\ 25 From 12a; 4- 6a — 46 4- 40 talver46 — 3a + 4a; 4- 6ri — 10 j A71S. 8a; + 9a — 86 — 6^ 4- 50. 26. From 2a; — 3a 4- 46 4- 6c — 50 ta];eV9a 4- a; 4- 66 — 6c -f 40]] A71S. X — 12a — 26 4- 12c — 10. 27. From Oa ~ 46 — 12c + 12a; take (2a; — 8a -f 46' — Q>c,, Ans. 14a — 86 — 6c 4- 10a;. 38. In Algebra, the term difference does not always, as in Arithmetic, denote a number less than the minuend. For, if* from a we subtract — 6, the remainder will be a 4- 6 ; and this is numerically greater than a. We distinguish between the two cases by calling this result the algebraic difference. 88. lu Algebra, as in Arithmetic, docs the term difference denote a DTimber less than the minuend ? How are. the results in the twc cases, distinguished from each_ other f 8UBTKACTI0N. 55 89. ^V^lcn a polynomial is to be subtracted from an al- gebraic quantity, we inclose it in a parenthesis, place the minus sh^n before it, and then ^Tite it ailer the minuend Thus, the expression, _ Ga^ — (Sab - Ih^ + Ihc), indicates that the polynomial, Zah — 2^^ ^ 25c, is to be taken from Ca^. Performing the operations indicated, by the rule for subtraction, we have the equivalent expression : T^ie last expression may be changed to the former, by changing the signs of the last three tenns, inclosing them in a parenthesis, and prefixing the sign — . Thus, 6a* — Zdh + 2^2 _ 26c = ^cO- — (3a6 - W- 4- 25c). In like manner any polynomial may be transformed, as in- dicated below : = Ta3 — 8a25 - {Wc - G*^). 8a3 — 7^2 Ac^ (p_ 8a3 — {W. — c + ^ ' = 8a3 _ 7^2 - (_ c + d), 95^ — a -f 3a2 — c? = 05^ — (a — Sa^ 4. ^7) = 953 - a — (- Sa^ 4- ^. Note. — The agn of every ^quantity is changed when it is placed within a parenthesis, roduci of two quajitities, • The quantity to be multiplied is called the Midtiplicand ; thai l)y which it is multiplied is called the Multiplier ; and the result is called tlie Product. The Multiplier and Multi- plicand are called Factors of the Produrt. G. If' a man's income is 3a dollars a w eek, how much \vill he receive in Ab weeks ? Za X Ab = \2ab. If we suppose a = 4 dollars, and b = 3 weeks, the pro- duct will be 144 dollars. Note. — It is proved in Arithmetic (Davies' School, Art. 48. University, Ait. 50), that the product is not altered by chajig- uig the arrangement of tlie factors ; that is, \2ab = a X b X 12 = b x a x \2 = a x \2 x b. MTn.TIPLICATION OF POSFm'B MONOMIALS. 4*i. Multiply Sa^ft^ by 2a^b, We write, 3^262 X 2a'^b = 3 X 2 X a^ X a^ X b^ X b = 3 X 2 a a a ab b b\ ui which a is a factor 4 times, and b a factor 3 times ; hence (Art. 14), Za^b"^ X 2a^b = 3 x 2a*b^ = Qa*b\ in which ice rmdtiply the cot'.ffi,cie7it3 together^ ajid add the exjx)?i€fits of the like letters. The j)roduct of any two positive monomials may be found in like manner ; hence the RI7LB. I. Multiply the coefficients together for a neio coefficiept: II. Write after this coefficient all the letters in both rnono- 42^ What id the rule fot multiplying oue mononiial by another? 3* 58 ELEMENTARY ALGEBRA, 7)iiaL% giving to each Utter an exponent equal to the sum of Us exponents in tJie tico fadors. EXAMPLES, Multiply by ^a^hc^ X lahd"^ = 5QaWc^d\ 21a^^cd X 8abc^ = 168a*Z>Vf?. 4ahc X Id/ = 2Sabcd/. 3a^b %a^h a^xy (5.) \2a^x, \2xhj (8.) (C.) Qxyz ay'iz^ Qaxy'^z^ (9.) 87aa;2y 3PxHf • 27a3^»V 10. Multiply 5a3^2aj2 j^y 6^^3,6, 11. Multiply lOa^b^c^ by Yac^Z. 12. Multiply ^QaWc^d^ by 20a52^3(7 13. Multiply 5«"' by SaS". 14. Multiply Za'^b^ by 6a2J^ 15. Multiply ea'"^*" by 9a^5^ 16. Multiply 5«'"5'»'by 2aPM. n. Multiply ha'^JP'C^ by 2a^>''c. 18. Multiply Ga^Zf^c" by ZaWc^. 19. Multiply l^a^b^cd by lla^x^y. Ans. 240a'^b^cdx^y 20. Multiply 14a456(^4y ]3y 20a'Va;22/. ^. 280a'^>VfZ»a;V 21. Multiply ^d^b^y^ by la'^bxy^. Ans. BGa'^b^xy^ 22. MuUipl)' Ibaxyz by ha^bcd^y"^. Ans, 315a^bcdx^y^z. 2Qlab^x''y\ Ans. 30«-^ZiVa^ yl/i5. I20a^b^c^d^ Ans. I5a'^'^^b Ans. ISa'^-^^b''-^ Ans. 54a'" + V>'*-^ Ans. 10a'" + pZ»'»-^r J[w5. 10a'" + ^Z>'»+V, Ans. 18a5^>'"-^2c« + 2 M CLT L I C ATIO N. 59 23. Multiply OAct^m'^x^yz Ly 8flr5V. A, 512a^5Vm«aj^cV8 by Sa?^»V. ^;i5. 648a^Z»Vc?» 20. Multiply lOa^'O'c^dyx by l^cfb^c^dxhf, Ans, 840a»*5^V c ac - - ^ 8 - 3 33 6 7 . . . 7 56 - 21 ~ 35 MULTIPLICATION OP POLYNOMIALS. 43. 1. Multii>ly a — ^ by c. It ia required to take the difference between a and b, c times ; or, to take c, a — b times. As w«f can not subtract b from c, we begin by taking a^ c times, which is ac; but this product is too large by b taken c times, which is be ; hence, the true product is ac — be. I If a, b, and c, denote numbers, as a = 8, 6 = 3, and c = 7, the operation may be written in figures. Multiply a — b l)y c — d. It is required to take a — h as many times as there are units in o - d. If we take a — b^ c times, ve have 04: — be \ but this product is too large hj a — b taken d times. But a — b taken d times, is ad—db. Subtracting this product from the preceding, by changing the signs of its terms (Art. 37), and we have, (« - «) X(^ - "^r = «* - a - -b c - - d ac- -be - ad -h bd acr- - be — ad + bd S -3 = 5 7 -2 = 5 56 - 21 -16+6 66 -37 + 6 = 25, be - - ad 4. bd. 60 ELEMENTARY ALGEBRA. Hence, we have the foUowmg RULEFOKTIIESIGNS. I. Whefi the factors have like signs^ the sign of their product will be -f .* II. When the factors have unlike signs, the sign of (heir irrodact will be — : Therefore, we say m Algebraic language, that + inrJti- plied by + , or — miiltiphecl by — , gives -f ; — muki- plied by -\- or -f- multiplied by — , gives — . Hence, for the miilti])licalion of polynomials, we have the foilowmg RULE. Multiply every term of the niultiplicand by each, term of the multijylier, observing that like signs give -f , cin^d mdike nans — / the7i reduce the result to its simplest form. EXAMPLES IN WHICH ALL THE TERMS ARE PLUS. 1, Multiply .... 'ia^ -{■ Aab + b"^ by 2a -t bb 6a3 -f ^d'b-^ lab"^ The product, after reducing, -f \oaVj-\- 20ab^ -f bb"^ becomes .... Ga^ + 2;]aV>+ 22a62 ^ 5^3^ 44. Note. — It will be found convenient to arrayige the terms of the polynomials with reference to some letter; that is, to write them down, so that the highest power of that letter shall enter the first term ; the next liigliest, the second term, and so on to the last term. 41. How are the terms of a polynomial arranged with reference to a particular letter ? What is this letter called ? It the leading letter in the multiplicand and multiplier is the same, which will be the leading letter 111 the product? MULTIPLICATION. 61 The letter with reference to which the arrangement is made, is called the leading letter. In tlig above example the leading letter is a. The leading letter of the product will always be the saine as that of the fActors. 2. MiiUii)ly x^ + 2aa; -f- a^ by a; -}- a. Ajis, x3 + 3ax2 + 3a2aj + a^ 3. Multiply x^ -\- y^ by x -\- y, Ans. aj< + a^^ -f aJ^y + y*. '4. Multiply 3a62_^6(/V by Sa^^^ + 3c/V. Ans. ^d'b^.+ lld'h^c^ + 18aV. 5. Multiply d^ly^ -f ed by a -\- b. / Ans. a^b^ + (K^^d + a^i^ + bcH. 6. Multiply 3aar^ + Oo^^ -f- cc/* by 6aV. ^7W. 18a\'2{c2 + 54a V63 + ^a?-L^d'>. 7. Multiply 64aV + lla'^x + 9ai by %a\d. Ajis. 5\2a^cdx^ -f 216a*c(iB f- 12a*b€d 0. Multiply a^ + J^a^aj 4- Gax^ + x3 by a + x. ^ Ans. a* + 4a^x + Ca222 ^ 4^,5^53 4. ^^ 9. Multiply aj2 _|_ y2 y^y g. _|. y^ -4?i5. ar' + a;y2 4- 3^2 y _j_ ys^ 10. Multiply a:* + xif -)- 7«aj by ax -\- box. A?2S. Qax^ -f 6c/xy ^ 42a2aj> 11. Multiply a^ + Oa^j 4. zab^ 4. 53 i,y ^ ^ ^^ Ans. a* + 4a36 + Qa^b^ -r 4a^3 + &*. 12. Multiply ar* + a^^y + ary^ 4- y^ by a; 4- y. Ans. X* 4- 2ar''y + 2x-y^ -f 2a^5 ^. y4^ 13. Multiply ar3 4- 2x2 4- a: + 3 i,y 3aj 4- 1. A7is. Xr\ 4 7.r3 -f 'yx"^' 4- lOx 4- 3, 62 ELEMENTARY ALGEBRA. GENERAL EXAMPLES. I. Multiply . .• 2ax — 2,ah by 3a; — h. The product ^ax^— ^ahx becomes after — ^abx + 3ffJ^ reducing ... , . . . .^ax^— Wahx + Zah"^. 2. Multiply a* — W hj a — b. . A?is. a^ — lab^ — a^h -^ 25*. 3. Multiply {«2 _ 3^. __ 7 ])y a; _ 2. An?, x^ — 5x'^ — jc + 14. 4. Multiply 3a2 — 5ab + 2b^ by a^ — lab. Ans. 3a* — 2Qw^b + Zla'^b'^ — Uab\ 5. Multiply b^ + b^ -\- b^ by b''- — 1. A71S. b^ - b\ 6. Multiply £c*— 2x^y-{-^x^y'^—Qx\f-\- IGy* by a;+ 2?/. Ans. x^ -\- 32y\ 1. Multiply 4a!^ — 2y by 2?/. A^is. Qx^y — Ay"^. 8. Multiply 2x + 4y by 2x — Ay. Ans. Ax^ — l&y\ 9. Multiply x^ + x^y -f- cc?/^ -f ^3 i,j y^ _ y^ Ans. X* — y*. . 10. Multiply a;2 _{_ jf-y _|_ y2 j^y 2^2 _ ^.y _l_ ^2^ ylws. x^ 4- a;2y2 _j_ 2/4 II. Multiply 2a^ — 3«£c + Ax"^ by 5«2 _ g^,^ _ 2^32. u4ws. 10a* — 2la^x 4- 34a2cc2 _ iQax^ — Sx\ 12. Multiply 3a;2 _ 2i:cy + 5 by a;^ + 2a:y — 3. Ans. 8a?* + Ax^y — 4a;2 — 4a;y + IQxy — 15. 13. Multiply dx^ H- 2.'c"i/2 + Sy^ by 2a;3 — Scc^y^ ^ 5^3 j 6a;« - 5a;V^ — 6ccy + 6iry -f ^^* I ISccV — ^^V + lOtc'y^ -f 15y\ 14. Multiply Sax — Gab — c by 2ax + ab -\- c. A71S. IGa^a;^ — Aa'^bx — ^a^-b"^ -h Gac^. — labv, — c'^. DIVISION. DO 16. Mullfply 3a2 - 5^2 -f Sc^ by a^ - b\ Ans. d(i* - Sarh- + 3aV + 5fi* - W(^. 16. 3a2 _ tihd 4- C/" iVo.rcd. - 15a* + Z1a^bd-2^a\'f-2QbhP^A^hcdf-^c^P 17. Multiply arx — a'^b'^ by a^a;". 16. Multiply «*"+ i" by a*"— ^>\ -4/i3. a^** — li^\ 19. Multiply rt- + i" by «»" + 6". DIVISION. 4, "5, Dn'isK^N is the operation of finding from two quan- tities a third, which being multiplied by the second, will produce the first. Tlie first is called the Dividend^ the second the Divisor^ and the third, the Quotient. Division is the converse of Multiplication. In it^ we have given the product and one factor, to find the other. The rules for Division are just the converse of those for Multi- plication. To divide one monomial by another, 46. Divide 72a' by 8a^. The division is indicated, thus : 8a3 * Tlic quotient must be such a monomial, as, being multiplied by the divisor^ will give the dividend. Hence, the coefficient 46. Wh\t is division ? What is the first quantity called? The 9ec0r.fl? llic third • What is given in division? What is required ? 40. What is the rule for the dlviHion of moiwmiial}*? 64 ELEMENTAKV ALGEBRA. of the quotient must be 9, and the literal part (i^ ; for tliese quantities multiplied by %a? will give 72a^. Hence, The coefficient 9 is obtained by dividing 72 by 8; and tlie literal part is found by giving to «, an exponent equal to 6 minus 3. Hence, for dividing one monomial by another, we have the following RULE. I. Divide the coefficient of the dividend hy the coefficient of th,e divisor^ for a new coefficient : II. Aft at' this coefficient icrlte all the letters of the dividend^ giving to each an exponent equal to the excess of its expo- ponent in the dividend over that in the divisor. SIGNS IN DIVISION. 47. Since the Quotient multiplied by the Divisor must l)roduce the Dividend : and, shice the product of two factors having the same sign will be -f ; and the product of two factors having different signs will be — ; we conclude: 1. When the signs of the dividend and divisor are Uke, the sign of the quotient Avill be -{-. 2. When the signs of the dividend and divisor are unlike, the sign of the quotient will be — . Again, for brevity, we say, -f divided by +, and — divided by — , give + ; — divided by -f , and -f divided by — , give — , 4- «^ , , —ah 4- a ~ ' — a ~ 47. What is the rule for the pigiw, in divisiou ? BI V I 8I0N. 65 EXAMPLES. (1.) = I- 2a^b, (2) (3.) - 24a^ 4- 3adc = - 8a\ — 5u^x 32a^b'^x^ 8a'b^ 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 26. Divide Divide Divide Divide Divide Divide Divide Divide Divnde Divide Divide Divide Divide Divide Divide Divide Divide Divide Divide Di\nde Divide 15aa^y3 by — Say. Siab^x by I2b\ — 3Ga^6V by da^b^c, — Qda'^b^x^ by 11 a^b^x\ lOSjc^y^s^ by 54a:'^z. 64ic''y*2« by — 16a;*'j^s*. — 9Ga^5V by I2a^bc. — '3Qa*b^d* by 2a^b\l. — Qia^b*c^ by 32a**c. \28a^3f"y' by IBoary*. ' — 2bQa'b^c\V by 16a^6c«. 200a^wi2/i2 by — 50a''m?i. 300aryz2 by COajy^^, 2na^b'^c^ by -^ 9a*c. G4ayz® by 32a2/V. — 88a^^»6c8 by 11 a^^^c*. Ila^y^z} by — llaV^*- 8\a^b'^c^d by — 42a*b'^c^d. — 8%a^b\^ by 8a''^.«c«. 1Gj:2 i^y _ 8iB. — 88a"6^ by lla"»6. (4.) = — 4<2aj*. Ana, — 5a;2y2^ -^?w. 7 a^ a' a^' a3> ^» ^^ *^c., and apply the rule for the exponents, we shall have, a a? ' a^ ' But since any quantity divided by itself is equal to 1, it fol- lows that, - .= ao ^ 1, ^ r= a2-2 ^ f^o ^ 1 &c. ; or, finally, if we designate the exponent by m, we have, . — = cr""* = a^ =.1; that is. The po'uber of any quantity is equal to 1 : therefore, Any quantity may he retained in a term^ or introduced bito a tenn^ by giving it the exponent 0. EXAMPLES. 1. Divide Ga^^V by laW. — ^y- = 3^2-2^2-2^4 ^ 3^050^4 ^ 3^, 2a-b^ 2. Divide Sa^J^c* by — ^a'^h'^c. Ans. — 2a"Z^"6'* — — 2c'*. 3. Divide - ?>1mVx^y'^ by Amhi^xy. Ans. — ^m^n^xy — — 8,Ty. 49. When the exponents of the same letter in the dividend and divisor are equal, what takes place ? May the letter still be retained ? With what exponent? What is the zero power of any quantity equal to? DIVISION. 69 4. Dii-ide — 96a. A. ^a%^c^d\f'^. 50. WTien the exponent of any letter is greater in the divisor tlian it is in the dividend, the exponent of that letter in the quotient may he written with a negative sign. Thus, ;-^, by the rale; b X a-^ = -^, I , «^ hence, •-=i Since, a - 3 = — , we have. a' that is, a in tlie numerator, wit!) a negative exponent, is equal to a in the denominator, with an equal i)08itive ex- ponent; hence, Any quantity having a negative exponent, is equal to tfi^ reciprocal of the same quantity with an equal positive ex- ponent. Ilence, also, Any factor may be transferred from the denominator to the numerator of a fraction, or the reverse, by changing the sign of its exponent. ^ EXAMPLES. ^5^ 1. Divide Z2a^bc by IGa^R , 32a2^>c ^ _-._, 2c ^^- 16^^ = '^ ^ '' = a^h- 50. When the exponent of any letter in the divisor is greater than in the dividend, how may the exponent of that letter be written in the quo- tient ? What is a quantity with a negative exponent equal to ? How may a factor be transferred from the numerator to the denominator of a fraction ? 70 ELEMENTARY ALGEBRA. 3. Reduce ^, f „ • ^;i5. , or — z I blx^y^ 3 3a;^ f 4. Ill 5ai/~^x~'^, get rid of the negative exponents. A ^^ ^' Iji _3t_5 ? get rid of the negative exponents. Aois. 3£C2. 6. Li _^ - _g _ , get rid of the negative exponents. da c cr 7. Reduce ^^ „,_, — Ans. , or --^^- • 9 8. Reduce l^a^Jy^ -i- 8a^&^ ^^s. 9a~i5~\ or — r. 15(^ — 4^6^—1 9. In -2h-\ ' S^^ ^^^ ^^ ^^® negative exponents. Ans. -^' 10. Reduce z^ — =— • Ans. Sab^c^, — ba~^b~^ To divide a polynomial by a monomial. 51. To divide a polynomial by a monomial : Divide each term of the dividend^ separately^ by the divisor ; the algebraic sum of the quotients will be the quo- ilent sought, EXAMPLES. 1. Divide ZaW - a by a. Ans. ZaV^ — 1. 51, How do you divide a polynomkil by a monomial? DIVISION. 71 2. Divide ba^b"^ — 2oa*b^ by 5a'b\ Ans, 1 — 5a, 3. Divide 35arb^ — 25ab by — 5ub. Ans, — lab + 5 i. Divide lOab — I5ac by 5a. A713, 2b - 3c. 5. Divide 6a^ — 8aa; -f 4a2y by 'ia. ^w*. db — 4x -i- 2ay, 6. Di\nde — ISaa;^ + coj^ by — 3a;. A7\s, 5ax — 2x\ 7. Divide - 2\xy^ -f Sba^b^i/ - Ic^j by - 7y. yl;!.*?. 3iry — ha'^b^ + c*. 8. Divide iOa^b* + 8a*6' - 32a^i*c* by Qa*b\ Ans. on* -\- b^ — 4c* DIVISION OF rOLYNOinALS. 52. 1. Divide — 2a + Ca^ _ 8 by 2 + 2a. Dividend. Divisor. 6a2 — 2a — 8 | 2a. + 2 ba^ 4- 6a 3a — 4 Quotient — 8a — 8 — 8a — 8 Bemainder, We first arrange the dividend and divisor with reference to a (Alt. 44), placing the dinsor on the left of the dividend. Divide the first term of the dividend by the first term of the divisor; the result will be the first term of the quotient, which, for convenience, we place under the divisor. The product of the divisor by this terra (Ba^ -t- Ca), being sub- tracted from the dividend, leaves a new dividend, which may ne treated in the same way as the original one, and so on tn the end of the operation. 52. What is the rule for dividing one polynomial by another ? Wlicn if the division exact t When is it not exact f 72 ELEMENTARY A L O E B E A . Since all similar cases may be treated in the same way, we have, for the division of polynomials, the following E ULE. I. Arraiige the dimdend and divisor with reference to the same letter: n. Divide the first term of the dividend hy the first term cf the divisor^ for the first term of the quotient. Multiply the divisor hy this term of the quotmit, and subtract the product from the dividend: ni. Divide the first term of the remainder hy the first terrn of the divisor^ for the second term of the quotient, Mxdtlply the divisor hy this term, and subtract the product froTR the first remahider, and so on: rV. Continue the operation, until a remainder is found equcd to 0, or one ichose first term is not divisible by that of the divisor. Note. — 1. When a remainder is found equal to 0, the division is exact. 2. "Wlien a remainder is found whose first term is not divisible by the first term of the divisor, the exact division is impossible. In that case, write the last remainder after the quotient found, placing the di\'isor under it, in the fonn of a fraction. SECOND EXAMPLE. Let it be required to divide 51a'^^>2.4. loa* - 4.Qa^b - 155^+ ^ab"^ hy ^ah - Sa^ -f- zh\ We first arrange the dividend and di\^sor vnth reference to a. DIVISION. 73 IXuidend, 1 Oa* - 4 M^b -h 5 1 a^h'^ + Aab^ - 1 55* 4 10a*— ^a^b- Qd'b^ Divisor, 5a' + 4ad 4- 86* — 40a^b-^51a'^b'+ 4ab^-^l5b' ~ iOa^ + 32a^5^4-24a5^ 25a^^—20ab^^l5b^ 2oa^b^—20ab^—l5b^ — 2a^+ Sab — 56» (3.) aj« 4- Jc3y 4- a;2y + ay2 _ 2y | g 4- y jc* 4- ar'y 0^4- scy ' 4- x^y -f a?/* 4- x^y 4 ccy^ 2y - 2y Here the division is not exact, and the quotient is frac- donoL (4.) 1 4- a 1 - a 1 4- 2a -I- 2a2 4- Sa^ + , &c. -f 2a 4- 2a — 2a^ 4- 2a2 4- 2a^ — 2a^ 4- 2a3 In this example the operation docs not terminate. It may be continued to any extent. BXAMP'^BS. 1. Divide a' -I- 2cw; + x^ by a 4- «. Ana. a -^ x, 2. Divide a' — 3a^y 4 3ay2 — y^ by a — y, Ajis. a* — 2ay 4- y*. 74 ELEMENTARY ALGEBKA. a. Divide 24a'^6 — \2aW'' — eab by — Cab. Ans. — 4a 4- 2a-cb -f 1. pi 4. Divide 6a^ — 96 by Sx — 6. Ans. 2x^ 4- 4a;^ 4- 8x -f- 16. ^- 6. Divide a^ — 5a*a; 4- lOa-^x^ — lOa'^ic^ 4- 5a2?* ~ x^ by a^ — 2gkb 4- a^. -4/is. a^ — 'Sa'^x + dax^ — vK / 6. Divide 48ic3 — 76«ic^ — 64a2a; 4- 105a^ by 2ic — 3a. Ans. 24x^ — 2ax — ^5a\ '7 Divide y^ — Sy^x"^ + Si/'^x'^ — x^ by y'-^ — Sy'^x 4- 3yx^ — x^. Ans. y^ + dy'^x + Zyx^ 4- x^, ^ 8. Divide 64a^^>«> — 2bd%'' by 8a-6^ 4- ^ab\ Ans. 8a^b'^ — bab^, 9. Divide Ca^ -\- 2^a^b + 22ab'^ + bb' by 3«-^4- 4^/^4-R ^l?^6\ 2a + 56. ^^ 10. Divide 6aaj« 4- Oc/ir^y^ 4- 42^/2,^.2 by^aic^^js^^ A') IS. x^ + crv/ 4- 'J(}X.^ 11. Divide — 15a^ 4- ^Icfbd — 2^da\f — 2()b'-d' 4- 446('4/ ' - 8cy^ ])y 3t<2 _ 5^ J ^ (.y^; ^y^^s-. - 5a^ 4- 4^^/ - 8c/. 12. Divide oc* 4- aY^ 4- y* by «'' — x,t/ + t/^. J.y/6'. £c^ + a;y/ 4- y'^, 13. Divide x* — y* by £c — y. ^/is. aj^ 4- x^y 4- £cy'^ + y\ 14. Divide 2a'- 8a^b''-\- 3aV4- ^b*- 36V by a2_ ^2 Ans. Sa^ — 56^ 4- 3c". 15. Divide 6^« - hxHf' - Cx^y^-k- Cx^y'^ IbxY— 9a;V -f 10a;'y 4- 15y' by 2x^ + 235^2/^ + 32/2. Ans. 2x^ - ^x^y'^ 4- 5y^ 16. Divide — c^-f- 16a"^a;''^— nabc — 4ry%/; — 6^/2*24 6a«z by 8aa; — 6a6 — c. Ans. 2ax 4- «6 -f c. lY. Divide 3x^ 4- ^o^'y — 4a;'' — ^x'y'- 4 lOa-y - 15 b> 2arx^ — 2iKy t 5 ' DIVISION. 75 18. Divide a^ 4- 32?/' by x -\- 2y. Ans. X* — 2»«y + 4a^y2 - Sjcy^ ^ jgyH^ 19. Divide 3a* — 2^a^b — Wab'^ + 37a2^»''' by 26^ — 6a6 ^ 8a^ ^n«. a^ — 7a^. 20. Divide a* - 6* by a^ + a^i + ab^ + i^. -4y^w5 the square of the second. 1. Find the square of 2a — b. We have, (2a ~ by = 4a2 — Aab + b'^- 2. Find the square of 4ac — be, Ans. 16aV — ^abc^ + b'^c^. 8. Find the square of la^b"^ — 12aR Ans. 40a*b* - USa^^ -f 144a2^»«. (3.) 56. Multiply a -{- b hj a — b. We have, (a -f- 6) X (a - 5) = a2 — ^>2, Hence, 77/e .«v7/? c>/* ^?c. ^7?.?. 81 a V — 9*V 66. What is the square of the difForcnce of two quantities equal to? 66. What is the sum of two quiintiticf) uultiplicd hy their differcnuo eqoal to ? 78 ELEMENTARY ALGEBRA. 3. Multiply 8a3 -f- ^ah"^ by Sa^ _ ^obK Ans. UaP - i-:)a,W 57 • IVrUtiply a^ ^- ah + 6^ by a - b. We have, {a? -{- ab ^ b"") (a - b) = a' - b\ (5.) 58. Multiply a^ — ab -\- b"^ hy a -\- b. We have, (a2 - ab ^ 62) {a ^ b) ^ a? -^ bK (6.) 59. Multiply together, a + J, a — b^ and a? -(- 5'. We have, c»/ ^ ^ (a + ^>) (a - J) (a2 + ^2) ::::, ej4 _ ^4^ 60. Since every product is divisible by any of its factors, each formula establishes the principle set opposite its number. 1. The Slim of the squares of any two quantities^ plus ttoice their product, is divisible by their sum. 2. The sum of the squares of any two quantities, minus twice their product, is divisible by the differ e7ice of the quantities. 3. The difference of the squares of any two quantities is divisible by the sum of the quantities, and also by their difference. 4. The difference of the C7d)es of any two quantities is divisible by the difference of the quantities ; also, by the sum of their squares, plus their product. 5. The sum of the cubes of any two quantities is divisi 60. By what is any product divisible ? By applying this principle, wha^ follows from Formula (1 ) ? What from (2)? What from (3) ? What from {4:)'i> What from (5)? What from (6)? FACTORING. 7JI We by the sum of the quantities ; also., by the sum of the.ii squares minus th^ir product, G. The difference between the fourth powers of any t^no quantities is divisible by the sum of the quantities^ by their differetice^ lyy the sum of their sqv^reSy and by the dif jcTcnce of their squares. FACTORING. 61. Factoring is the operation of resolving a quantity into factors. The principles eni])loy(Hl are the converse of those of INIultiplication. The operations of factoring a>e performed by inspection. 1. Wliat are the factors of the pol}'nomial ac 4- ob + dd. We see, by inspection, that « is a common factor of all the terms; lience, it may be ])laced without a parentliesia, ajid the otner pans within ; thus : ac -^ ah •\- ad — ^c + ^ + d\ 2. Find the factors of the polyL nmial a'^b'^ -f- fi'^d — a\f. Ans. a'^{b'^ -f d — f), 3. Find tne factors of the polynomial ^a^b — Ga'^b^ -I- b^d. A?is. b{^a^ - Qfi^b -f bd). 4. Find the factors of Sa^b — da'^c ~ \Sn^jy. Ans, Sd^(b — t^c — Qxy), 6. Find the factors of Qa^cx — 1 Sacoc^ + la&y — aoa^". Ans. 2ac{4ax — 9x^ + c*y — 15^/V**). 6. Fjictor 300**2^ _ eaWd^ -f- 1 Sal&^c^. Ans. Qa''b'^{5nr _ ,J* ^ ^c^). ^ 7. Factor I2c*bd^ — }5c\l* — 6c^d^f Ans. 3c2(/3(4c2^ _ 5cd - 2/). CI Whatip faotorinel' 80 ELEMENTARY ALGEBRA. 8. Factor \5a^hcf — lOabc"^ — Ihdbcd. Ans. babc{Za^f— 2S^ - bd). 62. When two terms of a trinomial are squares, and positive, and the third term is equal to twice the product of their square roots, the trmomial may be resolved into factors by Formula ( l ). 1. Factor a' + 2ah -f ly" A^is. [a + 5) (a -f- b), 2. Factor ia? + Vlab -f 9Z>2. ^ Aois. {2a + 31) (2a + 3^). 3. Factor 9a^ + I2ab + 4R A91S, (Sa + 2b) {3a + 2b). 4. Factor 4x^ -\- 8x -^ 4. Ans. {2x -f 2) {2x -f 2). 5. Factor 9^252 + l^abc + 4c2. A71S. {Sab + 2c) {dab -\- 2c). 6. Factor IQx^y^^ + IQxy^ + 4y\ Ans. {Axy + 2y'^) {4xy + 2y^). 63. When two terms of a trinomial are squares, and positive, and the third term is equal to minus twice their square roots, the trinomial may be factored by Formula (2). 1. Factor a^ _ 2ab + b\ Aiis. {a - b) {a - b). 2. Factor An?; — 4ab + R Ans. {2a — b) {2a — b). 3. Factor Oa^ _ Qac + €\ Ans. {3a — c) {3a — c). 4. Factor a'^x'^ — 4ax -f 4. A7is. {ax — 2) {ax — 2) 5. Factor 4x^ — 4:cy -f y"^- Ans. {2x — y) {2x — y) 62. Wbi'n may a trinomial be factored ? 68. When may a trinomial be factored by this method? KAOTORINO. 81 6, Factor ZQx^ — 24ay -f 4y2. .1;^.^. (Gx — 2.v) (6a; - 2y). 64. Wlien the twd terms of a l/momial are squares and have contrary signs, the biuoniial may l)e factored by FoiTiiula ( 3 ). 1. Factor 4c2 — b'i Ans, (2c -f b) (2c — b) 2. Factor 8la^c^ - db'c^. A?is. (0(fc + Uc) (Oac — Sbc). 8. Factor 64ntrary signs, the hinotnial may be fao'o'.oil l.y Formula ( 4 ). 1. Factor %d? — c^. Ans. (2a — c) (\a'^ a- e^), 2. Factor 2nd? — 64. Ans. (3a - 4) (Qa^ 4 1 _ / \- 16). 3. Factor a? — G^h\ Ans. (a — 4b) (a^ -f iab -f ^Gb^), 4. Factor a^ — 27R ^??.9. (a - 3^») (a' + 3or^ + Oi'O- r»4. Wlien may a binomial be factored ? <\6. When mnr a binomml bo fvjior<'3. ^??s. (a -f Ab) [a^ - iab 4- 166*^). 4, Factor a^ f 21 b\ A?}s. (a -f- 3*) (a^ _ 8or^> -f 9^>^). 67. When the terms of a binomial are 4th powers, and have contrary signs, the binomial may be factored by I Fonnula (6). 1 What are the factors of a" — b^? Ans. (a i- b) {a - b) [rf -f- -6^). 2 Wliat are the factors of 81a* - T6^;^ ? Ans. (3a + 2h) (3a - 2b) (Oa*^ + 4*^). 3. What are the factors of 16a-^/^^ — Slc^c/* ? Ans. {2(tb ^ 3c(0 (2^/^ — 3cc/) {ia'^b'^ + 9e''c/2)^ GUEATEST COMMON DTTTSOR. 68. A Common P«tvtsok ot two quantities, is a quantity that will divide them both wit^hout a remainder. Thus, ba-'^'J^is a common divisor of da'^b^c and 3aV/'* — Qa^P, fi^ When may a binomial be factored by this method? f>7. When may a binomial be factored by this method? 68. What is the oominon divisor of two anantitieg )r GREATEST COMMON DIVISOR. 83 69. A Simple or Prime Factor is one that cannot be resolved into any other factors. Every prime factor, common to two quantities, is a com- mon flivisor of those quantities. Tlie continued product of any number of piime factors, common to two quantities, is also a common divisor of those quantities. TO. The Greatest CoarMON Divisor of two quantities, IB the continued product of all the piime factors which are common to both. 71. When both quantities can be resolved uitb prinm fiictors, by the method of factoiing already given, the great- est conmion divisor may be found by the following rule. L Resolve both quantities into their prime factors : II. Fiiid tlie continued product of all the factors which are common to both / it will be the greatest common dv)i- 9or required. EXAMPLES. 1. Required the greatest common dinsor of I5a^^c and 25abd. Factoring, we have, Iba^b'^c = 3 X 5 X b'aabbc 2^nhd = 5 X babd. Tlie factors, 5, 5, a and i, are common ; hence, 5x5Xax6 = 25aft, IS the divisor sought. 69. What is a simple or prime factor? Is a prime factor, common to two quantities, a common divisor? 70. What is the grcHtcst common divisor? 71. If both quantities can be renolved into prime fnctore how do you 6ud the ijroatefit »onimoii divipory 84 EIEMENTARY ALGEBRA. VERIFICATION. nha^lP-c -^ Toah - Zahc Ihabd -V- 25aZ> = d\ and since the quotients lia\e no common factor, they cannot be further di\dded. 2. Required the greatest common divisor of a? — 2ab 4 h"^ and a? — IP-. Ans. a - b 3. Required the greatest common divisor of a^ + 2ab -\ b^ and a -i- b. A7is. a -\- b 4. Required the greatest common divisor of a^x^ — 4ax 4- 4 and ax — 2. Ans. ax — 2. 5. Find the greatest common divisor of 3a^& — Qa'^o •^ 1 %a^xy and b'^c — Sbc'^ — Qbcxy. Ans. b — Sc — Q>xy. 6. Find the greatest common divisor of Aa^c — iacx and iia^g — Zagx. Ans. a{a — x)^ or a^ — ax. 1. Find the greatest common divisor of 4c'^ — \2cx + Occ^ and 4c2 — Qx^. Ans. 2c — dx. 8. Find the greatest common divisor of x^ — y^ and a.2 _ y2^ Ans. X — y. 9. Find the greatest common divisor of Ac^ -f 45c -f b"^ and 4c2 — 6^. ^It?^. 2c -f- 5. 10. Find the greatest common divisor of 2ba\Z.— 9x^y* and r^acd^ + Sd^x^Y- ^^^' -«^^ -^- 3a;V. NoTE.-^To find the greatest con^mon divisor of three quantities. First find the greatest common divisor of two of tliem, and then the greatest common divisor betv>^een this result and the third. 1. What is the greatest common divisor of 4ax'^y, lQabx% and24aca.'^? Ans. ^ax^, 2. Of 3£c2— 6fc, 1x^— 4x\ and cc^ /_ ^xyl Ans. x"'— 2iC •72. When is one quauilty a tntiltiple of another? LEAST COMMON MULTIPLE. 85 LEAST COMMON MULTIPLE. 79, One quantity is a mlt>tiple of another, when it can be divided by that other without a remainder. Thus, Bd^d, U a multiple of 8, also of a% and of b. * 73. A quantity is a Common Multiple of two or moro quantities, when it can be divided by each, separately, witli- out a remainder. Thus, 24a^ar', is a common multiple of Goaj and 4a^£c. 74. The Least Common Multiple of two or more quan- tities, is the simplest quantity tliat can be divided by each, without a remainder. Thus, 12 a^b^x^, is the least common multiple of 2d^, 4ab\ and 6a^b^^. 75. Since the common multiple is a dividend of each of the quantities, and since the division is exact, tlie common multiple must contain every prime factor in all the quanti- ties ; and if the same factor enters more than once, it must enter an equal number of times into the common multiple. When the given quantities can be factored, by any of the methods already given, the least common multiple may be found by the following BULB. I. JResolve each of the qiicmtities into its prime factors : IT. Take each factor as many times as it enters any 07ie of the quantities^ and form the continued produi't of these factors / it will be the least common midtiple. 78. When is a quantity a common multiple of several others? 74. What is the lcj\st common multiple of two or m6re quantitins? 76. What does the common miiltiple of two or more qvianiities contain, ■IS factors? How may the least common multiple be found ? * Th« mulUpU + b"^. Ans. {a + ^')2 (a - b), 1. Sa^b\ Qa^x\ 18«y, Sa^y^. . Ans. IStrb^x'^yl 8. Qa%a i b), I5a\a - by, and 1 2^3(^2 _ ^2). . A/is. \20a^{a — b)'^ {a + S). FRACTIONS. 87 CILVPTER rV. FRACTIONS 76 If the iinit 1 be divided into any number of eqnal parts, tfacli pail is called a FRAcnoNAL umt. Thus, - , j, 11 2 4 ^ , y , are fractional units. 77. A Fraction is a fractional unit, or a collection of fractional units. Thus, - , - , - , ^ , are fractions. 7§. Every fraction is composed of two parts, the De- nominator and Numerator. The Denominator sliows into how many equal parts the unit 1 is divnded ; and the Nur inerator how many of tliese parts are taken. Thus, in the fraction i , the denominator J, shows that 1 is divided hito h equal parts, and the numerator a, shows that a of these parts are taken. The fractional unit, in aU cases, is equal to the reciprocal of the denominator. 7fi. If 1 be divided into any number of equal parts, what is each part called? 77. What is a fmction ? 78. Of how many parts is any fraction composed? What arc they oatled? What doesi the denominator show? What the numerator? VVhrii ie r.h»» frn<'tioii:il iiiiif eiju.il to? 88 E L E :M E N T A R Y ALGEBRA. 70. An Entire Quantity is one which contains no fractional part. Thus, 7, 11, a^x^ ix^- — 3y, are entire quantities. An entire quantity may be regarded as a fraction whose denominator is 1. Thus, 7 = - , a5 r= — • 80 A Mixed Quantity is a quantity containing both bx entire and fractional parts. Thus, 7p*o> ^^ » ^ H > ^^^ c mixed quantities. SI. Lf^t 7 denote any fraction, and q any quantity . d whatever. From the preceding definitions, - denotes that 7 is taken a times ; also, -r^ denotes that y is taken aq times ; that is, aq a , -~ — - X q\ hence, Multiplying the numerator of a fraction by any quan- tity^ is eqxdvalent to rnultiplymg the fraction by that quantity. We see, also, that any quantity may be multiplied by a fraction^ by multiplying it by the numerator^ and then dividing the result by the denonfiinator. 82. It is a principle of Division, that the same result will be obtained if we divide the quantity a by the product of two factors, ji? x 5', as would be obtained by dividing it 79. What is an entire quantity ? "When may it be regf.rded as a frao lion ? SO. What is a mixed quantity ? 81. How may a fraction be rauUipliod by any quantity ? 82 How may a traction be divided by any quantity ? TBANSFOKMATION OF FRACTIONS. 89 first by one of the fiictors, jt), and then dividing that result by the other factor, q. That is, — = (-)-f-^; or, — z= \-\ -^ p\ hence. Multiplying the denominator of a fraction by any quan iity^ is equivalent to dividing the fraction by that quantity 83. Since the operations of Miilti]>lication and Division are the converse of each other, it follows, from the preced ing principles, that, Dividi7ig the numerator of a fraction by any quantity^ is equivalent to dividing the fraction by that quantity ; and. Dividing the denominator of a fraction by any quantity ^ is equivalent to multiplying the fraction by t/uit qtiantlty, 84. Since a quantity may be multiplied, and the result divided by the same quantity, without alteiing the value, it follows that, Both terms of a fraction may be multiplied by any quaty- tity, or both divided by any quantity^ without changing the value of the fraction. TKANSFORMATION OF FRACTIONS. 85. The transformation of a quantity, is the operation of changing its form, without altering its value. The tenn reduce has a technical signification, and means, to IVaiw- form, 98. What follows from the preceding principles ? 84. What operations may be performed without alf^ring tho value of a fraction ? 86. What is the tranpformation of a quantity ? 90 ELEMENTARY ALGEBRA FIKST TRANSFORMATION. To reduce an entire quantity to a fractional form having a given denominator. 86. Let a be the quantity, and h the given denomi- nator. We have, evidently, a = — ; hence, the RULE. Multiply the quantity by the given denominator, and write the product over this given denominator, SECOND TRANSFORMATION. To reduce a fraction to its lowest terms. 87. A fraction is in its lowest terms^ when the numerator and denominator contain no common factors. It lias been shoAvn, that both terms of a fraction may be divided by the same quantity, without altering its value. Hence, if they have any common factors, we may strilie them out. RULE. Resolve each term of the fraction i?ito its prime fac- tors / then strike out all that are common to both. The same result is attained by dividing both terms of the fraction by any quantity that wilJ divide them, without a remainder ; or, by dividing them by their greatest common divisor. 86. How do you reduce an entire quantity to a fractional form having a given denominator ? 87. How do you reduce a fraction to its lowest terms ? TRANSFORMATION OF FRACTIONS. 91 EXAMPLES. I. Redace — — -, to its lowest terms. Canceling the common factors, 5, a, and c^ we have, 25ac5 — 1J A71S, 2. Reduce 8. Roe entire part by the denomirmtor of the frac- tion^ and add to the product the mmierator ; write the resuU over the denominator of the fraction, EXAMPLES. 1. Rednce 6^ to the form of a fraction. JO 6 X V = 42 ; 42 + 1 = 43 ; hence, 6^ =. y • Reduce the following to fractional forms : 2. X —^ ^ = X -a^) 2a? - a» Ans. < X 3. ax •+ Q^) la • . ax — Q? Ans. — 2a 4. -^- Am, — 3a; 5. ^ JB - a - 1 ^ a . 2a — a; -t- 1 Am, ^— a 6. 5a! A ns. 10.r2 4- 4ar + 3 hx 80. How do you reducp a mixed quantity to a fractional form? 94 ELEMENTARY ALGEBRA H ^ . r 3c 4- 4 ^ 16a -\- Sh -- Sc - 4 7. 2a + — • A?is. 8 8 ^ ^ . , Ga^a; — ad . 18a'^x + 5ab 8. 6aic + 6 Ans. 9. 8 + Sab — 4a 4a 8 + Qa'^b'^x\ I2abx* 9Qabx^ + 30^2520;* — 8 A?is. I2abx^ FIFTH TRANSFORAtATION. To reduce fractions having different denominators^ to equi- valent fractions having the least common denominator. 90. This transformation is effected by finding the least common multiple of the denominators. -10 c 1. Reduce -, -, and — , to their least common denomi- nators. The least common multiple of the denominators is 12, which is also the least common denominator of the required fractions. If each fraction be multiphed by 1 2, and the resiill divided by 12, the values of the fractions will not be changed, - X 12 = 4, 1st new numerator ; g - Xl2 = 9, 2d new numerator : 4 — X 12 = 6, 3rd new numerator ; hence, 4 9^5 ^ . , ^ . -— , -— , and — are the new equivalent tractions. \ i \ 2d J. Z 00. How (10 you reduce fractions having different denominators, to equi \ alent fractions having the least common denominator ? When the nu- merators have no common factor, how do you i educe tiiem ? TRAHSFOKMATION UF FKACTIONS. 95 BULB. L J^nid the least common multiple of the denomwatora : II. Multiply each fraction by it., and cancel the denon^ inaioT : m. Write each product over the common multiple, and tjuj res'ults will be the required fractions, GENE HAL BULE. Multiphj each, numeraior by all the denominators except its own, for t/ie new numerators, and all the denominati/rs togetfierfor a common denominator. EXAMPLES. a c 1. Reduce -r rs and — ; — = to their least common a} — b^ a -\- b denominator. The least common multiple of the denominators is (a + J) ('/-ft): ^-^2 X (a + 5) (« - 5) = a — — T X {a -\- b) {a — b) = c{a — b; hence, and —z \/ — T\y are the required (a + b) {a — b) {a + b) (a - b) fractions. Reduce the following to their least common denominators ; ^ 3a; 4 ^ 12«2 45a. 40 48a;» ^' 4' 6» '""^ ■T5-- ^^^'- -eo'e-o'lcr 8.C/, --, and -5-. Ar^. — , -, -- ^ Zx 2b ", . Of-a: 4ab Qacd 4. —- , — , and d. Ans. - — , - — , — — 2a' 3c 6ac 6ac Que 96 E L E M E N T A li Y A L G K B R A . (X. 3 2x 2x . 9a Sax I2a^ -\- 2ix '• 4' ¥' « + a • ^'''- 12^' m' ^^m"^' a .r-2 6. ' — : , -— rr, , and 1 - x\{l - xy^' (1 - xy- ' x(l — xY a;2(i - x) , x^ . c c — b , c 7. -^ , , and — -—=■ ' ac^ + bc^ 5a^c — 5a^b + 5ab c — 5ab^ 5ac^ 5tt-c + 5abd' 5a^c -\- babe ' ba^c + babe ' ex dx^ ' x^ o. , — - — , and a — a^'aH-a:' a -\- x . cxia-\-x) dx^ia—x) \ x^ia — x) Arts. -\--^ , — ^ ^ , and -\ ^ a^ — x^ a^ — x^ a^ — x*- ADDITION OF FEACTIOISTS. 91. Fractions can only be added when tliey have a com- mon unit, that is, when they have a common denominator. In that case, tlie sum of the numerators av411 indicate how many times that unit is taken in the entire collection. Hence, the RULE. I. Reduce the fractions to be added, to a common denom^ inator : n. Add the numerators together for a new numerator^ and write the sum over the common denominator, EXAMPLES. 6 4 2 1. Add -, -, and -, together. 2 3 5 91. Whftt ip the rule for adding fractions ? ADDITION OF FRACTIONS 97 By reducing to a common denominator, we have, 6 X 3 X 5 = 90, Ist numerator. 4 X 2 X 6 = 40, 2d numerator. 2x3x2 = 1 2, 3d numerator. » 2 X 3 X 6 = 30, the denominator. Hence, the expression for the sum of the fractions becomes 90 40 12 _ 142^ 80 30 "^ 30 "~ 30 *' which, being reduced to the simplest form, gives A\^. (Z C B 2. Find the sum of y> -t> and -• a f Here, a x d x f — acif ) c X b X f = cbf e X b X d = ebd bnd b X d xf = bdf „ adf , cbf , ebd adf + cbf + ebd . the new numerators. the conMnon denominator, sum. Add the following : 3aj2 _ ^ , 2flKB 3. a =- , and b + o c Ans. a + b + 2abx — 3ca^ X X 2' 3' SB and -2 - 4a; - — and r-» 3 7 6. a; H — and 3a; -\ — ^ . ^^ ^ « + « ^- -*«, — , and Ans. X + Ans. Ans. Ax + be X 12 19a; — 14 21 10a; — 17 8. 2a; 2a 1x 3' T and 2a; 2x 4- 1 Ans. Ax 4- Ans, 2x -h 12 5x^ + ax -^ a^ 2nx A9x -I- 12 60 9. 4(8, ~, and 2 -}- | Ans. 2 4- 4a 4- 44a: '45' 98 'elementary algebra. 10. 3a; + ^ and a; - ^, Ans. 3a; + — 5 9 '45 11. ac — -- and 1 -,- 8a <^ Ane, 1 + ac — >^ 8at? q q J ^715. -— {X - 1)3 13. rr— - — r, -7- r, and 777- ^. • -4?15. 4(1 + a)' 4(1 - a) ' " 2(1 ^- a^) I - a^ SUBTEACTION OF FKACTION'S. 92. Fractions can only be subtracted when they have the same unit; that is, a common denominator. In that case, the numerator of the minuend, miiius that of the sub- trahend, will indicate the number of times that the common unit is to be taken in the difference. Hence, the KULE. I. Reduce the two fractions to a common denomi- inator : n. Theii subtract the numerator of the subtrahend from that of the minuend for a new numerator^ and write the remainder over the common denominator. EXAMPLES. 3 2 1. VThat is the difference between - and - - 7 8 3 2_24_U_10_^ . 7 "~ 8 ~ 56 ~ 56 ~ 56 "" 28 ' ^^' 92. What is the rule for subtracting fractions MULTIPLICATION OF FEA0TI0N6. 99 ic — cb 2<2 — 4cc 2. Find the diiference of the fractions -—rr— and — 26 3c TT S (a; — a) X 3c = 3caj — 3ac ) . Here, 1 ,„^ , ( „, , , „. Khe numerator?, ' {{2a — 4a;) x 26 = 4a6 — 86a; ) and, 26 X 3c = G6c the common denominator. _- Sob— Sac 4a6— 86a; 3ca;— 3«c— 4a6-{-86a; . Hence, — --. -r = —: — -—^ Aiis, ' 66c 66c 66c 3. Required the difference of -r-- and — - • Ans, —^ • ^ '7 5 35 4. Required the difference of 6y and -^ • Ans. -~ • 6. Required the difference of — and — • Ans, -— • (J. From 't±JL subtract ^-=^ • Ans. ■ 4^, • a; — y x -{- y x^ — yK 7. From \'" subtract -r -• Ans. ^ — r- • y — z y^ — z^ y^ — '^ Find the differences of the folloTsing : ^ 3a; + a , 2a; 4- 7 . 24a; + 8a — 106a; — 356 8. — -£ — and — —— • Ans, rr^ 56 8 406 ^« .a; - a; — a . ^ , ex -^ bx -- ab 9. 3a; + J and x Ans. 2x -{ z » p c oc -^■^ ,a — a; ,a + a; . 4a; 10, a H — 7 — ; — r and —, :• Ans, a :; :^- a(a + X) a{a — a;) a^ — ar MUlfflPLIOATION OF FRACTIONa CL G 93» Let T and -^ represent any two fractions. It hsjt\ been sliown (Art. 81), that any quantity may be multiplied 98. What is the rule for the multiplication of fractions f iOO ELEMENTARY ALGEBRA. by a fraction, by first multiplying by the numerator, and then dividing the result by the denommator. To multiply j- by - , we first multiply by c, givmg ~ then, we di\^de this result by c7, which is done by multiply- CLG ing the denominator by d ; this gives for the product, j-^ ; that is, a c aQ , RULE. I. If there are mixed quantities^ reduce them to a frao lional form ; then^ n. Multiply the numerators together for a new numera- tor^ and the denominators for a new denominator. EXAMPLES. 1. Multiply a -\ hy -• First, a -\ = , a a a a a"^ + hx c a^c •\- hex . hence, X -, = ^ — • Ans. ^ a d ad Find the products of the following quantities : ^ 2a; 3a5 , Zac . 2. — , , and — =- Ans. 9ax, a' c ^ 2b ^ ■. , hx ^ a . ah -\- hx 3. H and -• Ans. • ax X , x^ — J2 ^ a;2 + 52 . a;* — 5* 4. — r — and -^ • Ans. be b + c ' bH -\- bc^ aj-fl ^ X — \ . ojx^ — ax 4- x^ —1 6. a H , and — — r* Ans. -r— — -, ■ a ' a -^ b a^ -\- ah ■ ■ , ax * , a^ — a;2 . a^ 4- «^ 6. a j ana — ; — r-- Ans. ; — ^ a — X x -\- x^ x -\- x^ MULTIPLICATION OF FBAOTIONB. 101 7 Multiply 7 by • Of — U O We have, by the rule, 2a g}- b^ __ 2a(a^-, &^) . . .^(^a. ^ .b\ (a - b) a - b ^ 3 ~~ S{a'— b) "^i ' '■ . •' >^aV; f/). = f (a + i). After indicating the operation, we factored both numera* tor and denominator, and then canceled the common factors, before performing the multiplication. This should be done., w/ienever there are common factors. 8. 2 . x^-y' ^m, 2(^ + 2'' X — y~ *' a a 9. a;2 _ 4 Ax 3 ^ a; + 2 A^. "<'-». 10. {a^by Ax^ 2x ^ {a + b) Arts, 2x{a + b\ 11. y3 ^ x-l y 12. 1 - a;2 ^ a -{- X 4 a — X Arts. • 1 — X 13. x^ 2.y ^ 2xy x-y x-Yy Arts. a^. 14. 2a - b 6a — 25 4a ^^ 62 _ 2a& . i - 3a Ans, ^ . ' ^ 2ab .'^ 14. x-?^' by ?f2/. a: "^ y a Ans, — — ^ ^y 102 ELEMENTARY ALGEBRA, DIYISIOlSr OF FRACTIOIJ^S. 94 Since. -::;=, p X - , >t follows that, dmding by a (juantity is equiialeut to multiplying by its reciprocal. Bnt c d the reciprocal of a fraction, -,, is - (Art. 28); conse- ci c quently, to divide any quantity by a fraction, we invert the terms of the divisor, and multij^ly by the resulting fraction. Hence, a c _ a d _ ad b ' d ~ b c ~ bo Wlience, the folloT\ang rule for dividing one fraction by another : RULE. I. Reduce mixed quantities to fractional forms : II. Invert the terms of the divisor, a7id m^idtiply the dividend by the resulting fraction. Note. — ^The same remarks as were made on factoring and reducing, under the head of Multiplication, are appli- cable in Division. EXAMPLES. 1. Divide a — — by - • 2c -^ (7 b 2ae a — — 2c 2c „ b f 2ac — b g 2acg — bg . Hence, a r- -^ = • — x '^ = — ^-z-— ^- ^'^^ 2c ^ 2g f 2cf 94. What iG the rule for the division of fractions? DIVISION OF FRACTIONS. #2 103 2. DiM(le -A_2_^Z by 2(a -4- y) « a ^ a.2 _ y2 2(a; 4- y) (a; -f y) (aj _ y) jc — y ^?i5. 3. Let — - be divdded by — • 5 13 4a.2 4. Let -TT- be divided by 6a;. 5. Let be divided by -— 6 3 8. Let be di\idcd by - a; — 1 2 7. Let — be divided by , 3 oo 8. Let -~j- be divided by —j ^n«. 91^ 60 A ^» Ans. Ans. g + 1 4a; 2 a; — 1 . bhx, Ans. — — 2a /l71S. X — h Cc^a; Divide the folloTiTng fractions: 4a;2 - 8a; , a;^ - 4 0. by — -— 10. 3 a;* - J* by x^ -\- hx, x^ — 2bx -{- b"^ "'' X - b 4a:2 11. 2a;(a -h b) by a + 6 y a; - 1 a^ — ax . 3(c — a) 13. be -^ bx ^^^ 4(a -h a;) J 4^ v^ a; H- 2 "^ Ans. X -{- --■ / (a + hy Ans 2x Ans. ^ ^r—^-. ^ yd . Zb(c^ — a;2) 104 BLEMENTAIir ALGEBEA. a — X . 1 -f « 14. -— by ^ 1 — a? ^ a + X 2x1/ 15 x^ by X X + y Ans, - 1 — x^ Ans. ^^±^. _ b — 3a Qa — 2b 16. ^ . Dy 2ab b^ - 2ab x^y y X Ans, A71S, 2a — b 4a x^— y^ X8./C^+l+^^V^ + ifl Ans. m -\ 1. m IP (x + f^) by (i -4^)' ^^' y- \ 1 + xyj '' \ 1 -h ccy/ ^ ^ 20. /i±-?^ -f ?) by (^±i^^-4-) • An& 1 BQUATIOMB OF THE FIRST DEGREE. 106 CHAPTER V. EQUATIONS OP THE FIRST DEGREE. 95. An Equation is tlie expression of equality between two quantities. Thus, X = b + Cy is an equation, expressing the fact that the quantity x, is equal to the sum of the quantities b and c, 96. Every equation is composed of two parts, connected by the sign of equality. Tliese parts are called members : the part on the left of the sign of equality, is called the^rs^ member ; that on the right, the second member. Thus, in the equation, X + a = b — Cy a; -h a is the first member, and b ^ c, the second member. 97. An equation of the Jirst degree is one which involves only the first power of the unknown quantity ; thus, 6a; + 3a; — 5 = 13; (1 ) and ax -\- bx -\- c = d; (2) are equations of the first degree. 95. What IP an equation? 96. Of how many parts is every equation composed? How are the partfl connected ? What are the parts called ? What is the part on the ■ left called? The part on the right? 97. What ip an equation of the first degree 5* ICG ELEMENTARY ALGEBRA. 98. A NUATERiCAL EQUATION is ODG in Trhich the coefR. cients of the unknown quantity are denoted by numbers. 99. A LITERAL EQUATION is One in which the coefficients of the unknown quantity are denoted by letters. Equation ( 1 ) is a numerical equation ; Equation ( 2 ) is a literal equation. EQaATIONS OP THE FIRST DEGREE CONTAINING BUT ONE UNKNOWN QUANTITY. 1 00. The Transfor:mation of an equation, is the opera- tion of changing its form without destroying the equality of its members. 101. An Axiom is a self-evident proposition. 102. The transformation of equations depends upon the following axioms : 1. If equal quantities he added to both members of an equation, the equality will not be destroyed. 2. If equal quantities be subtracted from both mewhers of an equation, the equality will not be destroyed. 3. If both members of an equation he multiplied by the same quantity, the equality will not be destroyed. 4. If both members of an equation be divided by the same quantity, the equality will not be destroyed. 5. like powers of the two members of an equation are equal. 6. Lilze roots of the two members of an equation are equal. 98. What is a numerical equation ? 99 What is a literal equation ? 100. What is the transformation of an equation ? 101 What is an axiom? 102, Name the axioms on which the transformation of an equatior depends. OLEABINO OF FHACTION8. 107 103. Two principal transformations are employed in the solution of equations of the first degree: Clearing of frac- tions^ and Transposing, CLR-VErNG OF FRACTIONS. 1, Take the equation, 2a; 3a; a; The least common multiple of the denominators is 12. If we multiply both members of the equation by 1 2, each term will reduce to an entire foim, giving, 8a; — 9a; 4- 2a; = 132. Any equation may be reduced to entire terms in the same manner. 104. Hence for clearing of fractions, we have the fol- lowing RULE. I. Find the least common multiple of the denominators : n. 3fultiply both members of the equation by it, reduc- ing the fractional to entire terms. Note. — 1. The reduction will be effected, if we divide the least common multiple by each of the denominatoi-s, and then multiply the coiTCsponding numerator, dropping the denominator. 2. The transformation may be effected by multiplying each numerator into the product of all the denominators except its ovnti, omitting denominators. 108. How many transformations Arc employed in the solution of equa- tions of the first degree? What are they? 104. Give the rule for clearing an equation of fractions ? In what three ways may the redtiction be cfTocted? 108 ELEMENTARiT ALGEBEA. 3. The transformation may also be effected, by multiplying both members of the equation by any multiple of the de- nominators, EXAMPLES. Clear the following equations of fractions: 1. ^ -h ^ — 4 = 3. Ans. 1x -{- 5x — 140 = 105 5 7 2. t; -\- - — ~ = S. Ans. 9x + 6x — 2x = 432. b 9 27 3. X X X X 2 + 3-9+Ii = ^°- Ans. 18a; -f 12a; — 4a; + 3a5 r= 720. 4. 1* o* d* - + z — - = 4. Ans. 14a; -j- 10a; — 3oa; = 280. 5. - — - -h - = 15. Ans. 15a; — 12a; -}- 10a; = 4 5b a; — 4 a; — 2 5 900 6. 3 6 ~ 3 Ans. — 2a;+8 — a;-|-2 - = 10 7. X 3 \ A — , An" 'it* I fin "Ot» — - 3a;. 3 — aj 5 8. ?_? + ?+?= 12, 4 6 ^ 8 ^ 9 ^715. 18a; — 12a; -f 9a; + 8a; = 864. 9. -0. ^ - 1 4- / = 5^. Ans. ad - bo + bdf = ax 2c'^x , , 4^c2a; 5a^ . 26^ -T- 7- + 4a = — z -Tz- -\ 3&. b ab a^ b^ a bdg The least common multiple of the denominators is a^b^ a*bx— 2a^bc^x + 4a*b^ = 4b^c^x — 5a« -j- la^b'^c'^ - ^a^bK TBAN8P06INO. 109 rBANSPOsmo. 105. TfiANSPOsrnoN is the operation of changing a terra from one member to the other, without destroying the equality of the members. ) . Take, for example, the equation, 6aj — 6 = 8 H- 2aj. I^ in the first place, we subtract 2a from both members- the equality will not be destroyed, and we have, 6a; — 6 — 2a; = 8. Whence we see, that the term 2a;, which was additive in the second member, becomes subtractive by passing into the first. In the second place, if we add 6 to both members of the last equation, the equality will still exist, and we have, 5a; — 6 — 2x + 6 = 8 + 6, or, since — 6 and -f 6 cancel each other, we have, 5a; — 2a; = 8 4- 6. Hence, the term which was subtractive in the first member, passes into the second member with the sign of addition. 106. Therefore, for the transposition of the terms, we have the follo^ving RULE. Any term may be transposed from one member of an equation to the other, if the sign be changed, 106. What is transposition ? 106. What is the rule for the tranHpoeitioD of the terms of an equation? 110 ELEMENTARY ALGEBRA. EXAMPLES. Transpose the unknown terms to the first member, and the known terms to the second, in the following : 1. 3a; + 6 — 5 = 2a; — 7. Ans. 3a; — 2a; = — Y - 6 -f 5 2. ax -\- b = d — ex, Ans. ax -\- ex = d ~ h 3. 4a; — 3 = 2a; + 5. Ajis. 4a; — 2a; ~ 5 + 3, 4. 9a; + c — ca; — d. A?is. 9a; — cjb = — d — e, 5. ax -i- f =z dx -{- b. Ans. ax — dx — b — f. 6. Qx — c = — ax -\- b. Ans. Qx -\- ax = b -{- c. SOLUTIOl^ OF EQUATIONS. lOT. The Solution of an equation is the operation of finding such a value for the unknown quantity, as mil satisfy the equation ; that is, such a value as, being sub- stituted for the unknown quantity, mil render the two mem- bers equal. This is called a root of the equation. A Hoot of an equation is said to be verified^ when being substituted for the unknown quantity in the given equation, the two members are found equal to each other. 1. Take the equation, |_4 = 1^^4-3. Clearing of fractions (Art. 104), and performing the operar lions indicated, we have, 12a; — 32 = 4a; — 8 + 24. 107. When is the solution of an equation? What is the found value of the unknown quantity called ? When is a root of an equation said to be verified. SOLUTION OF EQUATIONS. Ill Transposing all the unknown terms to the first member, and the known terms to the second (Art. IOC), we have, 1205 - 4a; = - 8 + 24 -h 32. Reducing the terms in the two members, 8iB = 48. Dividing both members by the coefficient of Xy 48 X = - = 6. VERITICATIOX. 3X6 , 4(6 - 2) , ^ -2--^= 8— + '' or, + 9 — 4 = 2 + 3 = 5. Hence, 6 satisfies the equation, and therefore, is a root. 108. By processes similar to the above, all equations of the first degree, containing but one uukno>vn quantity, may be solved. BULB. I. Clear the equation effractions^ and perform all tho indicated operations : II. Transpose all the unknown terms to the first member, and all the known terms to the second member : in. Reduce all the terms in the first member to a single term, one factor of which will be the unknown quantity, (fiyrl fhn other factor will be tJie algebraic sum of its coeffu- (■(■ /I '■• : IV. Divide both members by the coefficient of the unknown quantity : the second member will tJien be the value of t/ie wiknown quantity. 108. Give the rule for aoIviDg equations of the fir?t degree with ouf tinkDown quantity. 112 ELEMENTARY ALGEBRA. EXAMPLES. 1. Solve the equation, 5x 12 4x y 13 _ 1 "" 8 13aj • 6 Clearing of fractions, 10a; — 32a; — 312 = 21 - 52a;. By transposing, 10a; — 32a; + 52a; = 21 4- 312. By reducing. 30a; = 383; u ^^^ 111 hence, a; = — = — = 11.1; a result which may be verified by substituting it for x in the given equation. 2. Solve the equation, (3a — x) {a — b) + 2ax — 4l{x + a). Performing the indicated operations, we have, 3a2 _ ax — Sab + 6a; + 2<7a; = 46a; + 4a5. By transposing, — ax -{- bx -\- 2ax — 46a; — 4ab + Sab — Sa^, By reducing, ax — 36a; m lab — So?- ; Factoring, (a — 36)a; — lab — Sa^. Dividing both members by the coefficient of x, lab - 3^2 X = — — . a — 36 3. Given 3a; — 2 + 24 = 31 to find x, Ans. a; = 3. 4. Given a; 4- 18 = 3a; — 5 to find x Ans. a- =r 11^ 80LDTI0N OF EQUATIONS. U3 6. Given 6 — 2a; -f 10 = 20 — 3aj — 2, to fiud x. Arts, a; = 2. 6. Given a; + ^a; 4- i« = H, to find a?, xins. a: = 6, 7. Given 2aj — ^a; + 1 = 5a; — 2, to find x, Ans, X = ij Solve the fbllowing equations: a , , 6 — Sa 8. 3ax + 2 - 3 = &« - «. Ans. x = ^^^TYb' 9. ^^ + f = 20 - ^-=-^- ^1«5. X = 23}. 2 3 2 * ,^2;4-3,a; , a: — 5 , 10. -^ + 3 = ^ — r~' ^''^* "^^ = ^^' ,, 25 3a; . 4x „ . 11. - _ —■ + a; = — - 3. -4;/5. a; = 4. 4 2 o ,^ 3aaj 2bx . ^ . ^ 0^7/*+ 4c2 a^- 5* 16. — ; = -z r Ans. X = a + X a — X a^ — x^ c ^ Qax — h Sh — c , , 10. z r = 4—0. 1 2 56 -h 96 - 7c ^1715 X ~z 16a - aj 05 — 2 a; 13 , 17 ---^- + - =-. a,« « = 10. 114 ELEMENTARY ALGEBRA. 18. 5_f !-?_?-/. a he d -^ . ahcdf Ans. X bed — acd + abd — ahc Note. — ^Wliat is the numerical value of aj, when a = 1 , 5=2, c = 3, ^ = 4, and / = 6 ? 19. ? _ I _ ^3 = _ 1211. Ans. X = l^, Sx — 5 4x — 2 , , . 20 JK ^ ^_ 1 — = X -\- 1, Ans. a = 6, lo 11 Sy cC Q! 21. a + - + - — - = 203 — 43. A7is. x = 60. 4 5 6 o^ « 4cB — 2 3a; — 1 , 22. 2a; — = — Ans. x = S. o 2 «« « . ^a; — c? . . Sa -\- d 23. 3a; 4-, — - — = x -\- a. Ans. x = , • ax — b a __ hx hx — a ' 4 "^ 3 ~ T ^3 3d -4?25. X = 3a— 26 4a; 20 - 4a; 15 . „ 2 25. = A71S. X = 3— -• 6 — a; a; a; 11 2a; + 1 _ 402 — 3a; _ ^ _ 471 — Qx ' 29 12 ~ 2 A?is. a; = 72, ^^ (a -{-h)(x — b) „ 4«J - 52 - a^_ i,,^ 21 - ^ -^ ^ - 3a = —7 2a; -] , a — h a-\-b b a* + Sa^b -f 4a2J2 _ q^J)^ ^ 26" -4'i. 2 o 4 3. What number is that whose third part exceeds its fourth by 16? Let the required number be denoted by x. Then, -X — the third part, 3 -X z= the fourth part PB0BLBM8. 117 and, hj the conditions of the problem, ~x '- -X = 16. 3 4 This i£ the statement. Clearing of fractions, 4a; — 3aj = 192, and hence, x = 192. VERIFICATION. 192 192 i^ - i^ = 64 - 48 = 16. 3 4 4. Divide llOOO between A^ B^ and (7, so that A shall have $72 more than J?, and C llOO more than A. Let X denote the number of dollars which B received. Then, x = B''s number, JB 4- 72 = A'*s number, and, jc + 172 = (7's number; and their sum, Zx + 244 = 1000, the number of dollars. This is the statement. By transposing, 3aj = 1000 - 244 = 756 ; and, X = — - = 252 = B's share. 3 Hence, a + 72 = 252 + 72 = 324 = A's share, and, a; -f 172 = 252 + 172 = 424 = C'« share. VERinCATION. 252 + 324 4- 424 = 1000. 6. Out of a cask of wine which had leaked away a third part, 21 gallons were afterwards dra^m, and the cask being then gauged, appeared to be half full : how much did it hold? 118 ELEMENTARY ALGEBRA. Let X denote the number of gallons. gj Tlien, - = the number that had leaked away. 85 and, - + 21 = what had leaked and been drawn. d OR CC Hence, by the conditions, - + 21 = - • o ^ This is the statement. Clearing of fractions, 2x + 126 = 3x, and, — X =z — 126 ; and by changing the signs of both members, which does not destroy their equality (since it is equivalent to multi^Dlying both members by — 1), we have, X — 126. VERIFICATION. 1|5 + 21 = 42 + 21 = 63 = ij-^ 6. A fish was caught whose tail weighed g lbs., his head weighed as much as his tail and half his body, and his body weighed as much as bis head and tail together : what was the weight of the fish ? Let 2x = the weight of the body, ui pounds. Then, 9 + cc = weight of the head ; and since the body weighed as much as both head and tail| 2aj = 9 -}- 9 + JC, which is the statement. Then, 2x — X = 18, and x = 18. PROBLEMS. 119 Hence, we have, 2x = SQlb. = weight of the body, 9 + X = 21lb. — weight of the head, 9/^. = weiglit of the tail ; hence, I2lb, = weight of the Mu • 7. The Slim of two numbers is 67, and their difference 19 . what are the two numbers ? Let X denote the less number. Tlien, X + 19 z= the greater; and, by the conditions, 2aj + 19 = 67. This is the statement. Transposing, 2a; = 67 — 19 = 48 ; AH hence, a; = — = 24, and a; + 19 = 43. VERTFICATION, 43 + 24 = 67, and 43 — 24 = 19. ANOTHEK SOLUnOX. Let X denote the greater number. Then, x — 1 9 will represent the less, and, 2a: — 19 = 67; whence 2a; = 67 -f 19. r^ n 86 Therefore, x = — =43; and, consequently, a; — 19 = 43 — 19 = 24. GENERAL SOLUTION OF THIS PROBLEM. The eum of two numbers is 8, their difference is di what are the two numbers ? 120 ELEMENTARY ALGEBRA. Let X denote the less number. Tlien, X -{- d will denote the greater, and 2x + d = s, their sum. WTience, _ s — d _ s d ^ ^ ~ 2 ~ 2 ~ 2* and, consequently, 2 2 2 2 As these two results are not dependent on particular values attributed to s or , How much did A win ? Jjei X denote the number of dollars A won. Then, A rose with 84 -h a; dollars, and -B rose with 48 — a; dollars. But, by the conditions, we have, 84 + JB = 5(48 - x), hence, 84 + a = 240 — 5x; and, 6a! = 156, consequently, x = 26 ; or ^ won $26. VERIFICATION. 84 4- 26 = 110 ; 48 — 26 = 22; 110 = 5(22) = 110. 11.-4 can do a piece of work alone in 10 days, J? in 13 days ; in what time can they do it if they work together ? Denote the time by x, and the work to be done, by 1. Then, in 1 day, A can do — of the work, and i? can do — of the work ; and in 13 X X days, A can do — of the work, and ^ can do — of the work. 13 124 ELEMENTARY ALGEBRA. Hence, by tlie conditions, — - + — = 1 "v\-hich gives, 13aj + 10a; =:: 130; 10 13 130 hence, 23a; = 130, x = — — = 5|f days. 12 A. fox, pursued by a hound, has a start of 60 of his own leaps. Three leaps of the hound are equivalent to 7 of the fox ; but while the hound makes 6 leaps, the fox makes 9 : how many leaps must the hound make to overtake the fox? There is some difficulty in this problem, arising from the different units which enter into it. Since 3 leaps of the hoimd are eqjial to 7 leaps of the fox, 7 1 leap of the hound is equal to - fox leaps. * 3 Since, while the hound makes 6 leaps, the fox makes 9, 9 3 while the hound makes 1 leap, the fox will make - , or - leaps. Let X denote the numhcr of leaps which the hound makes before he overtakes the fox ; and let 1 fox leap denote the unit of distance. Since 1 leap of the hound is equal to - of a fox leap, x 1 . . leaps tvtU be equal to -a: fox leaps ; and this will denote the 3 distance passed over by the hound, in fox leaps. Q Since, wHle the hound makes 1 leap, the fox makes - 3 leaps, wliile the hound makes x leaps, the fox makes -a; leaps ; Ji and this added to 60, his distance ahead, will give g ~x -{- 60, for the whole distance passed over by the fox 2i PROBLEM 8. 125 Hence, from the conditions, 7 3 -a; = -a; + GO ; wnenco, 3 2 14a; = 9a; + 360; X = 72. The hound, therefore, makes 72 leaps before overtakmg le fo leaps. tlie fox; in the same time, the fox makes 72 x - = 108 VERIFICATION. 108 4- 60 = 168, whole number of fox leaps, 72 X ^ = 168. o 13. A father leaves his property, amounting to $2520, to four sons, A^ J5, (7, and D. (7 is to have $360, i? as much as C and D togetlier, and A twice as much as i?, less $1000 : how much do -4, i?, and D receive ? Aiu, ^,$760; ^,$880; 2>, $520. 14. An estate of $7500 is to be divided among awddow, two sons, and three daughters, so that each son shall receive twice as much as each daughter, and the widow herself $500 more than all tlie children : what was her share, and what the share of each child ? ( Widow's share, $4000. Am, \ Each son's, 1000. ( Each daughter's, 500. 15. A company of 180 persons consists of men, women, and cliildren. The men are 8 more in number than the women, and the children 20 more than the men and women together : how many of each sort in the company ? Al^s. 44 men, 36 women, 100 children. 126 ELEMENTARY ALGEBRA. 16. A father divides $2000 among five sous, so that each elder should receive $40 more than his next younger bro« ther : what is the share of the youngest? Afis. $320. 17. A purse of $2850 is to be divided among three per sons, A, B^ and C. A's share is to be to B''s as C to 11, and C is to have $300 more than A and B together : what is each one's share? A's, $450 ; B's, $825 ; C's, $1575. 18. Two pedestrians start from the same point and travel in the same direction ; tlie first steps twice as far as the second, but the second makes 5 steps while the first makes but one. At the end of a certain time they are 300 feet apart. Now, allowing each of the longer j)aces to be 3 feet, bow far will each have traveled ? Ans. 1st, 200 feet ; 2d, 500. 19. Two carpenters, 24 journeymen, and 8 apprentices received at the end of a certain time $144. The carpenters received $1 per day, each journeyman, half a dollar, and each apprentice, 25 cents : how many days were they em- ployed ? Alls. 9 days. 20. A capitalist receives a yearly income of $2940 ; four- fifths of his money bears an interest of 4 per cent., and the remamder of 5 per cent. : how much has he at interest ? Ans. $70000. 21. A cistern containing 60 gallons of water has three unequal cocks for discharging it ; the largest will empty it in one hour, the second in two hours, and the third, in three: in what time will the cistern be emptied if they all run to- gether ? Ans. 32^\ min. 22. In a certain orchard, one-half are api)le trees, one- fourth })each trees, one-sixth plum trees; there are also, 120 clicriy trees, and 80 pear trees: how many trues in tha orchard? Ans. 2400. 23. A farmer being asked how many sheep he had^ rfiOBLEMS. 127 answered, that he had them in five fields ; in the Ist he had J, in llic 2d, j, in the 3d, |, and in the 4th, j^, and in llie 5th, 450 : how many had he ? A}is. 1200. 24. My horse and saddle together are worth |132, and the horse is worth ten times as much as the saddle: wlmi is tlie value of the horse ? Ans. |120. 25. Tlie rent of an estate is this year 8 per cent, greater than it was last. This year it is $1890; what was it last year? Ans. $1750. 26. Wliat number is that, from which if 5 be subtracted, § of the remainder will be 40 ? Ans. 05. 27. A post IS \ in the mud, i in the water, and 10 feet above the water : what is the whole length of the post ? Ans, 24 feet. 28. After paying ^ and ^ of my money, I had 66 guineas left in my purse : how many guineas were in it at first ? Ans. 120. 29. A person was desirous 'of giving 3 pence apiece to some beggars, but found he had not money enough in his pocket by 8 pence; he therefore gave them each 2 pence and had 3 pence remauiing : required the number of beg- gars. Ans. 11. 30. A person, in play, lost | of his money, and then won 3 sliillings ; after which he lost ^ of what he then had ; and this done, found that ho had but 12 shilUngs remauiing: what had he at first ? Ans. 20s. 31. Two persons, A and i?, lay out equal sums of money in trade; A gains $120, and J3 loses $87, and A''s money is tlien double of JPs : what did each lay out? Ajis. $300. j 82. A person goes to a tavern with a certain sum of money m his pocket, where he spends 2 shillings: he then borrows as much money as he had left, and going to another tavern, he there spends 2 shillings also; then borrowmg 128 ELEMENTARY ALGEBRA. again as much money as was left, he went to a third tavern, where Ukewise he spent 2 shillings, and borrowed as much as he had left : and again spending 2 shillings at a fourth lavem, he then had nothing remaining. What had he at lirst ? Ans, 35. 9c?, 33. A tailor cut 19 yards from each of three equal pieces of cloth, and 17 yards from another of the same length, and found that the four remnants were together equal to 142 yards. How many yards in each piece? Ans, 54. 34. A fortress is garrisoned by 2600 men, consisting of infantry, artillery, and cavalry. Now, there are nine times as many infantry, and three times as many artillery soldiers as there are cavalry. How many are there of each corps ? Ans. 200 cavalry; 600 artillery ; 1800 infantry. 35. All the joumeyings of an individual amounted to 2970 mUes. Of these he traveled 3^ times as many by water as on horseback, and 2i times as many on foot as by water. How many miles did he travel in each way ? A71S. 240 miles; 840 m.; 1890 ra. 36. A sum of money was divided between two persons, A and Ji. A''s share was to JB^s in the proportion of 5 to 3, and exceeded five-nmths of the entire sum by 50. "What was the share of each? A7is. ^'5 share, 450; ^'5, 270. 37. Divide a number a into three such parts that tho second shall be n times the first, and the third m times a? great as the first. a , na ^ , ma ^^' I -^m + n* ^^' 1 + m + n' ' 1 -h m -h n 38. A father directs that 111 70 shall be di\ided among his three sons, in proportion to their ages. The oldest is twice as old as the youngest, and the second is one-third older :han the youngestc How much was each to receive ? Ans. $270, youngest; $360, second ; $540, oldest PROBLEM 8. 129 39. Three regiments are to furnish 594 men, and each to fiiruieh in proportion to its strength. Now, the strengtli of the fii-st is to the second as 3 to 5 ; and that of the second to the third as 8 to 7. How many must each furnish ? Am. 1st, 144 men ; 2d, 240 ; 3d, 210 40. Five heirs, -4, J5, C, jO, and ^, arc to divide an inhtr - itanco of $5600. J5 is to receive twice as much as Ay and |200 more ; C three times as much as Ay less $400 ; D tlie half of what £ and C receive together, and 150 more ; and JS the fourth part of what the four others get, plus $475. How much did each receive ? A% $500; JB'Sy 1200; C's, 1100; D*Sy 1300; U's, 1500. k' ,%, 1- r is/i 741. A person has four casks, the second of wliich being filled from the nrst, leaves the first fouj»-seventlis full. The third Jjcing filled from the second, leaves it one-fourth fulj, and when the thiid is emptied hito the fourth, it is found to fill only nine-sixteenths of it. But the first will fill the third and fourth, and leave 15 quarts remaining. How many gallons does each hold ? Ans. 1st, 35 gal. ; 2d, 15 gal. ; 3d, 11 J gal. ; 4th, 20 gal. 42. A courier having started from a place, is pursued by a second after the lapse of 10 days. The fii*st travels 4 miles a day, the other 9. How many days before the second will overtake the first ? Atis, 8. 43. A courier goes 31^ miles every five hours, and is fol- lowed by another after he had been gone eight hours. Tlie second travels 22^ miles every three hours. How many hours before he will overtake the first ? Ans, 42. 44. Two places are eighty miles apart, and a person leaves one of tliem and travels towards the other at the rate of 3j miles per hour. Eight hours after, a person departs from 6* 130 ELEMENTARY ALGEBRA. the second place, and travels at the rate of 5} miles per hour How long before they will be together ? Ans. 6 hoTirs, EQUATIONS CONTAINING TWO UNKNOWN QUANTmES« 1 10. If w^e have a single equation, as, 2£C + 3?/ = 21, containing tw^o unknown quantities, x and y, wo may find the value of one of them ui terms of the other, as. 21 -3y 2 ^ = — — (1.) Now, if the value of y is unknow^n, that of x will also be unknown. Hence, from a single equation, contaming two unknown quantities, the value of x cannot be determined. If w^e have a second equation, as, 5x + 41/ = 35, we may, as before, find the value of a; in terms of y, giving, 35 — 4v , , ^ = 1-^ (2.) Now^, if the values of x and ?/ are the same in Equations (1 ) and ( 2 ), the second members may be placed equal to each other, giving, 21 - 32/ 35 - 4y ,^„ ^^ ^^ ^ —-^ = ■ ^ , or 105 - ISy = 70 - 8y ; from Avhich we find, y = 5. 110. In one equation containing two unknown quantities, can yon find the vaiue of e^her ? if you have a second equation involving the same two unknown quantities, can you find their values ? What are such equa- tions called ? ELIMINATION. 131 Subtituting tliis value for y in Equations ( 1 ) or ( 2 j, we find X — Z. Sucli equations are called tSlmultaneoua equations. Hence, 111. Simultaneous Equations are those in which the values of the unknown quantity are the same in both. ELDIINATION. 113. ELniiNATioN is the operation of combining two equations, containini^ two unknown quantities, and deduciug tJierefrom a single equation, containing but one. Tlicre are three principal methods of eUnuuation : 1st. By addition or subtraction. 2d. By substitution. 3d. By comparison. We shall consider these methods separately. Elimination by Additio7i or Subtraction, 1. Take the two equations, 3aj — 2y = 7, 8a; + 2y = 48. If we add these two equations, member to member, we obtain, 11a; = 55; which gives, by dividing by 11, jc = 5; and substituting this value in either of the given equations, we find, y = *. 111. What are simultaneous cquatiou^P 112. What \s eliiiiinutiou? Ilow many methods of elimiDation ore there f What are they ? 132 ELEMENTAEY ALCJEBEA. 2. Again, take the equations, 8a; + 2y = 48, 3iB 4- 2?/ == 23. If we subtract the 2d equation from the 1st, we obtain, 5a; = 25; which gives, by dividing by 5, a; = 5; and by substituting this value, we find, 2/ = 4. 3. Given the sum ol two numbers equal to s, and their difierence equal to J, to find the numbers. Let X = the greater, and y the less number. Then, by the conditions, x -]- y — s. and, X — y ^ d. By adding (Art. 102, Ax. 1), 2a; = 5 4- d By subtracting (Art. 102, Ax. 2), . . , 2y =r: s -^ d. Each of these equations contains but one unknown quantity. From the first, we obtain, ..,.,. a; -^ — ■ — , 2 and from the second, , V = - These are the same values as were found in Prob. 7, page 120. 4. A person engaged a workman for 48 days. For each day. that he labored he was to receive 24 cents, and for each day that he was idle he was to pay 12 cents for his board. At the end of the 48 days the account was settled, when the laborer received 504 cents. Required the number of work- ing days, and the number of days he was idle. ELIMINATION. 133 Let X = the numbBr of working days, y = the number of idle days. Then, 24a; = what he earned, and, \2y =. what he paid for his board. Tlien, by the conditions of the question, we have, a; + y = 48, and, 24aj — 12y = 604. This is the statement of the problem. It has already been shown (Art. 102, Ax. 3), that the two members of an equation may be multiplied by the same num- ber, without destroying the equality. Let, then, the first equation be multiplied by 24, the coefficient of aj in the second ; we shall then have, 24a; -f 24y = 1152 24a; — 12y = 504 and by subtracting, 36y = 648 648 .'. V = = 18. ^ 36 Substituting this value of y in the equation, 24a; — 12y = 504, we have, 24aj — 216 = 504; vi'hich gives, 720 24aj -r-. 604 + 216 = 720, and x =^ —^ = 80. 24 VERIFICATION. jc + y = 48 gives 30 + 18 = 48. 24a: — 12y = 504 gives 24 x 30 - 12 X 18 = 504 134 ELEMENTAET ALGEBKA 113. In a similar manner, either unkno^^Ti quantity may be eliminated from either equation ; hence, the following EULE. I. Prepare the equations so that the coefficients of the quantity to be eliminated shall be numerically equal: II. If the signs are unlike^ add the equations^ member to member ; if alike^ subtract them^ member from mend^er. EXAMPLES. Find the values of x and y, by addition or subtraction, in the following simultaneo'j.s equations : - y = + 2x = n ibx ■\- 2y ■= 37 ) ^ J 2aj + 6y = 42 > * ( 8aj — 6y = 3 i" ■ 3 j 8:« - 9y = 1 ) ( 6a; — 3y = 4a; i Q j 14a; - 15y ^ 12 ) ' ( 1x -{- 8?/ — 37 ) 1 Ans. X = 2, y = 3. Ans. X = 5^ y — Q* Ans, a; = 4^, y — b\. Ans. a; = I, y = i. Ans. jc = 3, y = 2. 10. < 1^ + 3^^ 1,1 3^ + 2^ = 61 Ans. 6, y = 9. n.J" + §^ = ' A.ns. < a; = 14, y =: IG. y - 2 113. What is the rule for elimination by addition or subtractiou ? ELIMINATION. 135 12. Says A to B^ you give mo $40 of }oiir money, ami I sliall then liavo five times as much as you will have lell. Now they both ha^$120; how much had each? "^ Ans. Each had $60. 13 A father says to his son, " twenty years ago, my ago was four times yours; now it b just double:" what were their ages ? A \ Father's, CO years. * ( Son's, 30 years. 14. A father divided his property between his two sons. At the end of tlie first year the elder had spent ono-quarter of his, and the younger had made $1000, and their i)roperty was then equal. After this the elder spent ?^500, and the younger made $2000, when it appeared that the younger had just double the elder; what had each from the father? J Elder, $4000. ^^* I Younger, $2000. 15. If John give Charles 15 apples, they will have the tame number; but if Charles give 15 to John, John will have 15 times as many, wanting 10, as Charles will have left. How many hits each ? a \ John, 60. Ans. \ Charles, 20. 10. Two clerks, A and J?, have salaries which are together equal to $900. A spends ^\ per year of what he receives, and B adds as much to his as A spends. At the end of the year they have equal sums : what was the salary of each ? . S A's = $500. ^^^•Ii?', = $400. Elimination by Substitution, 114. Let us again take the equations, 5a!-f Vy = 43, (1.) lla^^- 9y = C9. (2.) 1 14 H've the rule for eluniDation by suhfititutioii. When \b thio n><>thod wfA to the greatmit advanta^ T 136 ELEMENTARY ALGEBRA. Find the value of a; in the first equation, which gives, "^ - 5 Substitute this value of a; in the second equation, and we have, 43 — 7v 11 X 5-^ + 92/ = 69; or, 473 — 11y + 452/ = 345 ; or, — 32y = — 128. Here, a has been eliminated by substitution. In a similar manner, w^e can eliminate any unknown quan- tity ; hence, the RULE. I. Find from either equation the value of the unJaiown quantity to be eliminated : n. Substitute this value for that quantity in the other equation. Note. —This method of elimination is used to great advan- tage when the coefficient of either of the unknown quantities is 1. EXAMPLES. Find, by the last method, the values of x and y in the following equations: 1. 3a; — 2/ = 1, and 3y — 2a5 = 4. Ans. a; = 1, 2/ = 2 2. 5y — 4a; = — 22, and Sy -^ Ax — 38. Alls, a; = 8, y = 2. 3. a; + 81/ = 18, and y — Zx — — 29. Ans. a; = 10, y = L ELIMINATION. 137 2 4. bx - y = 13, and 8a; + -y = 29. Ans. 05 = 3^, y r= 4i 6. 10a; ~ I = 09, and lOy - ^ = 49. 5 7 Ans. a = 7, y = 5. 6. « + |» - f = 10, and I + ^ = 2. Ans. a; = 8, y = 10. ». 1-1 + 5 = 2, « + ! = in. ^w«. a; = 15, y = 14. 1 + 1 + 3 = 6^ ana 1-1 = 1. ^?i5. a; = 3^, y = 4. ^/i5. aj = 12, y = 16. 10. ? - ? - 1 = - 9, and 5a; ~ ^ = 29. 7 2 ' 49 Ans. a; = 6, y = 7. 11. Two misers, -4 and -S, sit douTi to count over their money. They both have $20000, and 3 has three times aa much as A : how much has each ? , ^ a-^^^ . J -4, loOOO. ^''^' ] J?, $15000. 12. A person has two purses. If he puts $7 into the first, the whole is worth three times as much as the second purye: but if he puts $7 into the second, the whole is worth liv^e times as much as the first : '^hat is the value of each purse ? Ans. iBt, $2; ?d, $3. 138 ELEMEiq^TAEY ALGEBEA. 13. Two numbers have the following relations: if the first be multiplied by 6, the product will be equal to the second multiplied by 5 ; and 1 subtracted from the first leaves the same remainder as 2 subtracted from the second : what are the numbers ? Ans. 5 and 6. 14. Find two numbers with the following relations: the drst increased by 2 is 3J times as great as the second *, and the second increased by 4 gives a number equal to half the first: what are the numbers? A7is. 24 and 8. 15. A father says to his son, "twelve years ago, I was twice as old as you are now: four tunes your age at tliat time, plus twelve years, will express my age twelve years hence ; " what w^ere their ages ? ~o^ . ( Father, 12 years. ^''^- (Son, 30 " ^Elimination hy Comijarison. 115. Take the same equations, 5iB + 7y = 43 11a; + 92/ = 69. Finding the value of x from the first equation, we have, 43 — Vy aj := • « 5 , and finding the value of x from the second, we obtain, 69 — 9y X =: — • 11 115. Give the rule for elimiuatiO'n by cotnpsu-isoiu ELIMINATION. 139 Let these two values of x be placed equal to each other, and we have, 43 — It/ _ 09 — 9y 5 "■ 11 Or, 473 — ny = 345 — 45y; or, — 32y = — 128. llenco, y = 4. A 1 69 — 30 „ And, X = — — - — = 3. This method of elimination is called tlie method by com- fcarison, for which we have the following •RULE. L Find^ from each equation^ the value of the same unknown quantity to he eliminated': II. Place these values equal to each other, EXAMPLES. Find, by the last rule, the values of x and y, from the follo\i'ing equations, 1. 3a; + I + = 42, and y _ ^ = 14^. Ans. a; = 11, y = 15. 2. l-f + 5.6, and 1 + 4 = ^+6. Ans. ar = £8, y = 20. 3. ^ - f + f = 1, and 3y - »= = 6. Ans. a; = 9, y = 6. 4. y - 3 = „a; f- 5 and ^-^^ = y - 3^ • Ans. a; = 2, y = 9 140 ELEMENTARY ALGEBKA. 5. ^— ~ h - = y - 2, and - + ? = a - 13. 3 2 ^ ' 8 7 Ans. aj = 16, y = 1 . 6. ^^-5^ h ^ = 05 ^ , and a; 4- y = 16. ^725. a; = 10, y = 6. / 2 - 0. Ans. a; = 1, y = : 3. cc - 4 3 = y - X ' 5* t/i5. ar = 10, 2/ = 13. ^ 2aj — 3y 1. --^ = a - 2§, a- - 8. 2y 4- 3a; = y 4- 43, ?/ - 9. 4y - ^-^-^ = aj + 18, ^nd 27 - y = a; + y + 4. Ans. a; = 9, y = 1. 10. l_^ + 4 = y-16|, |-2 = |. -4w5. a; = 10, y = 20. 116 Having explained the principal methods of elimina- tion, wo shall add a feAv examples which may be solved by any one of them ; and often indeed, it may be advantageous to emj)loy them all, even in the same example. GENERAL EXAMPLES. Find the values of x and y in the following simultaneous equations : 1. 2a; 4- 3y = 16, and 3a; — 2y = 11. Ans. a; = 5, y =• 2. ELIMINATION, 1« 2, L _ii = _- and ■ — ~ z=. — ft ^ 4 20* 4^5 120 A 1 1 3. ? + 7y = 99, and ^ + 1x = 51. Ans. 25 == 7, y = 14 4. 1-12=1 + 8. ^l + l-S = 'l^ + .1. 6. 4o* * - ly + y = C} uln«. a; = 60, y = 40. a; = 6. --- + 7. = 41 ia; - 2^ + 4iy = 12} j 5 3y»~ g , 2a; - y 6 "^ 4 6a;-y + ?-^ = ^^^ j '3a; - 8 y — 6 Ans. Ans. Ans, + "—T- + y = 18 TT 8.^ 9. 8a - 3 — 6 - y 79 4j; — 4 V — 5 ix-iy + V-* Ans. Ans. y = 8 a; = 6. y = s. a; = 9. y=8. X = 10. y= 12 x=:e. y = 5. \/ / 142 10. ELEMENTARY ALGEIJllA. c -\- ab — bd X = A71S. ax — by = c a — y -^r X = d a — b y z= a^ + c — ad a — b J J j IS-T + Ty- 341 :=. 71^+4313;) \x=-\2 12 i(«^ + 5)(y+7) = (a;+l)(y-9) + 112) ( aj = • (2a:+10-32/-hl f ^''^- ( 2/ ^ 13. < 14. ^ 15. IG. < ax — by x-^ y — c ax -i- by = c fx +yy = h Ans. x = Ans. X = y = = 3. y — 5. be a^-b __ ac ~ a-\-b' eg — bh ah — ef ag-bf^ Ans. in, < b + y Sa -\- X ax + 2by = d bex z= cy — ^ ,, , it and the chain became worth three and a half times more than the silver watch ; but when he put the chain on the silver watch, they became worth one-half the gold watch and 15 dollars over : what was the value of each watch ? J j Gold watch, $80. ^"^^ (Silver « $30. 17. There is a cert^ number expressed by two figures, which figures are called digits. The sum of the digits is 11, and if 13 be added to the first digit the sum will be three times the second: what is the nimiber? Ans. 56. 18. From a company of ladies and gentlemen' 16 ladies retire; there are then left two gentlemen to each lady. After which 45 gentlemen depart, when there are left 5 ladies to each gentleman : how many were there of each at first ? J j ^^ gentlemen. ( 40 ladies. 1 9. A person wishes to dispose of his horse by lottery. If he seUs the tickets at $2 each, he will lose $30 on his borpe; but if he soils them at $3 each, he wall receive $30 148 ELEMENTARY ALGEBKA. more than his horse cost him. What is the value of the. horse, and number of tickets? . (Horse, $150. Ans. < -^ ' , ( No. of tickets, 60, 20. A person purchases a lot ot wheat at $1, and a lot of rye at 75 cents per bushel ; the whole costing him $117.50. He then sells | of his wheat and ^ of his rye at the same rate, and realizes $27.50. How much did he buy of each? . ( 80 bush, of wheat. -1 50 bush, of rye. 21. There are 52 pieces of money in each of two bags. A takes from one, and JB from the other. A takes twice as much as -S left, and J^ takes 7 times as much as A left. How much did each take? . j ^, 48 pieces. [A, 28 pieces. 22. Two persons, A and J?, purchase a house together, worth $1200. Says A to -B, give me two-thirds of your money and I can purchase it alone ; but, says ^ to A^ if you will give me three-fourths of your money I shall be able to purchase it alone. How much had each ? A71S. A, $800 ; -S, 23. A grocer finds that if he mixes sherry and brandy in the proportion of 2 to 1, the mixture will be worth 78s. per dozen ; but if he mixes them in the proportion of 7 to 2, he can get 795. a dozen. What is the price of each liquor per dozen? Ans. Sherry, 81s.; brandy, 72s. Equations containing three or more unknown quantities 117. Let us noT\' consider equations involving three or more unknown quantities. Take the group of simultaneous equations, 117. Give the rule for solving any group of simultaneous equations? EX AMTLES. 1^ 5a; - 6y + 42 = 16, . . (1.) 7aj 4- 4y - Sz = 19, . . (2.) 2a; + 2/ -h 63 = 46. . , . . (8.) 149 To eliminate 2 by means of tlie first two equations, multi- ply the first by 3, and the second by 4 ; then, since the coeflicients of 2 have contrary signs, acjd the two results together. Tliis gives a new equation : 43aj - 22, = 121 (4.) Multiplying the second equation by 2 (a factor of the coefficient of 2 in the third equation), and adding the result to the third equation, we have, 16a; + 9y = 84 (5.) The question is then reduced to finding the values of x and y, which will satisfy the new Equations ( 4 ) and ( 5 ). Now, if the first be multii^lied by 9, the second by 2, and the results added together, we find, 41905 = 1257; whence, a; = 3. We might, by means of Equations (4) and (6) deter- mine y in the same way that we have determined x ; but the value of y may be detennined more simply, by substi- tuting the value of a; in Equation ( 6 ) ; thug, 48 + 9y = 84. . • . y = ^^ ^ ^^ = 4. In the same manner, the first of the three given equations becomes, by substituting the values of x and y, 15 — 24 -I- 42 = 15 .-. z = ~ = 6, 4 In the same way, any gronr» cC «imiiltantous equations may be solved Hence, the 150 ELEMENTAEF ALGEBRA. KULE, 1. Combine one equation of the group with each of the others, by eliminating one unknown quantity ; there will result a new group containing one equation less than the original group: n. Combine one equation of this new group with each of the others, by eliminatirig a second unknown quantity ; there will result a new group containing two equations less than the original group : in. Conti9iue the operation until a shigle equation is found, containing but one unknown quantity : lY. Find the value of this unknown quantity by the preceding rules y substitute this in one of the group of two equations, and find tlie value of a second unknown quantity ; substitute these in either of the group of three^ finding a third unknown quantity ; and so on, till the values of all are found. Notes. — 1. In order that the value of the unknown quan- (;ities may be determined, there must be just as many inde- pendent equations of condition as there are imknown quan- tities. If there are fewer equations than unkno^vn quantities, the resulting equation will contain at least two unknown quantities, and hence, their values cannot be found (Art. 110). If there are more equations than unknown quantities, the conditions maybe contradictory, and the equations impossible. 2. It often happens that each of the proposed equations does not contain all the unknown quantitAes. In this case, with, a little address, the elimination is very quickly per- formed. Take the four equations involving four unknown quanti- ties: 2a; - 3y + 22 = 13. (1.) 4?/ + 23 = 14. (3.) 4w - 2a; =:= 30. (2.) 6y -f ^u = 32. (4.) BXi^MPLES. 151 By inspecting thase equations, we see that the elimination of z in the two Equations, ( 1 ) and ( 3 ), will give an equar tion involving x and y; and if we eliminate u in Equa- tions ( 2 ) and ( 4 ), we shall obtain a second equation, in- vohnng x and y. These last two unknown quantities may therefore be easily determined. In the first place, the elimination of z from ( 1 ) and ( 3 ) gives, 7y — 2aj = 1 ; That of u from ( 2 ) and ( 4 ) gives, 20y 4- 6a; = 38. Multiplying the first of these equations by 3, and adding, 41y = 41; Whence, y = 1. Substituting this value in Ty — 2a; = 1, we find, a; = 3. Substituting for x its value in Equation ( 2 ), it becomes 4w — 6 = 30. Whence, w = 9. And substituting for y its value in Equation (3), there reeulte, z = 6. EXAMPLES. « + y H- z = 29 SB -f 2y 4- 32 = 62 1. Given ^ . ^ ^ ^ + ^ + ^ = 10 A718. X = 8, y = 9, z = 12, ► to find JB, y, and e. 162 LEMENTARF ALGEBRA. r 2a; 4- 4y — 32 = 22 | 2. Given < 4x — 2y -{- 5z = 18 I to finrl x, y, and z. [ 6a; + Vy — s = 63 J Ans. a; = 3, y — 7, s = 4, 3. Given < 35 4- 2^ + 3^ = 32 o^^ + 72/ + ^2 = 15 r *^ fi^^ ^' y? and 2 o 4 5 Ans. a; = 12, y = 20, z = 30. 4. Given r a; + y + 2 = 29i < X + y — z = ISl I a; — y + s = 13| 294- , 4^ ^ to find a;, y, and z. Ans. a; = 16, y = Yf, s = 5J 3a; + 5y = 161 5. Given -{ Ya; + 22 = 209 }» to find a;, y, and z. 2y + 2 = 89 -4715. a; r= 17, y = 22, 2 = 46. V ri 1 6. Given ^ - -f — = 5 }- to find a;, y, and 2. L y s J a; = 2 a + b — c' y a -]- G — b' z = h-\-c — a Note. — In this example we should not proceed to clear the equation of fractions; but subtract immediately the second equation from the first, and then add the third ; we thus find the value of y. PROBLEM 8. 153 PROBLEMS. 1. Divide the number 90 into four such parts, that the first increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2, shall be equal each to each. This problem may be easily solved by introducing a new unlaio\Ma quantity. Let X, y, 2, and ?/, denote the required parts, and desig. nate by ni the several equal quantities which arise from the conditions. We shall then have, u X + 2 z=z m, y — 2 = r«, 2z = m, - = m. 2 X = m — 2, y = r7i4-2, z = — , u = 2m, From which we find, X = m — 2, y - And, by adding the equations. m SB-f-y + 24-w = m -\- m -\- — -\- 2m = 4\m, Si And since, by the conditions of the problem, the first member is equal to 90, we have, \\m = 90, or \m = 90; hence, m = 20. Having the value of m, we easily find the other values; viz.: jc = 18, y = 22, z = 10, u = 40. 2. There are three ingots, composed of different metals mixed together. A pound of the first contains 7 ounces of silver, 3 ounces of copper, and 6 of pewter. A pound of the second contains 12 ounces of silver, 3 ounces of copj)er, and 1 of pewter. A pound of the third contains 4 ounces of silver, 7 ounces of copper, and 5 of pewt.or. It is required 7* 154 ELEMENTARY ALGEBRA. to find how much it will take of each of the three ingots to form a fom-th, which shall contain in a pound, 8 ounces of silver, 3| of copper, and 4^ of pewter. Let jc, 2/, and ^, denote the number of ounces which it is necessary to take from the three ingots respectively, in order to form a pound of the required ingot. Since there are V ounces of silver in a pound, or 16 ounces, of the first ingot, it follows that one ounce of it contains ^^ of an ounce of silver, and, consequently, in a number of ounces denoted by (K, there is — ounces of silver. In the same manner, 1 2ii/ 4 z we find that, -^ , and — , denote the number of ounces of silver taken from the second and third ; but, from the enunciation, one pound of the fourth ingot contains 8 ounces of silver. We have, then, for the first equation, !^ 4. l?^ -J- 1^ ::^ 8 • 16 "^ 16 16 ' or, clearmg fractions^ 1x + 12y + 42 = 128. As respects the copper, we should find, 305 4- 3y 4- T2 = 60 ; and with reference to the pevki;er, Qx + y + 5z = 68. As the coefficients of y in these three equations are the most simple, it is convenient to eliminate this unknown quantity first. Multiplying the second equation by 4, and subtracting the first from it, member from member, we have, 5x + 242 = 112. PB0BLEM6. 155 Multiplying the third equation hj 3, and subtracting the second from the resulting equation, we have, 1525 + 83 = 144. Multiplying this last equation by 3, and subtracting the preceding one, we obtain, 40a; = 320; whence, a; = 8. Substitute this value for x in the equation, 16a; -f- 82 = 144; it becomes, 120 + 82 = 144, whence, 2 = 3. Lastly, the two values, a; = 8, 2 = 3, bemg substituted in the equation, 6aj + y -h 52 = 68, give, 48 + y + 15 = 68, whence, 2/ = ^* Therefore, in order to form a pound of the fourth ingot, we must take 8 ounces of the first, 6 ounces of the second, and 3 of the third. VEEIFICATION. K there be 1 ounces of silver in 16 ounces of the first ingot, in eight ounces of it there should be a number of ounces of silver expressed by 7x8 16 In like manner, 12 X 5 ,4x8 , and , 16 ' 16 ' will express the quantity of silver contamed in 5 ounces of the second ingot, and 3 ounces of the third. 156 ELEMENTARY ALGEBRA. Now, we have, 7X8 12 X 5 4 x3 _ 128 _ g. ~1l6 ^ 16 "^ 16 "" 16 "" ' therefore, a pound of the fourth ingot contains 8 ounces of silver, as required by the enunciation. The same conditions niay be verified with respect to the copper and pewter. 3. A''s age is double Ij\% and ^'s is triple of C»5, and the sum of aU their ages is 140 : what is the age of each ? Ans. A's — 84; B's = 42; and G's = 14. 4. A person bought a chaise, horse, and harness, for £80 ; the horse came to twice the price of the harness, and the chaise to twice the cost of the horse and harness : what did he give for each? ( £13 6s. 8c?. for the horse. Ans. \ £6 13s. 4f?. for the harness. ( £40 for the chaise. 5. Divide the number 36 into three such parts that \ of the first, \ of the second, and ^ of the third, may be all equal to each other. Aiis. 8, 12, and 16. 6. If A and B together can do a piece of work in 8 days, A and C together in 9 days, and B and C in ten days, how many days would it take each to perform the same work alone? Ans. A, 14ff; J5, l^H; G, S^/y. 7. Three persons, A^ J5, and C, begin to play together, having among them all $600. At the end of the first game A has won one-half of J5's money, which, added to his own, makes double the amount B had at first. In the second game, A loses and B wins just as much as G had at the be- ginning, when A leaves off with exactly Avhat he had at first : now much had each at the beginning ? Ans. ^,$300; i?, $200 ; G $100. 8. Three persons, A, B, and (7, together posscs.« $3640. PE0BLEM6. 157 It* B gives A $400 of his money, then A will have $320 more than B\ but if B takes $140 of C''s money, then B and C will have equal sums : how much has each ? Alls, A, $800 ;, B, $1280 ; C, $1560. 9. Three persons have a bill to pay, which neither alone is able to discbarge. A says to J5, " Give me the 4th of yonr money, and then I can pay the bill." B says to C •'Give me the 8th of yours, and I can pay it." But C says to A^ " You must give me the half of yours before I can pay it, as I have but $8 " : what was the amount of their bill, and how much money had A and B ? . j Amount of the bill, $13, • 1^1 had $10, and i? $12. 10. A person possessed a certain capital, which he placed out at a certain interest. Another person, who possessed 10000 dollars more than the first, and who put out his capital 1 per cent, more advantageously, had an annual income greater by 800 dollars. A third person, who possessed 15000 dollars more than the first, putting out his capital 2 per cent, more advantageously, had an annual income greater by 1500 dollars. Required, the capitals of the three per- sons, and the rates of interest. . j Sums at interest, $30000, $40000, $45000. * ( Rates of interest, 4 6 ' 6 pr. ct. 11. A widow receives an estate of $15000 from her de- ceased husband, with directions to divide it among two sons and three daughters, so that each son may receive twice as much as each daughter, and she herself to receive $1000 more than all the children together : what was her share, and what the «liare of each child? i The widow's share, $8000 Ans. \ Each son's, $2000 ( Each daughter's, $1000 158 ELEMENTAKY ALGEBRA. 12. A certain sum of money is to be divided between three persons, A, B^ and G, A is to receive |3000 less than half of it, B $1000 less than one-third part, and G to receive $800 more than the fourth part of the whole : what is the sum to be divided, and what does each receive ? Sum, $38400. Ans» A receives $16200 B " $11800. G " $10400. 13. A person has three horses, and a saddle which is worth $220. If the saddle be put on the back of the first horse, it will make his value equal to that of the second and third ; if it be put on the back of the second, it will make his value double that of the first and third ; if it be put on the back of the third, it will make his value triple that of the first and second : what is the value of each horse ? Ans, 1st, $20; 2d, $100 ; 3d, $140. 14. The crew of a ship consisted of her complement of sailors, and a number of soldiers. There were 22 sailors to every three guns, and 10 over ; also, the whole number of hands was five times the number of soldiers and guns to- gether. But after an engagement, in which the slain were one-fourth of the survivors, there wanted 5 men to make 13 men to every two guns: requked, the number of guns, soldiers and sailors. Ans. 90 guns, 65 soldiers, and 670 sailors, 15. Three persons have $96, which they wish to divide equally between them. In order to do this, A^ who has the most, gives to B and G as much as they have already ; then B divides with A and G in the same manner, that is, by giving to each as much as he had after A had divided with them • G then makes a division with A and i?, when it Is PB0BLEM8. 159 found that they all have equal sums : how m ach had each at first? Ans. 1st, $52; 2d, |28; 3d, $16. 16. Divide the number a into three such parts, that the first shall be to the second as m to w, and the second to the third as pto q. ^ amp _ anp _ a?iq ~~ mp-{-np-\-nq^ ^ ~ mp-\-np-\-nq^ ~ mp-{-np-\-nq 17. Three masons, -4, i?, and (7, are to build a wall. A and B together can do it in 12 days ; B and C in 20 days ; and A and (7 in 15 days: in what time can tach do it alone, and in what time can they all do it if they work together ? Ana. Ay in 20 days; -B, in 30 ; and C, in 00 ; all, in 10, 160 ELEMENTARY ALGEBRA.. CHAPTER VI. FORMATION OP POWERS 118. A Power of a quantity is the product obtained by taking that quantity any number of times as a factor. If the quantity be taken once as a factor, we have the first power ; if taken twice, we have the second power ; if three times, the third power; if n times, the n*^ power, n being any whole number whatever. A power is indicated by means of the exponential sign ' thus, a ^ a} denotes first power of a.* ax a = d?- (( square, or 2d power of a. ax ax a = a^ (( cube, or third power of a. axaxaxa = a* (( fourth power of a. axaxaxaxa = «® (( fifth power of a. axaxaxa — = a"^ (( m*^ power of a. In every power there are three things to be considered : 1st. The quantity which enters as a factor, and which is called the first power. 2d. Tlie small figure which is placed at the right, and a Uttle above the letter, is called the exponent of the * Since a« - 1 (Art. 49), a" X o = 1 X a = a^ ; so that the two factors of a\ are 1 and a. 118. What is a power of a quantity? What is the power when tte quantity is taken once as a factor ? When taken twice ? Three times f fi times? How is a power indicated ? In every power, how miny things are considered ? Name them. POWKBS OF MONOMIALS. 161 power, and shows how many times the letter enters as a fiictor. 3d. The power itself, which is the final product, or result of the multiplications. POWERS OF l»IONO>nALS. 119, Let it be required to raise the monomial 2a'5' to the fourth power. We have, (2a^b^y = 2a^b^ X 2a^h^ X '2a^b^ X 2a?h\ which merely expresses that the fourth power is equal to the product which arises from taking the quantity four times as a factor. By the rules for multiplication, this pro- duct is from which we see, l8t. That the coefficient 2 must be raised to the 4th power; and, 2d. That the exponent of each letter must be multiplied hy 4, the exponent of the power. As the same reasoning applies to every example, we have, for the raising of monomials to any power, the followmg RULE. I. Raise the coefficient to the required power : IT. Multiply the exponent of each letter hy the exponent of the power, EXAMPLES. 1. What is the square of 3ay ? Afis. 9a*i/^ 110. What is the rule for raising a monomial to any power? WLen the monomial is pogitive, what will be the sign of its powers ? When negative, what powors will be plitf' what miuua? 162 ELEMENTARY ALGEBRA. 2. What is the cube of Qa^y'^x'i Ans, 21Qa^^y^x^, 3. What is the fourth power of 2a^y^^? 16ai2yi2j2o^ 4. WTiat is the square of a^js^s p Ans. a^b^^y^. 5 6. 1, What is the seventh power of a^hcd^ ? Mns. a^^b'^c'd^^ What is the sixth power of a^b^c^d'i Ans. a^^b'^cH^ What is the square and cube of — 2a^'^? /Square. Cube. — 2a^b^ — 2a252 — 2a'^b^ - 2a^b^ + 4a^b\ -f 4a*5* - 2a2J2 By observing the way in which the powers are formed, we may conclude, 1st. When the monomial is positive, all the powers will be 2^ositive. 2d. When the monoTnial is negative, all even poicers loill be positive, and all odd will be negative. 8. What is the square of — 2^*5^ ? Ans. 4:a^b^\ 9. What is the cube of -^ da^'b^ ? Ans. — 125a^"b\ 10. What is the eighth power of —a^xy^ ? Ans. -f a^^x^y'^ 11. What is the seventh power of — a»»J"c ?. Ans. — a'''^b''^c^ 12. What is the sixth power of 2a¥y^ ? Ans. Ua^b'^^y^^, POWERS OF FRACTIONS. 163 18. What is toe ninth power of — a^hc^ ? Ans. — a^^b^c^\ 14. What is the sixth power of — Sab^d? Ans, I29a^b^^d^ 15. What is the square of — 10a'"5"c3 ? Ans. lOOa^-'J^'ce, 16. What is the cube of — Ga-^J" Jy^ ? Ans, — 729a3*i3'(fy«, 17. What is the fourth power of — 4a^Pc*d^ ? Ans, 256a205i2ci6^2o 18. What is the cube of — Aa'^'^b^^'c^d? A71S. — 64a«'"^»6«g9^j 19. What is the fifth power of 2a^b^xy ? Ans, ^2a^^h^^^y^. 20. What is the square of 20a?»y"c*? Ans, 400a;2'»?/2'«c^c^ 21. What is the fourth power of Sa^^^n^s? Ans. 81a*"68»c'2. 22. What is the fifth power of — c'^d^^x^y'^ ? Ans, — c*'«(?i5«^b102/^^ v»u/vifw - 23. What is the sixth power of — a'^b'^^c* ? ^^ Z^ 24. What is the fourth power of — laH'^d^, Ans, 16a«c«(?". POWERS OF FRACTIONS. lao. From the definition of a power, and the rule for e mult written, the multiplication effractions, the cube of the fraction ^, Is (aV _ a a a _ a^ b) - b ^ I ^ b ^ b^' lao. What is the rule for raising a fraction to any power t 164 ELEMENTARY ALGEBRA. and since any fraction raised to any power, may be written under the same form, we find any power of a fraction by the following RULE. liaise the numerator to the required poicer for a new numerator, and the denominator to the required power f2 _ Aa¥ 4- ** • 4th power. a -b at - Aa'b 4- Qa^b^ - 4^253 ^ ab* ._ a*b + 4aW - Qa'b^ 4- ^ab* - b" (jfi -. 5a*b + lOaW — lOa^b^ 4- 5«i* — b^' Am, P0WRB6 OF BINOMIALS. 167 In the same way the higher powers may be obtained. By examining the powers of these binomials, it is plain that four thmgs must be considered : Ist. The number of terms of the power. 2d. The signs of the terms. 3d. The exponents of the letters. 4th. The coefficients of the terms. Let us see according to what laws these are formed. Of the Terms, 123. By examining the several multiplications, we shall observ^e that the first power of a binomial contains two terms; the second power, three terms ; the third power, four terms ; the fourth power, five ; the fifth power, six, &c. ; and hence we may conclude : TJiat the number of terms in any power qf a hinomial^ is greater by one than the exponent of the power. Of the Signs of the Terms, 12 + &«. As the pupil should be practised in writing the terms with their proper signs, without the coefficients, we will add a few more examples. 125. "Which is the leading letter of a binomial? What is the exponent of this letter in the first term ? " How does it change in the terms towards the right ? What is the exponent of the second letter in the second term ? How does it change in the terms towards the right ? What is it in the last term ? What is the sum of the exoonents in any term equal to ? POWERS OF BINOMIALS. 169 1. {a-\-by . ,a'-\-a^+ab^ f b\ 2. (a -by . , a*-a^b-^a^b^— ab^ + b\ 8. ia + bY_, . a''-^a'b-\ra'b'^-{-a^b^^-ab'- -\- b\ 4. ((r-6)'. ,a?-ne multiplied by th^ expo- nent of the leading letter in that term, and the product divided by the number which marks the place of the term from the left. Vie quotient will be the coefficie^it of the next term. Thus, to find th*» coefficients in the example, (a-by , . , a'- a^b 4- a^l^-a'b'-\- d'b*- a'^b'' + ab^- b\ we first place the exponent 1 as a coefficient of the second term. Then, to find the coefficient of the third term, we multiply 7 by 6, the exponent of a, and divide by 2. The quotient, 21, is the coefficient of the third tenn. To find the coefficient of the fourth, we multiply 21 by 5, and divide the product by 3 ; this gives 35. To find the coefficient of tlic fifth term, we multiply 35 by 4, and divide the product by 4 ; this gives 36. The coefficient of the sixth tenn, found 126. Wliat is the coefficient of the first term ? What is the coofficicnt ol the second term ? IIow do you find the coefficient of the third term How do you find the coefficient of any term ? What are the coofficieiite of the first and last terms ? How are the coefficients of the exponente ot any two terms equally diftant from the two extremes T 8 170 KLEMENTAEY ALGEBEA. in the same way, is 21 ; that of the seventh, 7 ; and that of the eighth, 1. Collecting these coefficients, (a - hy = a' - la^b -I- 21a^b'^-35a^b^ + ^5a^b* - 21a'^b^ 4- 7a6« — b\ Note.— We see, in examining this last result, that the voeffisients of the extreme terms are each 1, and that the coefficients of terms equally distant from, the extreme terms are equal. It will, therefore, be sufficient to find the coeffi- cients of the first half of the terms, and from these the others may be immediately written. EXAMPLES 1. Find the fourth power of a + b, Ans. a* + Aa^b + ea^J^ + ^a¥ + &♦. 2. Find the fourth power oi a — b, Ans. a* — 4a35 + ^aW — 4a63 + 5*. 3. Fmd the fiilh power of a + 5. Ans. a^ + bO'b + lOa^^^ ^ loa^i^ + 5a6* + ¥. 4. Find the fifth power of a — J. Ans. a^ — ba^b + lOaW — lOa^ft' ~\- fiab' — ^. 6. Find the sixth power of a + 5. ^6 + 6^55 4. 15^452 j^20a^b^ + 16«^6' + 606^ + 66. 8. Find the sixth power of a — b. 127. When the terms of the binomial haye coefficients, we may still write out any power of it by means of the Binomial Formula. 7. Let it be requu-ed to find the cube of 2c + Sd» (a + 5)3 = a3 + ^^b + dab^ + b^ POWBEfl OF BINOMIALS. 171 Here, 2c takes the place of a in the formula, and Zd the place of b. Hence, we have, (2c+Zdy=: (2c)3+3.(2c)2.3(;+3(2c)(3df)2+(3cf)3 . (1; and l>y perfoiining the indicated operations, we have, (2c + 3J)3 = 8c3 + 3Cc^ 5 = .7745 + = Ans, 7 2. What is the square root of - ? Ans. 1.32287 +. • * 14 3. W^hat is the square root of — ? Ans. 1.24721 +, y 4. What is the square root of Utt? ^^5« 3.41869 -}-. 1 Q 5. What is the square root of 7— 7 ? A71S. 2.71313 +. 36 6. What is the square root of 8— ? Ans. 2.88203 +. ^ 7. What is the square roct of ~ ? -^ns. 0.64549 -f . d Q 8. What is the square root of lO— ? Ans. 3.20936 -f . 134. What is the rule when the numerator atd denominator arc not 5>erfect squares ? 6QDAKE ROOT OF MONOMIALS. 181 135. Finally, insteafl of the last method, we may, if wo please, Change the common fraction into a decimal, and continue the division until the 7iumher of decimal places is double the number of places required in tJie root Then extract the root of the decimal by the last rule, BZAMPLES. 1. Extract the square of — to within .001. Thisnum 14 ber, reduced to decimals, is 0.786714 to within 0.000001 ; but the root of 0.785714 to the nearest unit, is .886; hence, 0.P86 is the root of — to witliin .001. 2. Find the \/2— to within 0.0001. A7is. 1.6931 +. V 15 3. What is the square root of r^ ^ ^^' 0.24253 +. 7 4. What is the square root of - ? Ans, 0.93541 +. 8 5 6. What is the square root of - ? Ans. 1.29099 -\-, EXTRACTION OP THE SQUARE ROOT OP MONOMIALS. 136. In order to discover the process for extracting the Bquarc root of a monomial, we must see how its squaie \k formed. By the rule for the multiplication of monomials (Art. 42)t we liave, {ba'^b'^cY = ^o^h^c X oa^b'^c = 25a*Z»«c2 ; 186. What is a second method of finding the approximate root? 186. Give the rule for extracting the squar** root of mouomials? 182 ELEMENTARY ALGEBRA. that is, in order to square a monomial, it is necessary to square its coefficient and double the exponent of each of the letters. Hence, to find the square root of a monomial, we have the following RULE. 1 Bxtract tJie square root of the coefficient for a new coefficient : n. Divide the exponent of each letter by 2, and then annex all the letters with their new exponents. Since like signs in two factors give a plus sign in the pro- duct, the square of — a, as well as that of + «, will be + a^\ hence, the square root of a^ is either + a, oi — a. Also, the square root of 25a'^64, is either + hab'^y or — hab"^. Whence we conclude, that if a monomial is positive, its square root may be affected either with the sign 4- or — ; thus, -/Oo* = ±3^2; for, + Sa^ or — Sa^, squared, gives -\- 9a*. The double sign ±, with which the root is affected, is read plus and minus, EXAMPLES. 1. What IS the square root of 64a^6*? V'64a65* z:^ ^-%aW\ for -^^aWx +8a3J2_ _|_64a65* and, Veia^^: -%am\ for -^a^b'^y. -Sa^b^z= +64a«6* Hence, y^Ia^ = ± 8a^bh 2. Find the square root of Q25aWc^. ± 25a¥c\ 3. Find the square root of blQa^¥c^. ± ^ia^^cK 4. Find the square root of 196a;y2*. ± l^xhjz^. 5. Find the square root of Ula^b^c^^d^^ ± ^la^'b^c^d^ 6. Find the square root of lS4a^W^c^^d^ ± 2Sa^b''c^d, Y. Find the square root of 81«^i*c^ ± 9a'* />V. IMPERFECT SQUARK6. 188 Notes. — 137. 1. From tho preceding rule it follows, that wbeu a monomial is a perfect square, its numerical coefficient is a perfect square^ and all its exponmts even numbers. Thus, 25a*b^ is a perfect square. 2. If the proposed monomial were negativCy it would be impossible to extract its square root, since it iias just been shown (Art. 136) that the square of every quantity, whether positive or negative, is essentially positive. Therefore, are algebraic symbols which indicate operntions that cannot be performed. Tlicy are^ called imaginai 2c2 X bbd = Zahc^/bhd, -/864a2^»Vi = '/I44a"'=^»*ci" X Qbc = Uab'^c^ y/ebc. The COEFFICIENT of a radical is the quantity without the sign ; thus, in the expressions, 7^V2a, ^abc^/5bdy 12a5VV^, the quantities 7^^ 3a5c, 12aJV, are coefficieJits of the radicals, 141. Hence, to simplify a radical of the second degree, we have the following BULE. I. Divide the expression under the radical sign into two factorSy one of which shall be a perfect square : TL Extract the square root of the perfect square^ and then multiply this root by the indicated square root of ths remaining factor, 141, Give the rule for simplifying md^.^als of the second degree. How do you deiermine whether a given n : .\ factor which is a perfect square? 186 ELEMENTARY ALGEBKA Note. — ^To determine if a given number has any factor which is a perfect square, we examine and see if it is divi- sible by either of the perfect squares, 4, 9, 16, 25, 36, 49, 64, 81, &c.; tf it is not, we conclude that it does not contain a fiictcr which is a perfect square. EXAMPLES. Reduce the following radicals to their sunplest form : 1. ^5d^hc. 2. ^/l2^h^ahP. 3. ^S2aF¥c. Ans. ba-^Zahc. Ans. W-a^dJ2}). 4. ^/2h^d^l^C^, 5. ^/^m^(F8^, 6. \/l2^a'h^c^d. 1, ^i5a?b^cH, 8. y/lUba^cM^ 10. V^TsOoio^V. 11. ^Oba/b^d^. Ans. 4a*6* \/'2aG, Ans^lQcfPc\ Ans. Z2a'^Pc- -^/abc.' Ans. 2la'^h'^c^\^abd. Ans. 15aWc^abd. Ans. l^cfc'^d^^/da. Ans. 12a*d^m'^^72. 2. V— 128a*6^ 3. V— na^h''&, 4. y/- 48a?b(f, Ans. ^ah^ — ^. ADDITION OF RADICALS. 143. SnoLAE Radicals, of the second degree, are those in which the quantities under the sign are the same. Tims, the radicals 3y^, and bc^ are similar, and so also are 9v/2, and 1'/2. 144. Radicals are added like other algebraic quantities hence, the following 148. What are similar radicals of the second degree? 144. Give th< rule for the addition of radicals of the eccond degree ? 188 ELEMENTARY ALGEBRA KFLE. I. If the radicals are similar^ add their coefficients^ and to the sum annex the common radical : n. If the radicals are not similar^ connect them together with their proper signs. Thus, 3a v^ + bc^/h — (3a -h 5c) v^. In like manner, 7v'2a + Sy^ = (7 + 3)V5a = 10^/2a. Notes. — 1. Two radicals, which do not ajjpear to be sim- ilar at first sight, may become so by transformation (Art. 141.) For example, V48aP + h^/Wa = 4:b\/da + 5b^3a = Qb^/Sa; 2v^ + 3-/^ = Q^/5 + S^ = 9^5. 2. When the radicals are not similar, the addition or sub- traction can only be indicated. Tims, in order to add 3 \/h to 5-v/«, we write, 5ya 4- 3v^. Add together the following : 1. 'v/27a2 and x/48a\ Ans. *lay/^, 2. ^/hWd^ and ^2af¥. Ans, Ua^b^/2. 3. v/— - and \/-r-' Ans. 4a^ V 5 V 15 4. ^125 and y^OOa^. Aiis, (5.-{- 10a) y^ ^ /5o" , /Too . 10 r- 8UBTBA0TI0N OF BADI0AL8. 189 6. V98a2« and ySCar^ — 3Qa\ Ans, YayS + 6Vaj2__ a\ 1. -v/OSa^ and v^288a*x*. -4n«. {la -^ I2ah:'^)'/2x. J 8. V^ and -/TSs. 9. -^27 and -/liV- -4n5. U^. -4?is. lOv/2; ^ Ans, 30^ 3, (2a 4- 24aj2)y^. Ans. 119-v/3. 11. 2V^ and 3-/64&C*. 12. v^43 and loVsca. 13. y/^20a^^ and ^2450^. ^W5. (8aJ + 7a*53)v^ 14. v^5a*^ and -/SoOo^. -4n«. {5a^b^ -i- 10^3^.2)^5; D suBTRAcnorr of eadioai^. 145, Radicals are subtracted like other algebraic quan- tities ; henre, the following EU LE. I. If the radicals are similar, subtract the coefficient of the subtraJiend from that of the minuend^ and to t/ie differ^ eiice annex the common radical : n. Jf the radicals are not similar, indicate the operation by the mintis sign. EXAMPLES. 1. What is the difference between Say/b and ay^? Here, Sa-y/S — ay^ = 2ay/b, Ans, 145. Give the nile for the subtraction of mdlcals. 190 ELEMENT All Y ALGEBRA. 2. From 9a -^2^ subtract 6a ^27^. First, 9av^762 = 27a5'v/3, and ^a^/¥iW' = lSab\/3; and, 27 ab^ — ISab^ = 9ab\/3. Ans, Find the differences between the following : 3. ^15 and -v/iS- 4. -v/24a252 and \/E4b\ Ans. {2ab — db^)y^. 6. y/l2SaW and 'v/32a9. 7. v^48a3^ and Vdab. Ans. —Vis. 45 Ans. {8ab — 4a*) ^20] -4ws. 4abV3ab — S\/ab. 8. V242a52,5 and ^2a^K Ans, {Ua^^ — ab)\/2ab. ^- \/l ^^^ vl 10. V'320a2 and ^80a\ Ans. --/3. 6 Ans, 4a y^. 11. ^^200^ and ^24.5abc^d\ Ans, {12ab — Ycc?)-/5a6. 32. -v/oeSo^^ and v^OOo^^. u4w«. 12aJv^ 13. ■v/ll2a^ and '/28a86«. Ans. 2a^b^y/V, MULTIPLICATION OF RADICALS. E46. Radicals are multiplied like other algebraic quan* litics ; hence, we have the following BULE. I. Multiply the coefficients together for a new coefficient: 146. Give the rule for the multiplication of radicals. DIVISION OF EADI0AL9. 191 n. Multiply togetJier the quantities under the radical signs : in. Tlien reduce the result to its simplest form, 1. Multiply 3a V^ hj 2\/ab, da\/bc X 2^0^ = 3a X 2 X y^ X V^' which, by Art. 139, = Qa^/b^ac = Qab-^, Multiply the following : 2. S\/5ab and 4v^0a. Ans. 120a V^, 8. 2aV?c and 3aV?c. Ans, 6ii^bc, 4. 2a-v/aM^ and - Sa^a^+b\ A. - 6a'(a'^ + b\) 5. 2a5-/a + ^> and ocy^a — b. Ans. la^c^a^ — l^. 6. ny^ and 2-v/8. Ans. 24. ^ 7. \y/lc^ and y»y-/|^. ^n5. ^JyaJcyTs; 8. 2aj + v^ and 2a; — y^. Ans, Aa? — h 9. \/a 4- 2-/b and -v^a — 2-v/6. -4?i5. y^a^ — 45. 10. 3aV^7a3 by y^. -4w«. 9a3y/6" DIVISION OP RADICALS. 147. Radical quantities are divided like other algebraic quantities ; hence, we have the following BULB. L Divide the coefficient of the dividend by the coefficient of the. divisor^ for a new coefficient : 147. Give the rule for the division of radicals. 192 ELEMENTARY ALGEBRA. II. Divide tJie quantities under the radicals, in tJie same manner : m. The7i reduce the result to its simplest form, EXAMPLES. 1. Divide 8a y^ by 4a y^. 8a ^ «... ■— = 2, new coemcient. 4a hence, the quotient is 2 X - = • c c 2. Di\ide 5a\/h by 2by/c. Ans. -^v/-- 3. Divide 12ac-}/6bc by 4c y^. Ans. 3a ^3^. 4. Divide 6a-v/966* by SyW. ^?i5. 4aJV^ 5. Di\ide 4a^^/50b^ by 2a2y^. ^ns. 2b^^^, 6. Divide 26a^b\/81aFb^ by 13av/9a^. A. ea'^b^/ab. 1, Divide 84a35*-/27ac by 42a5-/3a. ^. ea^j^-y/c. 8. Divide ^/\c^ by V^. ^ws. la. 9. Divide Qa^b'^^/20a^ by 12-v/5a. Ans. a^b\ 10. Divide 6a -v/lO^ by Z^/b, Ans, 2ab^, 11. Divide 486* y^ by 25^-/^ ^m. Zmb\ 12. Divide 8a25*c3V^ by 2a^2M. Ans. 2a¥cH, 13. Divide OGa^c^y'gS^ by 48a5cV^. -4. Ua^^ic^. t^QUARE BOOT OF POLYNOMIALS. Wd 14. Divide 27a«i«-|/2ia3 by -/Ta- ^'*^- 27a«6°V5. 16. Divile ISa^^VSo* by Qaby/a\ Am. Qa^b^-y^. SQUARE ROOl OF POLYNOMLA.LS. I4H. Before explaiiimg the rule for the extraction of the bqiiare root of a polynoinial, let us first examine the squares of Beveral polynomials : we have, (a + by = a^ + 2rt6 4- b\ (a + b -^ cy = a^ + 2ab + b^ + 2{a -f- b)c -}- c», {a -h b + c -{- dy = a^ + 2ab -\- b^ -\- 2(a •}- b)c -\- cr" H- 2(a 4- * + c)d -r c^. The law by which these squares are formed can be enim ciated thus : The square of any polynomial is equal to the square of tfie Jirst term^ plus twice the j/roduct of the first tenn by the second^ plus tlie square of the second ; plus twice the first two terms midtipUed by the thirds plus the square of the third ; plus twice t/ie first three terms 7mdtiplled by the fourth, plus the square of the fourth; aiid so on. 149. Hence, to extract the square root of a pol}^lomial, w L' have the following BULB. L Arrange the polynomial with reference to mve of its letters^ arid extract tfie square root of the first term : this will give the first term of the root : 148. What is the square o^ a binomial equal to? What is the sqnaro of a trinomial equal to? To what is the square of any polynomial equal? 149. Give the rule for extracting the mjunre root of a polynomial? Wuat is the first step* Wl at the second ? WiuM the third ? What the fourth ? 9 IM ELEMENTARY ALQEBKA. n. Divide the second term of the polynorrdal by double, the first term of the rovt^ and the quotient will be the second term of the root : nL Then form the square of the algebraic sum of the two terms of the root founds and subtract it from the first polynomials and then divide the first term of the remainder hy double the first term of the root^ and the quotient will be the third term : rV. Form the doid)le product of the sum of the first a?ul second terms by the third, and add the square of the third ; then subtract this residt from the last remainder, and divide the first term of the result so obtained, by double the first term of the root, and the quotient will be the fourth term. Then proceed in a similar manner tofitid the other terms. EXAMPLES. 1. Extract the square root of the polynomial, 49^2^2 _ 24aZ>3 -f 25a* - ^Od'b + 16Z>*. First arrange it with reference to the letter a. 25a* - 25a*- - 30a^6 + 49a2^>2 _ - 30a^5 + Qd'b'^ - 24ab^ + IQb^ 5a2 _ Sab 4- 4b^ 10a2 40a^b^ - 40a'62 - - 24a53 + IGJ* - 2-iab^ + 16b^ . . . . . . 1st Hem, . . 2d Bem. After having arranged the polynomial with reference to a, extract the square root of 25a* ; this gives ba^, which is placed at the right of the polynomial : then divide the second terra, — SOa^S, by the double of bd^, or 10a-; the quotient is — ^ab, which is placed at the right of ba^. H(mce, the first two terms of t*ie root are ba^ — 3ab. Squaring this binomial, it becomes 25a* — SOd^b + 9a'^b% which, subtracted from the proposed polynomial, gives a remainder, of which the first terra is 40a,'^b\ Dividing this 8QDABB ROOT OP P O L I N ;> M I A t B . 195 first term by 10a , (the double of Sa^), the quotient ia ■f 4ft2 ; this is the third term of the root, and is written on the right of the first two terms. By forming the double product of 5a^ — Sad by 4^^^ squaring ib\ and taking the sura, we find the polynomial 40a'^b^ — 24ab^ -\- 106% which, subtracted from the first remainder, gives 0. There- fore, ba^ — 3ab + 4b^ is the requii*ed root. 2. Find the square root of a*+ 4a^x-\-Qa^x^+Aaa:^-{- jb*' Ans. a^-h 2aa; -h x^, 8. Fmd the square root of a*— Aa^x-^-Qa^x"^— 4ax^-\- x\ Ans, a} — 2ax + sc*. 4: Find the square root of 4x^ + 12x* -\- 5x* - 2x^+ 1x^ - 2a; + 1. Ans, 2x^ -\- dx^ — X -h 1, 5. Find the square root of 9a* - I2a^b + SSa^J' - IQab^ + 166*. Ans, 3a2 _ 2ab + 4^2. 6. Wliat is the square root of «♦ — 4ax^ + 4a2a;2 — 4x^ + Sax -\- 4? Ans. x^ — 2ax — 2. 7. What is the square root of 9a;2 — 12a; + Gary f y^ — 4y + 4 ? A71S, Sx f y — 2, 8. What is the square root of y* — 2y^ar* 4- 2a;2 _ 2y» + 1 + a:* ? Ans. if~9?^i, 9. What is the square root of 9a^6* — 30a^63+ 25a^b^? Ans. Za^b^ — 5ab, 10. Find the square root of 25a*A* - 40a362o -r 16a^^c^ - 4Sab^c^ f 3C6V — SOa^bc -f 24a36c* - SQa^bc^ -f 9a*c2. X A?is. ba^b - Za'^c — 4abc + 06o^ 196 ELEMENT AKY ALGEBKA. 150. W3 will conclude this subject with the following remarks : 1st. A binomial can never be a perfect square, since we know that the square of the most simple j)olynomial, viz. a binomial, contains three distinct parts, which cannot ex perience any reduction amongst themselves. Thus, the exj)ressi()n a^ -\- b"^, is not a perfect square ; it wants the term ± 2aby in order that it should be the square of a ± 6. 2d. In order that a trinomial, when arranged, may be a perfect square, its two extreme terms must be squares, and the middle term must be the double product of the square roots of the two others. Therefore, to obtain the square root of a trinomial when it is a perfect square : Extract the roots of the two extreme terms, and give these roots the sams or contrary signs, according as the middle term is positive or negative. To verify it, see if the double product of the two roots is the same as the middle term of the triJiomiaU Thus, 9a^ — 48^4^2 j^ 64a^^*, is a perfect square, smce, v^So^ = Sa^, and -/e^o^^^ = — Sad^ ; and also, 2 X 3a3 X — 8a52 = — 48a*^>2 = the middle term. But, 4a2 _}- 14^5 + 9Z>2 is not a perfect square : for, although 4^2 and + 96^ are the squares of 2a and 36, yet 2 X 2a X 36 is not equal to 14a6. 3d. In the series of operations required by the general rule, when the first term of one of th^ remainders is not exactly divisible by twice the first term of the root, we may 160. Can a binomial ever be a perfect power? Why not? When is h trinomial a perfect square ? When, in extracting the square root, wc find that the first term of the remainder is not divisible by twice the root, ifl the polynomial a perfect power or not? SQUARE ROOT OF POLYNOMIALS. 197 oonclude that the proposed polynomial is not a perfect square. Tliis is an evident consequence of the course of reasoning by winch vre have arrived at the general rule fo? extracting the square root. 4ih. When the polynomial is lot a perfect square, it may sometimes be simplilied (See Ait. 139). Take, for example, the expression, -^a^b 4- 4a^b^ -f 4ab^ The quantity under the radical is not a perfect square ; but it can be put under the form a5(a^ + 4ab -h 4h^.) Now, the factor within the parenthesis is evidently the square of a + 26, whence, we may conclude that, \^a^b + 4aV)^ + 4<(b' = (a + 2b) y/ab, 2. Reduce -/2a»6 — 4ab'^ -\- 2b^ to its simplest form. Ans, (a — h) -v/SS 198 ELEMENTARY ALGEBRA CHATTER Yin, BQUATIONS OP THE SECOND DEGREE. EQUATIONS CONTAINING ONE UNKNOWN QUANTITY. 151. An Equation of the second degree containing but one unknown quantity, is one in which the greatest exponent is equal to 2. Thus, ' x^ =z a, ax^ -\- bx = c, are equations of the second degree. 152. Let us see to what form every equation of the second degree may be reduced. Take any equation of the second degree, as. Clearing of fractions, and perforirdng indicated operations, Nve have, 4 -f 805 -h 4a;2 — 3a; — 40 = 20 — « + 2x\ Transposing the unknown terms to the first member, the kno\^Ti terms to the second, and arrangmg with reference to the powers of a?, we hftve, 4a;2 — 2a;2 + Sic - 3a; + a; rr: 20 + 40 — 4 ; 151. What is an equation of the second degree? Give an example. 152. To what form may every equation of the second degree be reduced? EQUATIONS OF THE SECOND DEGREE. 199 and, by reducing, 2x^ + Qx = 56 ; dividing by the coefficient of jc^, we have, a;2 + 3a; = 28 If we denote the coefficient of x by 2;>, and the second member by y, we have, a* + SjtxB = q. Tins is called the reduced equation. 153. When the reduced equation is of this form, it con- tains three terms, and is called a complete equation. The terms are, First Term. — ^Tlie second power of the unknown quan- tity, vnih a plus sign. Second Term. — ^The first power of the unknown quantity, with a coefficient. Third Term. — A known term, in the second member. Every equation of the second degree may be reduced to this form, by the following rule. L Ch"r tJii' equation of fractions^ and perform all the indicated operations : n. Tran,spose all theunhwy and v/^ . 208 ELEMENTARY ALGEBRA. 6. The sum of the squares of two numbers is 117, and the difference of their squares 45 : what are the numbers ? A?7S. 9 and 6. 6. The sum of the squares of two numbers is a, and the difference of their squares is b : what are the numbers ? . fa -^ b fa 1. What two numbers are those which are to each other as 3 to 4, and the sum of whose squares is 225 ? Ans. 9 and 12 8. Wliat two numbers are those which are to each other as m to n, and the sum of whose squares is equal to a^ ? ma na 9. Wliat two numbers are those which are to each other as 1 to 2, and the difference of whose squares is 75 ? Ans. 5 and 10. 10. What two numbers are those which are to each other as m to n^ and the difference of whose squares is equal to b"^ ? mb nb Ans. ^/wfi — n^ 'y/n 11. A certain sum of money is placed at interest for six months, at 8 per cent, per annum. Now, if the sum put fit interest be multiphed by the number expressing the interest, the product will be $562500 : what is the principal at m- tc^rest? Ans. $3750. 12. A person distributes a sum of money between a num- ber oi women and boys. The number of women is to the number of boys as 3 to 4. Now, the boys receive one-half as many dollars as there are persons, and the women, t^^ce as many dollars as there are boys, and together they receive COMPLETE EQUATIONS. 209 138 dollars : how many women were there, and how many boys? . j 36 women. ( 48 boys. COMPLETE EQUATIONS. t^H The reduced form of the complete equation (Art, 168) is, Comparing the first member of this equation with the pquare of a binomial (Art. 54), we see that it needs but the square of half the coefficient of a;, to make it a perfect square. Adding p^ to both members (Ax. 1, Art. 102)^ we have, x^ + 2px -\- p^ z=z q ■\- p\ Then, extractmg the square root of both members (Ax. 5), we have, X -\- p z=z ± 'v/^~T^. Transposing p to the second member, we have. X = — p ± vY-1- P^' Hence, there are two roots, one corresponding to the 2^U8 sign of the radical, and the other to the minus sign. De- noting these roots by a' and a", we have, cb' = — /) + \q-\- p\ and a" = — p — ^/q -\- p^. The root denoted by x' is called the first root ; that de- noted by x'* is called the second root, 168. What is the form of the reduced equation of the second to be negative, and q positi\'e we shall then have, x^ — 2px = q (2.) 3d, Let us suppose 2/> to be positive, and q negative ; we shall then have, jb2 4- 2;xc = - 5-. . . . ( 3.) 4th. Let us suppose 2p to be negative, and q negative ; we shall then have, x^ — 2px = — q. ... (4.) As these are all the combinations of signs that can take plao-e between 2p and y, we conchide that every complete equation of the second degree will be reduced to one or the other of these four forms : a^ + 2px = + q, . . 1st form, aj* — 2px = + J', . . 2d form. x^ + 2jyx z= ^ q, , . 3d form. x^ — 2/)ic = — q^ . . 4th form. EXAMPLES OF THE FIEST FORM. 1. What are the values of a; in the equation, 2a;2 + 8a; = 64 ? If we first divide by the coefficient 2, we obtam a;2 4- 405 = 32. and 7 in the first form? What in the second? What in the third? V-Tiat in the fourth ? 212 ELEMENTARY ALGEBEA. Then, completing the square, a2 -f- 4a; + 4 rzi 32 + 4 = 36. Extracting the root, a + 2 = ± V 36 = + 6, and — 6. Hence, x' =. — 24-6 — +4; and, a" = — 2 — 6 = — 8. rience, in this form, the smaller root, numerically^ is positive and the larger negative. VERIFICATION. If we take the positive value, viz. : a' = +4, the equation, a^ _^ 4a. _ 32^ gives 42 -h 4 X 4 = 32 ; and if we take the negative value of jc, viz. : jc" — — 8, the equation, q(? -\- ^x =: 32, gives (— 8)2 + 4(— 8) = 64 - 32 r=z 32; from which we see that cither of the values of a, viz.j k' = +4, or x" = — 8, will satisfy the equation. 2. What are the values of x in the equation, 3a;2 -f 12a; — 19 = — x^ — 12a; + 89? By transposing the terms, we have, 3a;2 4- ^2 + 12a; + 12a; = 89 4- 19 ; and by reducing, 4a;2 4- 24a; =108; and dividing by the coefficient of x^^ aj2 4- 6a; = 27. COMPLETE EQUATIONS. 21b Now, by coinpletbig the square, ar^ + 6a; -I- 9 = 36 ; extracting the Bquarc root, a; 4- 3 = ± -/s'c. = + 6, and - 6: hence, a:' = -f6 — 3 = 4-3; and, a;" = — 6 — 3 = — 9. VEUIFICATION. If we lake the plus root, the equation, x^ + Gx = 27, gives (3)'^+ 0(3) = 27; and for the negative root, a:2 -h 6a; = 27, gives (- 9)2 + 0(- 9) = 81 - 64 = 27. 3. What are the values of a; in the equation, a;2 - lOa; -f 15 = - — 34a: + 165 ? 5 By clearing effractions, we have, 5^2 - 50a: 4-75 = x^ — 170a; 4- 776; by transposing and reducing, we obtain, 4a^ 4- 120a; =700; then, dividing by the coefficient of a^y we have, a;2 4- 30a; = 175; and b} completing the square. ar' 4- 30d; 4- 225 = 400 *214 ELEMENTARY ALGEBRA. and by extracting the square root, a -j- 15 = iy'loo = + 20, and — 20. Hence, x' — +5, and x" — — 35. VERIFICATION. For the plus vahie of a;, the equation, a;2 _f- 30a; = 175, gives, (5)2 4- 30 X 5 = 25 + 150 = 176. And for the negative value of a;, we have, (_ 35)2 ^ 3o(_ 35) - 1225 — 1050 ~ 175. 4. "VvTiat are the vahies of a; in the equation, Clearing of fractions, we have, 10a;2 _ 6a; + 9 = 96 — 8a; — 12a;2 -f 273; transposing and reducing, 22a;2 -f- 2a; = 360 ; di\dding both members by 22, 2 _ 360 "^ "^ 22^ - 22 ' Add I — ) to both members, and the equation becomes, o . 2 . / 1 \2 360 / 1 \2 '^ + 22='+ (22) =^ + (25)' whence, by extracting the square root, 22 ~ "" V 22 "^ \ '^2 / ' COMPLETE EQUATIONS. 216 therefore, . 1 , /3C0 . /T\2 ^= -2-2 + V-22- + 12-2)' and, ."= __l_,/!^+(iy^ 22 V 22 V 22/ It remains to perform the Dmnerical opemtions. In the first place, 360 22 "^ (y. mast be reduced to a single nmnber, having (22)^ for its denominator. Now, 3G0 / 1 Y 22 \22/ ~ 3C0 X 22 4- 1 7921 (22)2 - (22)2' extracting the square root of 7021, we llnd it to be 89 j therefore. /3G0 / i~y _ ?? V 22 "^ \22/ ~ "^ 22* Consequently, the plus value of x is, * ~ 22 22 ~ 22 ~ ' and the negative value is, "_ - 2 _ ?? — _ 1^. ^ ~" 22 22 ~" ~ 11 ' that is, one of the two values of x which \»t11 stitisfy the proposed equation is a positive whole number, and tlie other a negative fraction. Note. — Let the pupil be exercised in writing the roots, in the last five, and in the following examples, without com- plcting the square. 216 ELEMENT AEY ALGEBRA, 6. What are the values of x in the equation, j JC' =: 5. \x"- - r>% Ans. 6 What are the values of a; in the equation, 2a;2 + 8a; + 7 = ^ ~ ¥ "^ "^^^^ A71S. i „ ( x" 7. What are the values of x in the equation, -4 15 = ^ - 8a; 4- 95i ? = 8. -''' {J- 9. -64f 8. Wliat are tlie values of x in the equation, a;^ 5a; £c ^ ^ o ____8 = --7a: + 6i? ( x"= - 7f 0. What are the values of a; in the equation, x^ x _ x^ X 13 A j a;' = 1, Ans. -j^,,^ _ % KXAMPI^S OF THE SECOND FORM. I. What are tbe values of x in the equation, a;2 _ 8a; + 10 =: 19? COMTLKTE EQUATIONS. 217 By transposing, aj2 - 8a; = 19 - 10 = 0; then, by completing the square, jc2 _ 8aj -f 16 = 9 -h 10 = 25 ; and by eicti-acting the root, a; - 4 = ± V^ = + 5, or -6. Hence, a' = 4 + 5 = 9, and a" = 4 — 6 =. - 1. That is, in this form, the larger root, numericallyj ie positive, and the lesser negative. VERIFICATION. If we take the positive value of cc, the equation, x^ — 8x = 9, gives (9)^ — 8x9 = 81 — 72 = 9; and if we take the negative value, the equation, a^ - 8aj = 9, gives (- 1)2 — 8 (— 1) = 1 + 8 = 9; from which we see that both roots alike satisfy the equa- tion. 2. What are the values of x in the equation, X^ X 3*^ By clearing of fractions, we have, 6a;2 + 4a; _- leo = 3x^ + 12aj — 177 ,* and by transposing and reducing, Sx^ — 6x z= 3 ; and dividing by the coefficient of cb*, we obtmn. 218 ELEMENTAKY ALGEBRA, Then, by completing the square, we have, 2 8 ,16 , , 16 25 3^9 ^9 9 ' and by extracting the square root, 4 ^ /25 5-6 a; — -= ±\/ — = +-, and -- - 3 V 9 3' 3 Hence, «^=5 + -3 = +«'-^»"=S-i = -i VERIFICATION. For the positive root of x, the equation, 2 8 x---x = 1, Q gives 32 — -x3 = 9 — 8 = 1 3 and for the negative root, the equation, 2 8 a' - 3^ = 1, / IV 8 1 1,8 ^^^^^ l-3J-3^-3 = 9 + 9 = ^- 8. What are the values of a; in the equation, ?-! + Vf = 8? 2 3 * Cleanng of fractions, and dividing by the coefficient of x', we have, cc^ - |c = H. COMPLETE EQUATIONS. 219 Completing the square, we have, 2 2,1 ,,1 49 ^ -3^ + 9 = '* + 9 = 3a' then, by extracting the square root, we have, X = +\/ — = H — , and — - ; 3 "^ V 36 6 ' . 6 * hence, ^ =3 + 6 = 6 = '^' ^"^ ^ =3-6 = -6' VERIFICATION. If we take the positive root of x, the equation, gives OiY-l X li = 2i - 1 = li; and for the negative root, the equation, / 6V 2 6 25 . 10 45 ^'''' r e) - 3 ^ - 6- = -3l^ + rs = 36 = ^' 4. What are the values of JC in the equation, 4^2 _ 2a;2 4- 2az - I8ab — 186^? By transposing, changing tlie signs, and dividing by 2, the equation becomes, x^ - ax = 2a'^ — 9ab -^ 9^ ; 220 ELEMENTARY ALGEBRA. whence, completing the square, 4 4 extracting the square root, a /9a^ 9a^ Now, the square root of — ■ — 9ab + ^b'\ is evidently - ^ 35. Therefore, '" = 2±(t-'*)'«°'^ ia="=- a + Sb. What will be the numerical values of x, if we suppose a = 6, and 5 = 1? 5. What are the values of a; in the equation. 1 4 jB — 4 - a;2 4- 2a; — -a;2 = 45 — 3a;2 + 4a; ? . J a;' = 1.12 ) to ^ ^''''\x-= -5.73f within 01. 6. What are the values of a; in the equation, 8a;2 — 14a; + 10 = 2a; + 34 ? 7. What are the values of x in the equation, J - 30 + a; = 2a: - 22 ? ^"^- { J' Z !i 1 8. Wliat are the values of a; in the equation, 2 a;2- 3a; -f ^ "^^ 9a; + 13^? Afis. \x' =9, I a-" = - 1. COMPLETE EQUATIONS. 221 9. What are the values of a; in the equation, Caa; — a;2 = — 2ah — ft^? ( x' = 2a -f 6. ^""''ix^'^ -6. 10. What are the values of x m the equation, 0» + 52 __ 2525 + JT* = — Y-? Atis. EXAMPLES OP THE THIRD POEM. 1. What are the values of a; in the equation, a:2 + 4a; = — 3 ? First, by completing the square, we have, a^ 4. 4a; 4-4 = -3 + 4 = 1; and by extracting the square root, a; -f 2 = ± -/l = +1, and — 1 ; hence, a;'=— 2 + l = -l; and a;"=— 2-l = -3 That is, in this form both the roots are negative. VERIPICATION. If we take the first negative value, the equation, a;2 4. 4a; = — 8, gives (- 1)24- 4(- 1) = 1 - 4 = - 8; and by talong the second value, the equation, a;' -f 4a; = ~ 3, 222 ELEMENTARY ALGEBRA. gives (— 3)2 +4( - 3) = 9 - 12 = ^ 3 ; hence, both values of x satisfy the given equation. 2. Wliat are the vahies of aj in the equation, - |- - 5a; - 16 = 12 + ia;2 ^ 6a5? By transposing and reducing, we have, — a;2 — lla; = 28; then, dividing by — 1, the coefficient of x\ vre have, a;2 + lla; = — 28; then, by completing the square, a;2 -h Ux -\- 30.25 = 2.25; hence, a; + 5.5 = ± -/2.25 = + 1.5, and — 1.5 ; consequently, a' = — 4, and a" z= — 7. 3. Wliat are the values of x in the equation, x^ 1 — - — 2x — 5 = -x'^ + 5x + 5? 8 8 . j a;' = - 2. Ans. ^ ''\t' ~ 6. 4. What are the values of a; in the equation. 2a;2+ 8a; = - 2| - ?a;? . (a;' = ~i ( a;" = --4 5. What are the values of a; in the equation, - 4a;2 4a;2 -f- ?aj + 3a; = - 14a; - 3| - 4a;2? Ans. J-- - ^- COMPLETE EQUATIONS. 233 6. What are the values of as in the equation, 4 2 ^-iJ.iiJ; 1. What are the values of jb in the equation, ^ + 7a; -h 20 = - ?z2 - 11a; - 60 ? 9 a ^'"•1J'= ' - 8. - 10. 8. What are the values of a: 1 in the equation. *-■ -.! = - 9ia:- 6 2 - 8. 9. What are the values of x in the equation, lar»-f6a;+i=~^2- 5^x - ? ? 5 4 O 4 ^"^^ ( a;" = - 10. 10. What are the values of a in the equation, 11. What are the values of a; in the equation, aJ + 4a; — 90 = — 93 ? A J*' = -1. BTAMPLES OF TOE POITRTH FORM. I. What are the values of x in the equation, a;2 - 8a; = - 7 ? 224 ELEMENTARY ALGEBRA. By completing the square, we have, a;2 - 8a; + 16 = - 7 -f 16 = ,9 then, by extracting the square root, a; — 4 = ± y^ = + 3, and — 3 ; hence, x- = 4-7, and aj" = +1. That ig, in this form, both the roots are positive. VERIFICATION. It* we take the greater root, the equation, a;2 _ 8a5 = — 7, gives, 7^ — 8 X 7 = 49 — 56 := — 7; and for the lesser, the equation, x^ — 8a; = — 7, gives, 1^ — 8x1 = 1 — 8= -7 hence, both of the roots will satisfy the equation. 2. What are the values of x in the equation, — \\x^ 4- 3a; - 10 = \\x^ - 18a; + 17? By clearing of fractions, we have, — 3a;2 4- 6a; — 20 = 2>x^ — 36a; 4- 40 ; then, by collecting the similar terms, — ^x^ 4- 42a; = 60 ; flien, by dividing by the coefficient of a;'^, which is — 6, we have, a;2 - 7a5 = - 10. By completing the square, we have, a;^ — 7a; f 12.25 — 2.25, COMPLETE EQUATIONS. 225 and by extracting the square root of both members, a; - 3.5 = ± y^^ = + 1.5, and — 1.5; hence, x' — 3.5 4- 1.5 = 5, and aj" = 3.5 — 1.5 = 2. VERIFICATION. If we take the greater root, the eqnation. SB* - 7x = - 10, gives, 52 - 7 X 6 = 25 - 35 = - 10: and if we take the lesser root, the equation, k2 — 7a; = — 10, gives, 2^ — 7 X 2 = 4 — 14 - — 10. 3. What are the values of x in the equation, -^ 3a; 4- 2a;2 + 1 = l7|a; - 2a;2 - 3 ? By transposing and collecting the terms, we have, . 4a;2 — 20|a; = — 4 ; then dividing by the coefficient of ai^, we have, a;2 - 6}a; = - 1. By completing the square, wc obtain, , ^, 169 , , 169 144 ^-^i^+^ = -^ + -25- =^5 and by extracting the root, hence, , «, . /r44 12 , 12 a:' = 2| + -^ = 5, and, a;" = 2J - \^ = 1. a 5 5 VERIFICATION. If we take the greater root, the eqnation, i' — 5|a: = — 1, gives, o* — 5| x 5 = 26 — 26 — — 1 ; 226 ELEMENTARY ALGEBRA. and if we take tLe lesser root, the equation, ^-H^=-^ gives, (^j - 51 X -=---=:- 1 4. What are the values of x in the equation, \^-^- + l = -f' + l--l-' Ans. 5. WTiat are the values of a; in the equation, 1 1 ja' = 3 — 4a;2 _ a; 4- i| = _ 5x^ + Sx? ^^^•1^-= t 6. WTiat are the values of x in the equation, _ 4a;2 _}. l_x = — 3a;2 — --a; + — ? ^ 20 40 20 ^ 40 ^^^•{J'= t 7. What are the values of x in the equation, a;2 _ lOJ^JB = — 1 ? . ja;' =^ 10. 8. What are the values of x in the equation, 1 7'r2 97*2 - 27a; H ^ + 100 = ^ 4- 12a; - 26? 5 5 Ans j a;' =7. * ( a" -^- 0. 9. What are the values of x in the equation, ja;' = 0. ( a:" r= I. 22a; + 15 = '— + 28a! - 30 3 3 Ans. PR0PEBTIK8 OF EQUATIONS. 10. Wliat are the values of as in the equation, 2aj»-30a;-f3 = - x' + 3tV« - ^^ Arts. \^ =-- 11, ( X" = A. PROPERTIES OF EQUATIONS OF THE SECOND DEOREK FIRST PROPERTY. 161. We have seen (Art. 153), that every complete equation of the second degree may be reduced to the form, a^ -I- 2px = q (1.) Completing the square, we have, transposing q + p^ lo the first member, x2 + 2pa; + /)2 - (^^ -h p") = 0. . (2.) Now, since x^ + 2/>a; + p^ \& the square of sc 4- 7>, and q ^ p^ the square of \/q~+ jt>*, we may regard the first member as the difference between two squares. Factoring, (Art. 66), we have, (aj + /> -h -/ ? 4- p') {x + p - V7+~P) = 0. . ( 3.) Tliis equation can be satisfied only in two ways : 1st. By attributing such a value to a; as shall render the first factor equal to ; or, Ifll. To what form may every equation of the second degree be re- duced? What form will this equation take after completing the square and transposing to the first member? After factoring? In bow many ways may Equation ( 8 ) be satisfied ? What are they ? How many roots has every equation of tho «orftT '' -i-^-, ■> ? 228 ELEMENTARY ALGEBRA. 2d. By attributing such a value to x as shall render the Becond factor equal to 0. Placing the second factor equal to 0, we have, aj ^-p — ^/q ■\-i9- — 0; and a' = — p -\- Vq~+^* (4.) Placing the first factor equal to 0, we have, ■i-p 1' yq+2^^ = ^'i ^^^ a;" = —p — y/q ^p\ (6,) Since every supposition that will satisfy Equation ( 3 ), will also satisfy Equation ( 1 ), from which it was derived, it fol- lows, that x' and a" are roots of Equation ( 1 ) ; also, that Every equation of the second degree has two roots, and only two, Note. — ^The two roots denoted by a;' and a", are the same as found in Art. 158. SECOND PROPERTY. 162. We have seen (Art. 161), that every equation of the second degree may be placed under the form, {x -\- p + -y/q + p^) {x -^ p — Vq~^^^) = 0- By examining this equation, we see that the first factor may be obtained by subtracting the second root from the unknown quantity a; ; and the second factor by subtracting the j^rs^ root from the unknowoi quantity jb; hence, JEoery equation of the second degree may he resolved into two binomial factors of the first degree, the first terms, in both factors, being the unknoicn quantity, and the second terms, the roots of the equation, taken loith contrary sigyis. 162. Into how many binoniinl factors of the first degree may every equation of the second degree be resolved? What are the first terms of thcee factors ? What the second ? FOKMATION OF EQUATIONS. 229 TIIIUD PROPERTY. 163. Il' we add Equations (4) and (5), Art. 161, we have, a' = — p 4 ^/q -j- p^ ." = - /> - V? + i>^ as' 4- a:" = — 2/> ; that is. In every reduced equation of (he second degree^ the sum of the two repots is equal to the coefficient of the second term \a^/^ taken with a contrary sign. FOURTH PPvOPERTY. 164. If we multiply Equations (4) and (5), Art. 161, member by member, we have. x' X x" = (— 7> 4- Vq 4- p^) {- P — V5-KP^) = jo2 - (2- + jt)2) = _ 5^; that is, In every equation of the second degree^ the product of the two roots is equal to the knoimi term in the second mem- ber^ taken with a coiitrary sign. FORMATION OF EQUATIONS OF THE SECOND DEGREE. 165. By taking the converse of the second property, i \y\. 102), we can form equations which sliall have given iliat is, if they are known, \\ <• cm liiid the corre- gpouding equations by the following RULE. ^ L Subtract each root from the tmknown quantity : ^ 163. What is the algebraic ecm of the roots equal to in every eqiiatioii of the second degree ? 164. What is the product of the roots equal to? 166. Ifow will yon 6nd th#«!qiifttion whor the roots ar€ known ? 230 ELEMENTARY ALGEBRA. n. Multiply the results together^ and place their prodiict equal to 0. ^, EXAMPLES. Note. — Let the pupil prove, in every case, that the roots will satisfy the third and fourth properties. 1. If the roots of an equation are 4 and — 5, what is the equation ? Ans. x^ -{- x = 20. 2. What is the equation when the roots are 1 and — 3 ? A71S. cc2 -f 2a; = 3. 3. What is the equation when the roots are 9 and — 10 ? A?is. x'^ -\- X = 90. 4. What is the equation whose roots are 6 and — 10? Ans. x^ -\- 4:X = 60. 6. Wliat is the equation whose roots are 4 and — 3 ? Ans. x"^ — X =z 12. 6. What is the equation whose roots are 10 and — y^ ? Ans. x^ — 9-i-\£c = 1. 7. What is the equation whose roots are 8 and — 2 ? Ans. a;2 — 6a; =: 16. 8. What is the equation whose roots are 16 and — 5 ? Ans. x^ — 11a; z= 80 9. What is the equation whose roots are — 4 and — 5 ! Ans. a;2 4- 9a; = — 20. to. What is the equation whose roots are — 6 and — 7 ? Ans. a;2 + 13a; = — 42. g 11. What is the equation wliose roots are — - and -- 2 ? q Ans. a;2 -f 2|a; = — -. 12. What is the equation whcse roots are — 2 and — 3 ? Ans. a;^ 4- 5a; = — 6. NUMERICAL VALUES OF THE ROOTS. 231 18. What is the equation whose roots are 4 and 3 ? Ans. x^ — 1x z= — 12. 14. What is th^ equation whose roots are 12 and 2 ? Afis. x^ - \4x = — 24, 16. What is the equation whose roots are 18 and 2? A^^s. x^ — 20a; = — 30. 16. Wliat jp the equation whose roots are 14 and 3? Ans. a;2 — 1 7a; = . — 42. 4 9 1 1, What is the equation whose roots are - and — t ? An^: x^ H X = 1, ' 2 18. What is the equation whose roots are 5 and — „ ? Ans. x^ -X — - • 3 u 19. What is the equation whose roots are n ^ nnd h ? Ans. a;2 — (a -f b)x = — ab. 20. What is the equation whose roots are c and — d? Ans. a;* — (c — d)x = c€L TRINOMIAL EQUATIONS OF THE SECOND DEGREE. 165.* A trinomial equation of the second degree con- tains three kinds of terms : 1st. A term involving the unknown quantity to the second degree. 2d. A term mvolving the unknown quantity to the first degree ; and 3d. A known term. Thus, a;2 - 4a: - 12 = 0, 10 a trinomial equation cf the second degree. 232 ELEMENTARY ALGEBRA. FACTOEING, 165.** What are the factors of the trinomial equation, a;2— 4a; — 12 = 0? A trinomial equation of the second degree may always be reduced to one of the four forms (Art. 160), by simply trans- posing the known term to the second member, and then solving the equation. Thus, from the above equation, we have, x^ — Ax =z 12. Resolving the equation, we find the two roots to be +6 and — 2 ; therefore, the factors are, x — 6, and a; 4- 2 (Art. 162). Since the sum of the two roots is equal to the coefficient of the second term, taken with a contrary sign (Art. 163) ; and the product of the two roots is equal to the known term in the second member, taken with a contrary sign, or to the third term of the trinomial, taken with the same sign : hence it follows, that any trinomial may be factored by inspection, when two numbers can be discovered whose algebraic sum is equal to the coefficient of the second term^ and whose product is equal to the third term, EXAMPLES 1. Wbat are the factors of the trinomial, cc^ _ g^. _ 3gp It is seen, by inspection, that — 1 2 and + 8 will fulfil the conditions of roots. For, 12 — 3 — 9; that is, the co efiicient of the second term with a contrary sign ; and 12 X — 3 = — 36, the third term of the trinomial; hence, the factors are, jb — 12, and a; + 3. 2. What are the factors of a:^ _ 7^; — 30 = ? Ans. ic — 1 0, and ar + TRINOMIAL EQUATIONS 233 8. What are the factors of x^ 4-1535+36 == 0? , Ans. X + 12, and a; -h 3. 4. What are the factors of a^ — 12a; — 28 = ? Ans, X — 11, and a; -f- 2 5. What are the factors of a;^ _ ^a; — 6 = ? Ana, a; — 8, and a; + 1. TRINOMIAL EQUATIONS OF THE FOEM x^* 4- Ipx"" — q. In the above equation, the exponent of a, in the first term, is double the exponent of aj in the second term.* a;fi — 4a;3 = 32, and a* + 4a;2 = 117, are both equations of this form, and may be solved by the rules already given for the solution of equations of the second degree. In the equation, aj2« -f 2/xc" = S and Jc" = V-io - V- P ^ P" 234 ELEMENTARY ALGEBKA. • EX A MPLES. 1. What are the values of a; in the equation. a;6 -f Qx3 = 112? Completing the square, aj6 + 6053 + 9 = 112 + 9 = 12I ; then, extracting the square root of both members, 053 + 3 = ± -/m ~ ± 11 ; hence, a' = 3y_ 3 ^ 11^ and a" = ^— 3 — 11 ; hence, 35' = 3^8 = 2, and a" = y- 14 = - y/lA. 2. What are the values of 05 in the equation, X" - Sx^ = 9 ? Completing the square, we have, 05* — 8052 + 16 = 9 + 16 = 25. Extracting the square root of both members, a;2 _ 4 _ -j- y/25 = ± 5 ; hence. 05' = ± V4 + 5, and 05" = =h -v/4 — 5 ; hence. a;' = + 3 and — 3 ; and 05'' = + y — 1 and — y^ 1. 3. What are the values of 0; in the equation, 0J6 + 20;k^ = 69 ? Completing the square, 056 + 20o;3 +100 = 69 + 100 = 169. Extracting the square root of both members, 053 + 10 = ± -/leg = ± 13 ; hence, x" = \J- 10 -f 18, and 05" = Ij- 10 ~ 13. 05' = \fz, and 05" — ^''— 23. TRINOMIAL EQUATIONS. 235 4. What are the values of a; in the equation, »♦ — 2ib2 = 3 ? Ans, x" z= ± y^, and a/' = ± y/^, 6. What are the vahies of 2 in the equation, Ans. x' = 1, and x" = y--^. 6. Given x ± yOa 4- 4 = 12, to find x. Transposing x to the second member, and then squaring, 9a; -h 4 = a;2- 24a; + 144; .-. a;2 — 33a; = — 140; and, x' = 28, and x" — 5. 7. 4a; ± Ay/x -f 2 = 7. ^/w. aj' = 4^, x" = i. 8. a; ± -/5a; -|- 10 = 8. Ans. x* = 18, a;" = 3. NUMERICAL VALUES OF TIIE ROOTS. 166. We have seen (Art. IGO), that by attributing all possible signs to 2/> and ^, ^^t' liave the four following forms: a;2+ 2;xB = ^ (1.) a;2 — 'Ipx — q (2.) a;2_,. 27a = - ^ (3.) 7? — 2jyx = — q (4.) 166. To how many forms may every equation of the second degree be reduced ? What are they ? 236 ELEMENTAR-S ALGEBRA. First Form, 167. Since q is positive, we know, from Property Fourth, that the product of the roots must be negative • hence, the roots have contrary signs. Since the coefficient 2p is positive, we know, from Property Third, that the alge- braic sum of the roots is negative ; hence, the negative root is numerically the greater. Second Form. 168. Since q is positive, the product of the roots must be negative; hence, the roots have contrary signs. Since 2/? is negative, the algebraic sum of the roots must be posi* tive ; hence, the positive root is numerically the greater. Third Form. 169. . Since q is negative, the product of the roots is positive (Property Fourth) ; hence, the roots have the same sign. Smce 2p is positive, the sum of the rooih must be negative ; hence, both are 7iegative. Fourth Form, 170. Since q is negative, the product of the roots is positive ; hence, the roots have the same sign. Since 2p is negative, the sum of the roots is positive ; hence, the roots are both positive. 167. What sign has the product of the roots in the first form? IIow are their signs? Which root is numerically the greater? Why? 168. What sign has the product of the roots in the second form f How are the signs of the roots ? Which root is numerically the greater ? 169. What sign has the product of the roots in the third form? How are their signs ? 170. What sign has the product of the roots in the fourth form ? How are the signs of the roots ? NUMEEICAL VALUE OF THE BOOTS. 237 I^irst and Second Forms, ITl, If we make $^ = 0, the first form becomes, a;2 4- 2/XB = 0, or x{qc +2/)) = ; ttrhich shows that one root is equal to 0, and the other to —2/1 Under the same supposition, the second form becomes, 7? — Ipx = 0, or x{x — 2/?) = ; which shows that one root is equal to 0, and the other to 2p. Both of these results are as they should be ; since, when 7, the product of the roots, becomes 0, one of the factors must be ; and hence, one root must be 0. Third and Fourth Forms, 172. If, in the Third and Fourth Forms, q^p^^ the quantity under the radical sign will become negative ; hence, its square root cannot he extracted (Art. 13 7). Under this supposition, the values of x are imaginary. How are these results to be interpreted ? If a given number he divider into tico imrts^ their pro- duct will he the greatest possible^ when the parts are equal. Denote the number by 2p, and the difference of the parts by d\ then, p -\- - =. the greater part, (Page 120.) and, jt) — - = the less part, and, p^ — — = P, their product. 171. If we make y = 0, to what does the first form reduce? What, then, are ita roots ? Under the same supposition, to what does the second form reduce ? What are, then, its roots ? 1*72. If ^ > p\ in the third and fourth forms, what takes place? If a number be divided into two parts, when will the product be the greatest possible ? 238 ELEMENTARY ALGEBRA It is plain, that the product P will increase^ as d dimirir ishes, and that it will be the greatest possible when d = 0; for then- there will be no negative quantity to be subtracted from jo^, in the first member of the equation. But when d = Oj the parts are equal ; hence, the product of the two parts is the greatest when they are equal. In the equations, a;2 + 2pa; = — <^, a^ — 2j!%c = — g', 2/> is the sum of the roots, and — q their product ; and hence, by the principle just estabhshed, the product q^ can never be greater than p^. This condition fixes a limit to the value of q. If, then, we make q > j^^^ ^ye pass this limit, and express, by the equation, a condition which cannot be fulfilled ; and this incompatibiUty of the conditions is made apparent by the values of x becoming imaginary. Hence, we conclude that, "When the values of the unknown quantity are imaginary ^ the conditions of the proposition are incompatible with each other, EXAMPLES. 1. Find two numbers, whose sum shaU be 12 and pro- duct 46. Let X and y be the numbers. By the 1st condition, x -\- y = \1\ and by the 2d, xy = 46. The first equation gives, X = \2 — y. Substituting this value for x in the second, we have. 12y -2/2 = 46; and changing the signs of the terms, we have, y^ - \1y = — 46 NUMEBICAL VALUE OF THE KOOT6. 239 Then, by completing the square, y2 - 12y + 36 = - 46 + 36 = - lOj which gives, y' = ^ + V^— 10, and. y" = 6 — -/- 10; !)Oth of which values are imaginary, as indeed they shoold l>e, since the conditious are incompatible. 2. The sum of two numbers is 8, and their product 20 : what are the numbers ? Denote the numbers by x and y. By the first condition, 35 + 2/ = 8; and by the second, sey = 20. The first equation gives, a; = 8 — y. Substituting this value of as in the second, we have, Sy -y^ = 20 ; changing the signs, and completing the square, we have, y2- 8y -f 16 = -4; and by extracting the root, y' = 4 -t- \/^^, and y" = 4 - y/"^^. These values of y may be put under the forms (Art. 142), y = 4 + 2-/^, and y = 4 — 2>/^^. 3 What are the values of sc in the equation, «2+ 2a; = - 10? ix" = - 1 240 ELEMENTAKY ALGEBKA. peoble:ms. 1. Find a number such, that twice its square, added to three times the number, shall give 65. Let X denote the unknown number. Then, the equation of the problem is, 2-052 4- 3a; =: 65 ; whence, 3 , /G5 , 9 3 , 23 ^= -i^Vy + Ie = -4±T- Therefore, 3 23 ^ , ,, 3 23 13 x^ =-- + - = 5, and x-= ----= --. Both these values satisfy the equation of the problem. For, 2 X (5)2 4-3x5=:2x25 4-15 = 65; :i J 13\2 , ^ 13 169 39 130 ^d, 2(- -) + 3 X - - = — - - = - = 65. Notes. — 1. If we restrict the enunciation of the problem to its arithmetical sense, in which " added " means numer- ical i7icrease, the first value of x only will satisfy the con- ditions of the problem. 2. If we give to " added," its algebraical signification (when it may mean subtraction as well as addition), the problem may be thus stated : To find a number such, that twice its square diminished by three times the number, shall give 65. Tlie second value of x will satisfy this enunciation ; for, m 13 169 39 PBOBLBMS. 24] 8. The root which results from giving the plus sign to the mdical, is, generally, an answer to the question in its arith- metical sense. The second root generally satisfies the pro- blem under a modified statement. Thus, in the example, it was required to find a number, of which twice the square, added to three times the num- ber, shall give 65. Now, in the arithmetical sense, added means increased ; but in the algebraic sense, it implies dimi- nution when the quantity added is negative. In this sense, the second root satisfies the enunciation. 2. A certain person purchased* a number of yards of cloth for 240 cents. If he had purchased 3 yards less of the same cloth for the same sum, it would have cost hhn 4 cents more per yard : how many yards did he buy ? Let X denote the number of yards purchased. 240 Then, — will denote the price per yard. H for 240 cents, he had purchased three yards less, that v\. a; — 3 yards, the price per yard, under this hypothesis, 240 would have been denoted by • But, by the condi- tioiiK, tliis last cost must exceed the first by 4 cents. There- fore, we have the equation, 240 240 _ JB- 3 a; - *' whence, by reducing; 7^ — 2x = 180, and, x= -±^- + 180 = —^i therefore, jc' = 15, and jb" = — 12. Notes. — 1. Tlie value, a;' = 15, satisfies the enunciation in its arithmetical sense. For, if 16 yards cost 240 cents, 11 2 42 ELEMENTARY ALGET- RA. 240 -7-15 = 16 cents, the price of 1 yard ; and 240 -- 12 = 20 cents, the price of 1 yard under the second supposition. 2. The second value of x is an answer to the following Problem : A certain person purchased a number of yards of cloth for 240 cents. If he had paid the same for three yards more, it would have cost him 4 cents less per yard : how many yards did he buy ? This would give the equation of condition, 240 240 x^ — Sx = 180; the same equation as found before ; hence, A single equation will often state tioo or more arith- metical problems. T'liis arises from the fact that the language of iVlgebra is more comprehensive than that of Arithmetic. 3. A man having bought a horse, sold it for $24. At the sale he lost as much per cent, on the price of the horse, as the horse cost him dollars : what did he pay for the horse ? Let X denote the number of dollars that he paid for the horse. Then, a; — 24 will denote the loss he sustained. But X M he lost X per cent, by the sale, he must have lost -— - opon each dollar, and upon a; dollars he lost a sura denoted 3.2 by ; we have, then, the equation, •^ 100 7 J » — = SB - 24; whence, x^ - 100a; = - 2400, rROBLEMS. 243 and, X =: 50 ± y^OO — 2400 = 50 ± 10. nierefore, 7f = CO, and a" =40. Botli of these roots will satisfy the problem. For, if the man gave $60 for the horse, and sold him for |24, he lost $36. From the enunciation, he should have lost GO per cent, of $60 ; that is, 60 . ^^ 60 X 60 — of 60 = = 36 ; 100 100 ' therefore, $60 satisfies the enunciation. Had he paid $40 for the horse, he would have lost by the sale, $16. From the enunciation, he should have lost 40 pei cent, of $40 ; that is, 40 ^ ,^ 40 X 40 ,^ — of 40 = = 16 : 100 100 therefore, $40 satisfies the enunciation. 4. The sum of two numbers is 11, and the sum of their squares is 61 : what are the numbers? Ans, 5 and 6 5. The difference of two numbers is 3, and the sum of their squares is 89 : what are the numbers ? Ans. 5 and 8. 6. A grazier bought as many sheep as cost him £60, and after reserving fifteen out of the number, he sold the re- mainder for £54, and gained 25. a head on tliose he sold : how many did he buy ? Ans. 75. 7. A mercliant bought cloth, for which he paid £33 15a., which he sold again at £2 8«. per piece, and gained by tlie liargain as much' as one piece cost him : how many pieces did he buy? Ans. 15. 8. The difference of two numbers is 9, and their sum, mnltiplied by the greater, is equal to 266: what are the numbers? Ans. 14 and 5 244 ELEMENTARY ALGEBKA. 9. To find a number, such that if you subtract it from 10^ and multiply the remainder by the number itself, the pro- duct will be 21. A71S. 7 or 3. 10. A person traveled 105 miles. If he had traveled 2 miles an hour slower, he would have been 6 hours longer in completing the same distance : how many miles did he travel per hour ? A7is. 1 miles. 11. A person purchased a number of sheep, for which he paid $224. Had he paid for each twice as much, plus 2 dol- lars, the number bought would have been denoted by twice what was paid for each : how many sheep were purchased ? Ans. 32. 12. The difference of two numbers is 7, and their sum multiplied by the greater, is equal to 130 : what are the numbers? A?is. 10 and 3. 13. Divide 100 into two such parts, that the sum of their squares shall be 5392. A7is. 64 and 36. 14. Two square courts are paved with stones a foot square ; the larger court is 12 feet larger than the smaller one, and the number of stones in both pavements is 2120 : how long is the smaller pavement ? Ans. 26 feet. 15. Two hundred and forty dollars are equally distributed among a certain number of persons. The same sum is agam distributed amongst a number greater by 4. In the latter case each receives 10 dollars less than in the former: how many persons were there in each case. A7is. 8 and 1 2. 16. Two partners, A and B, gained 360 dollars. A^s money was in trade 12 months, and he received, for prin- cipal and profit, 520 dollars. B^s money was 600 dollars, and was in trade 16 months: how much capital had A ? Ans. 400 dollars. MOBS THAN ONB UNKNOWN QUANTITY. 245 ■QUATIONS INVOLTrNG M :RE THAN ONE UNKNOWN QUANTITT. 173. Two simultaneous equations, each of the second degree, and containing two unknown quantities, will, when combined, generally give rise to an equation of the fourth degree. Hence, only particular cases of such equations can be solved by the methods already given. FIRST. 7\co simidtitneouR eqtiations^ involviiig two unknown quantitiesy can always he solved when one is of the first and tlie other of tlie second degree, E X A jr P L E s . 1. Given \ ., ^ ~ \ to find x and y. By transposing y in the first equation, we have, JB = 14 - y; and by squaring both members, xt = 196 — 28y + y'\ Substituting this value for 7? in the second equatioDy we have, 196 — 28y + y2 4. y2 _ joq. from wliich we have, y2 _ i4y = ~ 48. By completing the square, y2 — 14y -f 49 = 1 ; 178. When nuiy two simultaneous equations of the second degree be eoived ? 246 ELEMENTARY ALGEBRA. and by extracting the square root, y - 7 = ± -/l = + 1, and - 1 ; hence, y' = 7 + 1 = 8, and y" — 7 — 1 = 6. If we take the greater vahie, we find a; = 6 ; and if we take the lesser, we find a; = 8. {x' = 8, x" =r. 6 VERIFICATION. For the greater value, y = 8, the equation, X -\- y z=z 14, gives 6 + 8 = 14; and, a;2 + 2/2 _ iqq, gives 36 + 64 — 100. For the value y — 6, the equation, X -\- y = 14,. gives 8 + 6 = 14; and, x^ + y"^ = 100, gives 64 + 36 = 100. Hence, both sets of values satisfy the given equation. 2. Given i ^ '^^ . . r to find x and y. ( a;2 — 2/2 _ 45 ) i^ Transposing y in the first equation, we have, a; = 3 + y ; • then, squaring both members, x' =: 9 + 6?/ + y\ Substituting this value for £c^, in the second equation, w^ have, 9 + 62/ + 2/'- y^ -= 45; whence, we have, Qy — 36, and y = 6. 8IMDLTANEOU8 EQUATIONS. 24:7 SnbBtituting this value of y, in the first equation, we have, aj — 6 = 8, and, consequently, aj' = 3 + 6 = 9. VERIFICATION. X — ij z= 3, gives 9 — 6 = 3 ; and, a;2— 2/2 _ 45^ giygg 81 — 36 = 45. Solve the follo\ving simultaneous equations : ] ar^+ 2/2 ^ 117 f ( y' = 6, y"= - 9. ja;-f-y = 9 I ja;'=5, ^"= 5. • -1 x^- 2X7/ + y2^ i\ ^^^- I y' = 4, y"=: 4. (flj-y = 5 I • ( jbM- 2a^ + y2 ^ 225 f ix' = 10, a;"= — 5. I y' = 5, y"= - 10. SECOND. 174, 7\ro simtUtaneous equations of the second degree^ which are homogeneous with respect to the unknown quari' tity^ can always be solved. EXAMPLES* , p. J Jc2_^ 3a^ _ 22 (1.) 1. l^iven ]a.2 4. 3ay + 2y2 = 40 (2.) to find X and y. 174. When may two sunuiuueotis eqrataons of the second degree be solved ? 24S ELEMENTARY ALGEBRA. Assume oi — ty^ t being any auxiliary unknown quantity. Substituting this value of x in Equations ( 1 ) and ( 2 ), we have, «y+3«y^=22, ... y2=_-_; (3.) 40 ey 4 3^y^ + 2y2 ^ 40, .'. y2 ^ ___^___ , (4) 22 40 nence, ^2 + 3^ — f^ + Zt + 2' hence, 22«2 + QQt + 44 = 40^^ _|_ 120^; 22 reducing, f^ ■{- ^t = 2 11 whence, t' = - , and t''= • ' 3' 3 Substituting either of these values in Equations ( 3 ) or ( 4 ), we find, 2/' = +3, and y" = — 3 Substituting the plus value of y, in Equation (1), ^e have, x^ -{- 9x z= 22 ; fi'om which we find, x' = +2, and cc" = — 11. If we take the negative value, y" =^ — 3, we have, from Equation ( 1 ), a;2 — 9ic = 22 ; from which we find, a;' = + 11, and a" = — 2. VERIFICATION. For the values y' = +3, and x' = +2, the given equation, aj2 4. 3a;y = 22, SIMULTANEOUS EQUATIONS. 249 gives, 22 +3x2x3 = 4 + 18 = 22; and for the second value, a" = — 11, the wime equation, a;2 + Zxy = 22, gives, (- 11)2+ 3 X - 11 X 3 = 121 - 99 = 22. If, now, we take the second value of y, that is, y" = — 3, and the corresponding values of a^ viz., x' = +11, and a:" = — 2; for x' = +11, the given equation, a;2 + 3icy = 22, gives, 112 + 3 X 11 X — 3 = 121 — 99 = 22 ; and for a;" = — 2, the same equation^ a;2 + dxt/ = 22, gives, (— 2)2 + 3 X - 2 X - 3 = 4 + 18 = 22. The verifications could be made in the same way by em- ploying Equation ( 2 ). Note. — In equations similar to the above, we generally find but a single pair of values, corresponding to the values m this equation, of y' = + 3, and a;' = + 2. The complete solution would give four pairs of values. ar» - y2 _ _ 9 I y2 - ary = 5 f j ay - y2 = _ 7 J *"• I y2 4. a.2 33 85 f j 2a^ + 3x7/ = 470 ) j y» - ay = - 9 f j 5ay - 3y2 = 32 ) ^- "1 ar + y2 + 3ay = VI f 11* Ana \y = z 4. = 5. Ans X = = 6, = 7. Ana. X — ' y = 10. 9 Ans, \y = 7. 1. 250 ELEMENTARY ALGEBRA. THIRD. PAETIOILAR CASES. I'yS Many other equations of the second degree may be so transformed, as to be brought under the rules of sohition already given. The seven following formulas will aid in such transformation. (1.) ' When the sum and difference are known: X + y = s X — y — d. Then, page 132, Example 3, s + d 1,1, ,• s - d 1 1, ^ = -Y- = 2' + 7/^ ^nd 2/ = __ = -.- -^ (2.) When the sum and product are known: x-\- y = s (1.) xy = p (2.) a;2 ^- 2xy -\- y"^ = s\ by squaring ( 1 ) ; 4xy = 4p, by mult. ( 2 ) by 4. x^ — 2xy -f 2/2 — g2 __ 4p^ }jj subtraction. But, hence x — y = ± x/s"^ — 4p, by ext. root. X + y = s; X = -± - ^/s^- 4p and, 2/ = 2 "^ 2 V«^^- 175. What is the first formula of this article? What the sncond? Third? Fourth? Fifth? Sixt): ? Seventh? 8IMDLrANEOU8 EQUATIONS. 251 (3.) When the diifercnne and product are known: X- ij =z d (1.) ^y'== P (2.) a^ — 2xy + y^ = <^, by squaring ( 1 ) • Axy = 4/>, mult. ( 2 ) by 4. sc* + 2xy -^ y^ z= (P -f 4p, by adding. X + y = ± y/(P + 4p aj — y = d 1 a; = ^d ± ^ Vo?2 -f 4/). 4.) When the sum of the squares and product are kno^ni . a52 + y2= ,..(1.) xy=p,.(2.) .'.2xy = 2p..{3.) Adding ( 1 ) and ( 3 ), jc^ + 2a;y + y^ = s -\- 2p; hence, x -\- y = ± ys -\- 2p (4.) Subtracting (3) from ( 1 ), x^ — 2xy + y^ = s —2p; hence, a; — y = ± -/^ — 2/> (6.) Combining (4) and (5), a; = ^-/« + 2p + |V« — 2/>, an^ y = iVs -h 2p — iV* — 2/). When the sura and sum of the squares are known : X -{- y = 8 (1.) a* 4- y' = «' (2.) ffi* -I- 2a:y -f y^ = «* by squaring ( I ) 2xy = 8* -^ 8' g2 _ g' cry = — — - = p. (3.) 252 ELEMENTAF. Y ALGEBRA By putting xy = p^ and combining Equations (1) and (3 ), by Formula (2), we find the values of x and y. (6.) When the sum and sum of the cubes are known : a + 2/ = 8 .... (1.) jc3 + 2/3 1= 152 . . . . (2.) 7^ 4- 3a;2y + Sicyz + y^ -512 by cubing ( 1 ). Zx^y + 3jr?/2 =: 360 by subtraction. ^xy{x -[- y) = 360 by factoring. 3a;y(8) = 360 from Equa. ( 1 ) 14:xy = 360 hence, xy — 15 . . . , (3.) Combining ( 1 ) and ( 3 ), we find a; = 5 and ?/ = 3. When we have an equation of the form, (x + yY + {x + y) = g- Let ns assume a; 4- y = s. Then the given equation becomes, z'^ z = q; and z = - -± sj q + ^• 1 ^ + y = vA^ 1. Given EXAMPLE! xz ^ y'^ ( 1 ) r xz^ y^ {\)\ jaj + y +s r= 7 (2) [ (a;2 -t- y2 -f 22 = 21 (3) ) to find Xj y, and z. SIMULTANEOUS EQUATIONS. 253 Transposing y in Equation ( 2 ), we have, a + z = 7 — y; ... (4.) then, squaring the members, we have, jc2 4- 2a;2 + 22= 49 — 14y -f y"^. If now we substitute for 2iBz, its value taken from £qua^ lion ( 1 ), we have, a^^ + 2?/2 -I- 22 _ 49 _ 14y ^ y2 . and cancelling y^^ in each member, there results, a^ + 2/^ -f 2^ = 49 — 14 y. But, from Equation ( 3 ), we see that each member of the last equation is equal to 21 ; hence. 49 -14y = 21, and. 14y = 49 — 21 = 28, hence. y 28 ~ 14 = 2. Substituting this value of y in Equation (1), gives. JC2 = 4 ; and substitutmg it in Equation ( 4 ), gives, jc -f z = 5, or X z= b — z. Substituting this value of a, m the previous equation, wo obtain 52 - z2 = 4, or 2^ — 52 = — 4 ; and by completmg the square, we have, 2^ - 52 + 6.25 = 2.25. anrl, 2 — 2.5 = ±'^2^b= -\- 1.5, or — 1.5 hence, z = 2.5 -f 1.5 = 4, and 2 = -f- 2.6 — 1.5 = 1 254 ELEMENTARY ALGEBRA. 2. Given x + ^/xy -{- y = ^^ \ xq find 3. and and a;2 _j_ xy -{- y"^ — l^Z ) ^ ^* Dividing the second equation by the first, we have, x — ^/xy -{- y = 1 but, X -\- -/xy + y r= 19 hence, by addition, 2x -{- 2y = 26 or, X -{- y = IS and substituting in 1st Equa., \/xy + 13 = 19 or, by transposing, ^^ z= 6 and by squaring. xy =z 36. Equation 2d, is a;2 4. jcy + 2/2 _ 133 and from the last, we have. .ixy =108 Subtracting, a;2 — 2xy -f y'^ = 25 hence. X — y z= ±5 hut. X -^ y = 13 hence, a; = 9 and 4 ; and y =z 4 and 9. PROBLEIMS. 1. Find two numbers, such that their sum shall bo 15 and the sum of their squares 113. Let X and y denote the numbers ; then, a; -f- 2/ = 15, (1.) and x^ -i- y^ = 113. (2.) Front Equation ( 1 ), we have, a;2 = 225 — SOy -f y^ Substituting this value in Equation ( 2 ), 225 — dOy + y^ -^ y^ = 113; PR0WLEM8. 255 hence, 2y2 - 30y = - 112, 2/2 — loy = — 50, hence, y' = 8, and y" — 7. Tlio first value of y being substituted in Equation ( 1 ), gives jc' = 7 ; and the second, x" = 8. Hence, the nunt* here are 7 and 8. 2. To find two numbers, such tliat their [»roduct added to their sum shall be 17, and their sum taken from the sum of their squares shall leave 22. Let X and y denote the nmnbere; then, from the con- ditions, (X -\- y) + xy = 11. ... (1.) x' + y^- (x + y) = 22, ... (2.) Multiplying Equation ( 1 ) by 2, we have, 2xy -f 2(x + 2/) = 34. ... (3.) Adding ( 2 ) and { 3 ), we have, x'^ 2xy + y^-\- (x-\- y) = 56 ; hence, {x + yY -\- (x -{- y) = 56. . . (4.) Regarding (x + y) as a single unknown quantity (page 248), x + y = -l± y/sT + \ = 1. Substituting this value in Equation ( 1 ), we have, T -\- xy = 17, and y — 5, Hence, the numbers are 2 and 5. 3. Wliat two numbers are those whose sum is 8, and suni of Uieir squares 34 ? Ans, 6 and 3. 256 ELEMENTARY ALGEBRA. 4. It is required to find two such numbers, tLat the first shall be to the second as the second is to 16, and the sum ol whose squares shall be 225 ? Ans. 9 and 12. 5. What two numbers are those which are to each other as 3 to 5, and whose squares added together make 1666 ? Ans. 21 and 35. 6. There are two numbers whose difference is 7, and half their product plus 30 is equal to the square of the less number: what are the numbers? A7is. 12 and 19. 7. What two numbers are those Avhose sum is 5, and the pum of their cubes 35 ? Ans. 2 and 3. 8. What two numbers are those whose sum is to the greater as 11 to 7, and the difference of whose squares is 132 ? Ans. 14 and 8. 9. Divide the number 100 into two such parts, that the product may be to the sum of their squares as 6 to 13. Ans. 40 and 60 10. Two persons, A and j5, departed from different places at the same time, and traveled towards each other. On meeting, it appeared that A had traveled 18 miles more than i? ; and that A could have gone J3''s journey in 15f days, but J^ would have been 28 days in performing A'>s journey ; how far did each travel ? . \ A, 72 miles. Ans. j J?, 54 miles. 11. There are two numbers whose difference is 15, and half their product is equal to the cube of the leeser number : wliat are those numbers? A?2S. 3 and 18. 12. What two numbers are those whose sum, multiplied '^by the greater, is equal to 77 ; and whose difference, multi- plied by the less, is equal to 12 ? _ Ans. 4 and 7, or 1^2 and V \/2. PROBLEMS. 257 13. Divide 100 into two such parts, that the sum of their square roots may be 14. Ans. 64 and 36. 14. It is required to divide the number 24 into two such parts, that their product may be equal to 35 times their dif. ference. Ans, 10 and 14, 15. The sum of two numbers is 8, and the sum of tlioir cubes is 152 : what are the numl)ers ? Ans. 3 and 5. 16. Two merchants each sold the same kind of stuff; the second sold 3 yards more of it than the first, and together they receive 35 dollars. The first said to the second, "I would have received 24 dollars for your stuff;" the other replied, "And I should have received 12^ dollars for yours :" how many yards did each of them sell ? . \ 1st merchant a' = 15, a;" = 5. ^"*-J2d " y' = 18, "'■' y"=8. 17. A widow possessed 13,000 dollars, which she divided into two parts, and placed them at interest in such a manner that the incomes from them were equal. If she had put out the first portion at the same rate as the second, she "would have drawn for this part 360 dollars interest ; and if she had placed the second out at the same rate as the first, she would have drawn for it 490 dollars interest : what were the two rates of interest ? Ans. 7 and 6 per cent. 18. Find three numbers, such that the difference between the third and second shall exceed the difference between the second and first by 6 ; that the sum of the numbers shall be 33, and the sum of their squares 467. Ans. 5, 9, and 19. 19. What number is that which, being divided by the product of its two digits, the quotient will be 3 ; and if 18 be added to it, the resultmg number will be expressed by the digits inverted ? A?is. 24. ELEMENTARY ALGEBliA. 20. What two numbers are those which are to each othei as W2 to w, and the sum of whose squares is 5 ? m^ n-y/b Ans, ^/m? + m^ -}- n^ 21. What two numbers are those which are to each oilier as m to 71, and the difference of whose squares is 6 ? Ans. -v/m^ i^ y/m^ — w^ 22. Required to find three numbers, such that the product of the first and second shall be equal to 2 ; the product of the first and third equal to 4, and the sum of the squares of the second and third equal to 20. Ans. 1, 2, and 4. 23. It is requfred to find three numbers, whose sum shall be 38, the sum of their squares 634, and the difference betw^een the second and first greater by 7 than the difference between the third and second. Ans, 3, 15, and 20. 24. Required to find three numbers, such that the product of the first and second shall be equal to a ; the product of the first and third equal to h ; and the sum of the squares of the second and third equal to c. Ans. X — y R + 62 G ^''s/a^ c ^Va2 4. 52 25. What two numbers are those, whose sum, multiplied by the greater, gives 144 ; and whose difference, multiplied by the less, gives 14 ? Ans, 9 and 1. PBOPOBTIONS AND PROGRESSIONS. 259 CHAPTER IX. OF PROPORTIONS AND PROGRESSIONS. I T6. IHvo quantities of the same kind may be compared, the one with the other, in two ways : Ist. By considering how much one is greater or less than the Other, which is shown by their difference ; and, 2d. By considering how many times one is greater or less than the other, which is showTi by their quotient. Tims, in comparing the numbers 3 and 12 together, with respect to their difference, we find that 12 exceeds 3, by 9 ; and in comparing them together wuth respect to their quo- tient, we find that 12 contains 3, four times, or that 12 is 4 times as great as 3. Tlie first of these methods of comparison is called Arith- metical Proportion^ and the second, Geometrical Fropor- lion. Hence, Arithmetical Proportion considers the relation of quantities with respect to tJieir difference^ and Geometrical Proportion the relation of quantities with respect to their quotient. 176. In how many ways may two quantities be compared the one with the other? What does the first method consider? What the seconl? Wliat is the first of these methods called ? What b the second called P How then do you define the two proportions ? 260 ELEMENTARY ALGEBEA. OF AEITHMETICAI. PEOPOETION AOT) PKOGEESSION. I'yy. If we have four numbers, 2, 4, 8, and 10, of which tlie difference between the first and second is equal to the difference between the third and fourth, these numbers are said to be in arithmetical proportion. The first term 15 is called an antecedent^ and the second term 4, with wliich it is compared, a consequent. The number 8 is also called an antecedent, and the number 10, with which it is com-* pared, a consequent. When the difference between the first and second is equal to the difference between the third and fourth, the four numbers are said to be in proportion. Thus, the numbers, 2, 4, 8, 10, are in arithmetical proportion. ITS. When the difference between the first antecedent and consequent is the same as between any two consecutive terms of the proportion, the proportion is called an arith' metical progression. Hence, a progression by differences^ or an arithmetical pyrogression., is a series in which the suc- cessive terms are continually increased or decreased by a constant number, which is called the common difference of the progression. Thus, in the two series, 1, 4, 7, 10, 13, 16, 19, 22, 25, . . . 60, 56, 52, 48, 44, 40, 36, 32, 28, . . . 177. "When are four numbers in arithmetical proportion ? "What is the first called? What is the second called? What is the third called? What is the fourth called ? 178. "What is an arithmetical progression ? "What is the number called by which the terms are increased or diminished ? What Is an increasing progression ? What is a decreasing progression ? Which term is only an antecedent ? "Which only a consequent ? AKI'lUMBTICAL PKOOKK88ION. 261 the first is called an increasing progression^ of which tho common difference is 3, and tke second, a decreasing pro- ffressioti, of which the common difference is 4. In general, let a, ^, c, c/, e, /", ... denote the terms of a progression by differences ; it has been agreed to write them thus : a,b,c,d.e,/.g.h,i.k... This series is read, a is to J, as 5 is to c, as c is to (7, as df is to e, roduct a!Ud the first term ; the sum will be the last term. EXAMPLES. The formula, I — a -\- {n — \)d^ serves to find any term whatever, without determiningr all those Avhich precede it. "** 1. If we make 7i = 1, we have, I =z a; that is, the series will have but one term. 2. If we make n — 2^ we have, I = a -\- d\ that is, the series will have two terms, and the second term is equal to the first, plus the common difierence. 3. If a = 3, and d =z 2^ what is the 3d term? Ans. 1. 4. If a = 5, and c? = 4, what is the 6th term? A92S. 25 5. If a = 7, and d — 5, what is the 9th term? Ans. 47. 6. If a = 8, and c? = 5, what is the 10th term ? A71S. 5'i 7. If « = 20, and d = 4, what is the 12th term? Ans. 64, 8. If a = 40, and d = 20, what is the 50th teim? A71S. 1020. 9. If a = 45, and d = 30, what is the 40th term ? Ans. 1215. ARITHMETICAL PKOOKKSSION 203 10. If a = 30, and d = 20, what is the 60th term? Atis. 1210. 11. ir a — 50, and d = 10, Mli:it is the 100th tcrni? A/ts. 3 010. 12. To find tlie 50th tenn of tlie jjrugression, 1 . 4 . 7 . 10 . 13 . 16 . 19 we have, ^ = 1 + 40 x 3 = 148. 13. To find the 60th temi of the progression, 1 . 5 . 9 . 13 . 17 . 21 . 25 . . . we have, / = 1 + 59 x 4 = 237. 180. It' the progression were a decreasing one, we should !i:ive, / =r a — {n — l)d. Ileim, t(i ilnd the last term of a decreasing progression, we have tiie followmg RULE. L Multiple/ the common difference by the number of terms less one : n. Subtract the product from. Vie first term ; the ro- mainder will be the last term. EXAMPLES. 1. The first tenn of a decreasing progression is 60, the number of terms 20, and the common difierence 3 : wliat is the last term? l=za-(n-\)d, gives i = 60 - (20 — 1)3 = 60 -57 = 3 180. Give the rule for finding the laat term of a series, nhtn the pro grcssion is decreasing. 264 ELEMENTARY ALGEBRA. 2. The first term is 90, the common difference 4, and the number of terms 15 : what is the last term ? Ans. 34. 3. The first term is IGO, the number of terms 40, and the common difference 2 : what is the last term ? Ans. 22. 4. The first term is 80, the number of terms 10, and the common difference 4 : what is the last term ? Ans. 44. 5. The first term is 600, the number of terms 100, and the common difference 5 : what is the last term ? Ans. 105. 6. The first term is 800, the number of terms 200, and the common difference 2 : what is the last term ? Ans. 402. 1§1. A progression by differences being given, it is pro- posed to prove that, the sum of any two terms^ taken at equal distances from the two extremes^ is equal to the sum of the two extremes. That is, if we have the progression, 2 . 4 . 6 . 8 . 10 . 12, we wish to prove generally, that, 4 + 10, or 6 + 8, is equal to the sum of the two extremes, 2 and 12. Let a.b.c,e.f... i . k . I, be the proposed progression, and n the number of terms. We will first observe that, if x denotes a term which has p terms before it, and y a term which has p terms after it, we have, from what has been said, 181. In every progression by differences, what is the sum of the two extremes equal to? What is the rule for finding the sum of an arith- tnctical series? ABITHMETIOAL PBOOBEBSION. 2G5 X = a -\- p X d, and, y = ' — p X d; whence, by addition, x + y = a + I, which proves the proposition. Referring to the previous example, if we suppose, in the first place, x to denote the second terra 4, then y will de- note the term 10, next to the last. If x denotes the third term 6, then y will denote 8, the third term from the last. To apply this principle in findmg the sum of the tei-ms of a progression, WTite the terms, as below, and then again, in an inverse order, viz. : I . k . i c , b , a. Callmg 8 the sum of the terms of the first progression, IS will be the sum of the terms of both progressions, and we shall have, 2S={a+t) + (h+k)-^{c+i) . . . +(i-hc)+(A;+*) + (^+«). Now, since all the parts, a -^r I, h -\- k, c -{- i , . , ^re equal to each other, and their number equal to w, 28= (a + l) X w, or S - y-^) X w- Ilonce, for finding the sum of an arithmetical series, we have the following RULE. L Add the two extremes together^ and take haXfth^ir sum : U. Mtdtiply this half -sum by the number of terms ; the product wiU be the sum of the series. 12 266 ELEMEI?rTAEY ALGEBRA, EXAMPLES. 1. The extremes are 2 and 16, and the number of terms 8 : what is the sum of th.e series ? S = [—^) X ^^: gives S = -— ^ X 8 Y2. 2. The extremes are 3 and 27, and the number of tenna 12 : what is the sum of the series ? Ans, 180. 3. The extremes are 4 ard 20, and the number of terms 10: what is the sum of the series? Ans. 120. 4. The extremes are 100 and 200, and the number of terms 80 : what is the sum of the series r A7is. 12000. 5. The extremes are 500 and 60, and the number of terms 20 : what is the sum of the series ? Ans. 5600 6. The extremes are 800 and 1200, and the number of terms 50 : what is the sum of the series ? Ans. 60000. 1 82. In arithmetical proportion there are five menibers to be considered : 1st. The first term, a. 2d. The common difierence, d, 3d. The number of terms, n. 4th. The last term, I. 6th. The sum, S. The formulas, I = a -^ {n — l)d, and >S' = | — - — J x n, contain five quantities, a, d, w, I, and S, and consequently give rise to the following general problem, viz. : Any three 162. How many numbers are considered in arithmetical proportion? What are they ? In every arithmetical progression, what is the common difference equal to ? AUITHMETIOAL PROOEESBION. 267 of these five quantities being given, to determine the other tioo. We already know the value of aS' in terms of a, w, and L From the formula, I = a -^ (n — 1)(7, we find, a = I — {n — \)d. That is : The first term of an increasing arithmetical pro- gression is equal to the last term, minus the product of the common difference by the number of terms less one. From the same formula, we also find, I — a d = n Tliat is : In any ai'ithmetical progression, the common dif- ference is equal to the last term, minus the first term, divided by the numbar of terms less one. The last term is 16, the first tcnn 4, and the number of tenns 6 ; what is the common difierence ? Tlie fonntda, d = ^ n — 1 7 16-4 gives, d = — ~ — = 3. 2. The last tenn is 22, the first term 4, and the number of terms 10 : what is the common difference? Ans. 2. 183. The last principle afibrds a solution to the follow- Ing question : To find a number m of arithmetical means between two given numbers a and b. 1S8. How do jou find any number of ftrithroctical means betweci two 0\ien n'unbere ? 268 ELEAtENTAIiY ALGEBRA. To resolve this question, it is first necessary to find the common difierence. Now, we may regard a as the first term of an arithmetical progression, h as the last term, and the required means as intermediate terms. The number of terms of this progression will be expressed by m 4- 2. Now, by substituting in the above formula, h for Z, and >?i -f 2 for ?2, it becomes, ,_ h — a _ ^ ~ ^ . ~ 7)1 -{- 2 — \ ~ m + 1 ' that is : The common difference of the required progression is obtained hy dividing the difference between the given numbers^ a and b, by the required number of means plus one. Having obtained the common difference, d^ form the second term of the progression, or the first arithmetical mean^ by adding d to the fii?st term a. The second mean is obtained by augmenting the first mean by c?, &c. 1. Find three aritlimetical means between the extremes 2 and 18. b — a The formula, d m + l' ^ 18-2 gives, d = — — =4', hence, the progression is, 2 . 6 . 10 . 14 . 18. 2. Find twelve arithmetical means between 12 and 77. b — a Tlie formula, d = m + 1 ' ^ 77-12 gives, d =^ — = 5 ; hence, the progression is, 12 . 17 . 22 . 27 ... 77. ARITHMETICAL PBOORE88ION. 2(>9 184. Remark. — If the same number of arithmetical means are inserted between all the terms, taken two and two, these terms, and the arithmetical means united, will form one and the same progression. For, let a.b.c,e.f,,'. be the proposed progression and m the number of means to be inserted between a and i, b and c, c and e . . . . &c. From what has just been said, the common difference of each partial progression will be expressed by b — a c — b e — c m + 1 ' wi + 1 ' m 4- 1 * * ' expressions which are equal to each other, since a, 5, c . . . are in progression ; therefore, the common difference is the same in each of the partial progressions ; and, since the last term of the first fomis the Jirst term of the second, &c., we may conclude, that all of these partial progressions form a single progression. EXAMPLES. 1. Find the sum of the first fifty terms of the progression 2 . 9 . 16 . 23 . .. For the 60th term, we have, / = 2 4- 49 X 7 = 345. 50 Hence, S = (2 -\- 345) x y = 347 X 25 = 8676. 2. Find the 100th term of the series 2 . 9 . 16 . 23 . . . A?is, 695 3. Find the sum of 100 terms of the series 1.3.5.7. 9 A71S 10000. 270 ELEMENTARY ALGEBRA. 4. The greatest term is 10, the common difference 3, and the number of terms 21 : what is the least term and tiie fiimi of the series ? A71S. Least term, 10 ; sum of series, 840. 5. The first term is 4, the common difference 8, and the number of terms 8 : what is the last term, and the sum of the series ? Ans. j Last term, 60 ( Sum = 256. 6. The first term is 2, the last term 20, and the number of terms 10 : what is the common difference ? Atis. 2. 7. Insert four means between the two numbers 4 and 19 : what is the series ? Ans. 4 . 7 . 10 . 13 . 16 . 19. 8. The first term of a decreasing arithmetical progression is 10, the common difference one-third, and the number of terms 21 : required the sum of the series. Ans. 140. 9. In a progression by differences, having given the com- mon difference 6, the last term 185, and the sum of the /*^ terms 2945 : find the first term, and the number of terms. Ans. First term =z 5 ; number of terms, 31. 10. Find nine arithmetical means between each antecedent and consequent of the progression 2. 5. 8. 11. 14... Ans. Common diff., or d = 0.3. 11. Find the number of men contained in a triangular battalion, the first rank containing one man, the second 2, the third 3, and so on to the n^^, which contains n. In other words, find the compression for the sum of the natural num- bers 1, 2, 3 . . ., from 1 to w inclusively. Ans. S = ^^t.1) 2 12. Find the sum of the n first terms of the progression of uneven numbers, 1.3.6.7.9,... Ans. JS — «^ GEOMETRICAL PBOrOBTION. 271 18. One hundred stones being placed on the ground in a straight line, at the distance of 2 yards apart, how flir will a person travel who shall bring them one by one to a basket, placed at a distance of 2 yards from the first stoiu- *.* Ans, 11 mile-', SIO yards. GEO^VIETRICAL PROPORTION A^T> I'ilOORESSION. 185. liatio is the quotient arising from dividing one quantity by another quantity of the same kind, regarded as a standard. Thus, if the numbers 3 and 6 have the same unit, the ratio of 3 to 6 will be expressed by I"- And m general, if A and J^ represent quantities of the same kind, the ratio of ^ to i? will be expressed by B a' 186. The character oc indicates that one quantity is proportional to another. Thus, A oc B, is read, A proportional to B. If there be four numbers, 2, 4, 8, 16, having such values that the second di^^ded by the first is equal to the fourth divided by the third, the numbers are 185. What is ratio ? What is the Mtio of 3 to 6 ? Of 4 to 12 ♦ 186. What is proportion? How do you express that four numbers arc in proportion ? What are the numbers called ? What are the firsi and fourth terras c\Iled ? What the second and third ? 272 ELEMENTARY ALGEBRA said to form a proportion. And in general, if there be foni quantities, A, J3, (7, and J9, having such values that, :? _ :? A ~ C then, A is said to have the same^ratio to J5 that C has to Di or, the ratio of A to J5 is equal to the ratio of C to D. When four quantities have this relation to each other, com* pared together two and two, they are said to form a geo- metrical proportion. To express that the ratio of ^ to J5 is equal to the ratio of C to J9, we write the quantities thus, A : J^ :: G : D; and read, ^ is to .B as C to D. The quantities which are compared, the one with the other, are called terms of the i:>roportion. The first and last terms are called the two extremes^ and the second and third terms, the two means. Thus, A and D are the extremes, and _Z? and C the means. 1§7. Of four terms of a proportion, the first and third are called the antecedents^ and the second and fourth the consequents ; and the last is s;i)il to be a fourth proportional to the other three, taken in order. Thus, in the last pro- portion A and G are the antecedents, and B and D the con- sequents. I §8. Three quantities are in proportion, when the first has the same ratio to the second that the second has to the 187. In four proportional quantities, what are the first and third called ? What the second and fourth ? 188. When are three quantities proportional? What is the middle one called f OKOMETEICAL PEOPOBTION. 273 third ; and then the middle term is said to be a mean pro- portional between the other two. For example, 8 : 6 :; 6 ; 12; / IM^ and 6 is a mean proportional between 3 and 12. 18!>. Four quantities are said to be in proportion by in- version^ or mveisely^ when the consequents are made the antecedents, and the antecedents the consequents. Thus, if we have the proportion, 3 : 6 : : 8 • 16, the inverse proportion would be, 6 : 3 : : 16 . 8. 190. Quantities are said to be in proportion by altema- ^Ay^ tion^ or alternately^ when antecedent is compared with ante- cedent, and consequent with consequent. Thus, if we have the proportion, 3 : 6 : : 8 : 16, the alternate proportion would be, 3 : 8 : : 6 : 16. 191. Quantities are said to be in proportion by contpo eition^ when the sum of the antecedent and consequent if compared either with antecedent or consequent . Thus, if we have the proportion, ^^^^ 2 : 4 ; : 8 : 16, 189. When are quantities said to be in proportion by inversion, or to versely ? 190. When arc quantities in proportion by alternation? 191. When are quantities in proportion by composition? 12* 274 ELEMENTARY ALGEBRA. the proportion by composition would be, 2 + 4 : 2 : : 8 + 16 : 8; and, 2 + 4 : 4 : : 8 -h 16 : 16. 192. Quantities are said to be in proportion by division, when the difference of the antecedent and consequent it^ compared either with antecedent or consequent. Thus, if we have the proportion, 3 : 9 : : 12 : 36, the proportion by division will be, 9 — 3 : 3 : : 36 -- 12 : 12; and, 9 — 3 : 9 : : 36 — 12 : 36 ; 193. Equi-multiples of two or more quantities are the products which arise from multiplying the quantities by the same number. Thus, if we have any two numbers, as 6 and 6, and mul- tiply them both by any number, as 9, the equi-mulfiples -will be 54 and 45 ; for, 6 X 9 = 54, and 5 X 9 = 45. Also, m X A^ and m x B^ are equi-multiples of A and B^ the common multiplier being m. 194. Two quantities A and B^ which may change theii values, are reciprocally or inversely proportional^ when one is proportional to unity divided by the other ^ and then the^ft product remains constant. 192. When are quantities in proportion by division ? 1&8. What are equi-jQultipIes of two or more quantities ? 194. When are two quantities said to be reciprocally proportional P OKOMETRICAL PROPOBTION. 275 We express this reciprocal or inverse relation thus, in which A is said to be inversely proportional to B, 195. If we have the proportion, A : B :: C : By B B we have, -r = y,, (Art. 186); and by cleaiing the equation of fractions, we have, BC = AB, That is : Of four proportional quantities^ the product of the two extremes is equal to th^ product of the two means. This general principle is apparent in the proportion be- tween the numbers, 2 : 10 : : 12 : 60, which gives, 2 x 60 = 10 x 12 = 120. 196. If four quantities, A^ J5, C, 2>, are so related to each other, that A X B =z B X C, we shall also have, -j = -?^ ; and hence, A : B i : C i B, That is : If the product of two quantities is equal to the product of two other quantities, two of them may be made the extremes^ and the other two the means of a proportion. 196. If four quantitiea are proportional, what is the product of the two means equal to ? 196. If the product of two quantities is equal to the product of two olLcr quantities, WAy the four be placed in a proportion f How ? 276 ELEMENTARY ALGEBRA. Thus, if we have, 2X8 = 4X4, we also have, 2 : 4 : : 4 : 8. 1 97. If we have tliree proportional quantities, A ', B :i B '. C, B C we have, -j = -^; hence, B^ = AC. That is : If three quantities are proportional^ the square of VA^ the middle term is equal to the product of the two extremes Thus, if we have the pro])ortion, 3 : 6 : : 6 : 12, we shall also have, 6 X 6 = 62 = 3 X 12 = 36. 198. If we have, A : B : : (7 : X>, and consequently, -^ = -^, c multiply both members of the last equation by ^, and we then obtain, C _ B A ~ B' and, hence, A : C : : B : B. That is : If four quantities are proportional^ they will he in proportion by alternation, 197. If three quantities are proportional, what is the product of the jxtremes equal to ? 198. If four quantities are proportional, will they De in proportion by alternation ? OEOMETRIOAL PROPORTION. 277 Let 116 take, as an example, 10 : 15 : : 20 30. Wp shall have, by alternating the terms, 10 : 20 • : 15 : '30. 1 99. If we have, A : B :: C : D, and A : J^ : : E : T, we shall also have, A= C^ and ^ =^; J) F hence, -^ = -= , and C : D : : E i F That is : If there are ttoo sets of projyortions having an an tecedent and consequent in the one^ equal to an antecedeiit and consequent of the other^ the remaining terms will be proportional. If we have the two proportions, ♦ 2 : 6 : : 8 : 24, and 2 : 6 : : 10 : 30, we shall also have, 8 : 24 : : 10 : 30. 200. If we have, JS J) A '. B '.'. C : B^ and consequently, -^ = -^, we have, by dividing 1 by each member of the equation, A C -= = ^ and consequently, B : A : : B : C. 199. If you have two sets of proportions having an antecedent and coti- Dtquent in each, equal ; what will follow ? 200. If four quantities are in proportion, will thej be In proportio?' when tftkeu inveri»elv ? 278 ELEMENTARY ALGEBRA. That is : Four proportional quantities will he i7i p'opwtion^ when taken inversely. To give an example in numbers, take the proportion, V : 14 :: 8 : 16; then, the inverse proportion will be, 14 . 7 : : 16 : 8, in which the ratio is one- half 201, The proportion, A : B \\ C \ I), gives, A y D =z B x C, To each member of the last equation add B x D, We shall then have, {A ^ B) X B = (C + D) X B; and by separatiiif? the factors, we obtain, L ^ 4 A ■\- B : B :i C + B '. B, ^- '* ^+^^ If, mstead of adding, we subtract B x B from both members, we have, [A- B) X B = [C - B) X B', which gives, A ~ B '. B '.', C - B \ B. That is: If four quantities are proportional^ they wiU be in proportion by composition or division. Thus, if we have the proportion, 9 : 27 : : 16 : 48, 201. If four quantities are in proportion, will they be in proportion by composition ? Will they be in proportion by division ? What ib tb*> ^ifTorcncc between composition and division ? GEOMETRICAL PEOPORTION. 279 we shall have, by composition, 9 + 27 • 27 : ; 16 -f 48 : 48 ; that is, 36 : 27 : : 64 : 48, in which the ratio is three-fourths. The same proportion gives us, by division, 27 - 9 : 27 :: 48 - 16 : 48; that is, 18 : 27 : : 32 : 48, in which the ratio is one and one-half. 202. If we have, B _D A- C and multiply the numerator and denominator of the first member by any number m, we obtain, — J = -= , and mA : mB : : C : D, mA C Tliat is : Equal multiples of two quantities have the sanie ratio as the quantities themselves. For example, if we have the proportion, 6 : 10 : : 12 : 24, and multii)ly the first antecedent and consequent by 6, we have, 30 : 60 : : 12 : 24, in which the ratio is still 2. 203. The proportions, A : B : . C : D, and A : B : : B : JF, aoa. Have equal multiples of two quantities the aame ratio as the quantities ? 203. Suppose the antecedent and consequent be augmented or dimiu' ifalied by quantities having the same ratio ? 280 ELEMENTARY ALGEBEA. give, A X JD = B X (7, and AxF=BxE\ adding and subtracting these equations, we obtain, A(D±F) = B{C^E), or A . B . . C ±E i D ±F, That \^'. If C and Z>, the antecedent and consequent^ be augmented or diminished hy quantities E and F^ which have the same ratio as C to 2)^ the resulting quantities will also have the same ratio. Let us take, as an example, the proportion, 9 : 18 : : 20 : 40, ill which the ratio is 2. If we augment the antecedent and consequent by the numbers 15 and 30, which have the same ratio, Ave shall have, 9 -f- 15 : 18 + 30 : : 20 : 40; that is, 24 : 48 : : 20 : 40, in which the ratio is still 2. If we diminish the second antecedent and consequent by these numbers respectively, we have, 9 : 18 •: 20 — 15 : 40 — 30; that is, 9 : 18 : : 5 : 10, in which the ratio is till 2. 204. If we have several proportions, A : B A : B A : B : C : I), which gives A x I) = B x C, : F : F, which gives A X F = B x E^ : G : JI, which gives A x II = B x G^ &c., &c.. 204. In any number of proportions having the same ratio, bow \<'il] ony one antecedent be to its consequent ? OEOMKTRICAL PROPORTION. 281 we shall have, by addition, A(D + F+ IT) = B{C ^- E'\- GO; and by separating the factoi-s, A : B II C ■\-E-\- G I i>+i^4 H, That is: In any 7nimher of proportio7i8 having the same rtUio^ any antecedent will be to its consequent as the sum of the antecedents to the sum of the consequents. Let us take, for example, 2 : 4 : : 6 : 12, and 1 : 2 : : 3 : 6, &o. Tlien 2:4::64-3:12-f-6; that is, 2 : 4 : : 9 : 18, ill which the ratio is still 2. a05. If we have four proportional quantities, A : B \ : C : J)^ we have, -^ = --^ ; and raising both members to any power whose exponent is n, or extracting any root whose index is w, we have, -J- =z -z^ J and consequently. That is: Jf four quajitities are proportional, their liht powers or roots will be proportional. If we have, for example, 2 : 4 : : 3 : 6, we shall have 2^ : 4^ : : 32 : 6^ ; 806. In four proportional quantities, how aro like poweiT or roots ? 282 ELEMENTARY ALGEBRA. that is, 4 : 16 : : 9 : 36, ill which the terms are proportional, the ratio being 4. 206, Let there be two sets of proportions, B J) A : JB : '. C : D, which gives -j — -^; F If E \ F w G \ H, which gives -^ = -^ . Multiply them together, member by member, we have, B X F _ JD X H Ax E " G X G' A X E \ B X F w G X G -. D x H. That is : In two sets of proportional quayitities^ iheproduct& of the corresponding terms are p/roportional. Thus, it* we have the tuvo proportions, 8 : 16 : : 10 : 20, and, 3 : 4 : 6 : 8, we shall have. 24 : 64 : : 60 : 160. GEOMETRICAL PROGRESSION. 207. We have thus far only considered the case in which the ratio of the first term to the second is the same as that of the third to the fourth. 206. In two sets of proportions, how are the products of the correspond Ing terras ? 207. What is a geometrical progression ? What is the ratio of the progression ? If any term of a progression be multiplied by the ratio, what will the product be ? If any term be divided by the ratio, what OBOMKTEIOAL PBOGBESBION. 2S3 If we have the farther condition, that the ratio of the second terra to the third shall also be the same as that of the first to the second, or of the third to the fourth, we shall have a series of numbers, each one of which, divided by the preceding one, will give the same ratio. Hence, if any term be multiplied by this quotient, the product will be the Bucccoding term. A series of numbers so formed, is called a geometrical progression. Hence, A Geometrical Progression^ or progression by quotients^ IS a series of terms, each of which is equal to the preceding terra multiplied by a constant number^ which number is called the ratio of the progression. Tims, 1 : 3 : 9 : 27 : 81 : 243, &c., is a geometrical progression, in which the ratio is 3. It Is written by merely placing two dots between the terms. Also, 64 : 32 : 16 : 8 : 4 : 2 : 1, is a geometrical progression in which the ratio is one-half. In the first progression each term is contained three times in the one that follows, and hence the ratio is 3. In the second, each term is contained one-half times in the one which follows, and hence the rartio is one-half. The first is called an increasing progression, and the second a decreasing progression. Let a, J, c, (?,€,/*, . . . be numbers, in a progression by quotients • they are written thus : a:6:c:e?:e:/:<7... and it is enunciated in the same manner as a progression by differen-ces. It is necessary, however, to make the distinc* will the quotient be ? How \» a p'^gression by quotients written ? Which of the terms is only an antecedent t Which only a consequent? How nmy each of the others be considered? I obli: 1 find 284 ELEMENTARY ALGEBIJA.. tion, that one is a series formed by equal differences, and the other a series formed by equal quotients or ratios. It should be remarked that each term is at the same time an antecedent and a consequent, except the first, which is only an antecedent, and the last, which is only a consequent. 20§. Let r denote the ratio of the progression, a : b : c : d , . . 7 being > 1 when the progression is increasing^ and r< 1, when it is decreasing. Then, since, h c d e o~ we have, b = ar^ c = hr — ar^, d = cr = ar^j e = dr = ar*^ f — er = ar^ . . . that is, the second term is equal to «r, the third to ar^, the fourth to ar^, the fifth to ar*, &c. ; and in general, the nih term, that is, one which has ?i — 1 terms before it, is ex- pressed by ar''~^. Let I be this term • we then have the formula. by means of which we can obtain any term without being ged to find all the terms which precede it. Hence, to the last term of a progression, we have the following E u L E . I. liaise the ratio to a potcer whose exponent is one less than the numper of terms. II. Multiply the pov^er thus fonnd by the first term: the lyrodtict loill be the required term. 208. By what letter do we denote the ratio of a progression? In rd Increasing progression is r greater or less than 1 ? In a df»creasing pro GBOMKTlilCAL rKOOKEJJBION, 285 EXAMPLES. 1 , Find the 6th term of the progression, 2 : 4 : 8 : 16 . . . in wliich the first term is 2, and the common ratio 2. 6th term = 2 x 2* = 2 x 16 = 32. Ans. 2. Find the 8th term of the progression, 2 : 6 : 18 : 64 . . . 8th term = 2 x 3' = 2 x 2187 = 4374. Ana. 8. Find the 6th term of the progression, 2 : 8 : 32 : 128 . . . 6th term = 2 x 4^ = 2 x 1024 = 2048. An8 4. Find the 7th term of the progression, 3 : 9 : 27 : 81 . . . 7th term = 3 x 3« = 3 x 729 = 2187. Ans, 5. Find the 6th term of the progression, 4 : 12 : sa : 108 . . . 6th term = 4 x 8* = 4 x 243 = 972. Ans, 6. A person agreed to pay his servant 1 cent for the first day, two for the second, and four for the third, doubling every day for ten days: how much did he receive on the tenth day? A71S, $5.12. gresfiion is r greater or less than 1 ? If a is the first term and r the ratio, what is the second term equal to ? What the third ? What the fourth ? What is the Uwt terra equal to ? Give the rule for finding the lR8t term. ^Wv 286 ELEMENTARY ALGEBRA. 1, What is tlie 8th term of the progression, 9 ; 36 : 144 : 576 . . . 8th term = 9 X 4' = 9 X 16384 = 147456. Aiis, 8. Find the 12th term of the progression, 64 : 16 : 4 : 1 : i . . . 4 12th term = 64(J)" = i! = J, = ^. A,^. 209. We will now proceed to determine the smn of n terms of a progression, a : b : c : d : e : f : . , . : i : k : l] I denoting the ^ith term. We have the equations (Art. 208), b = ar, c -— br^ d = cr, e = dr, . , . k = ir, I = Jcr, and by adding them all together, member to member, we deduce, Sinn of l6i members. Sum of 2d tnenibere. b+c+d-{-e-\- . . . -h7c-\-l={a + b + c-\-d+ . . . ^.-^^-^>; in which we see that the first member contains all the terms but «, and the poIjTiomial, within the parenthesis in the second member, contaias all the terras but I. Hence, if we call the sum of the terms /S, we have, S^- a = {S - l)r = Sr -Ir, r . Sr - S = Ir -^ a whence, JS = • Kj.A^ r - 1 209. Give the rule for finding the sum of the series. What ie the first step? What the second? What the third? GEOMETRICAL PBOORESSION. 2S7 Therefore, to obtain the sum of all the terras, or wim of the series of a geometrical progression, we have the RULE. I. Multiply the last term by the ratio : n. Subtract the first term from tJie product : in. Divide the remainder by the ratio diminished by 1 and the quotient will be the sum of tlie series, 1. Find the sum of eight terms of the progression, 2 : 6 : 18 : 54 : 162 . . . 2 X 3' ;= 4374. Ir - a 13122 - 2 ^^^^ S = = r = 6560. r — \ 2 2. Find the su^i of the progression, 2 : 4 : 8 : 16 : 32. 5 = ?L=^ = «i^2 ^ 62. r — 1 1 3. Find the sum of ten terms of the progression, 2 : 6 : 18 : 54 : 162 ... 2 X 33 = 39366. Ans. 5904a 4. What debt may be discharged in a year, or twelve months, by pajdng $1 the first month, $2 the second month, $4 the third month, and so on, each succeeding payment being double the last ; and what will be the last payment ! Ans. j Debt, . . 14095 ( Last payment, $204^ 6. A daughter was married on New- Year's day. Her Cither gave her l5., with an agreement to double it on the first of the next month, and at the beginning of each succeed- ing month to double what she had previously received. How m'lch did she receive? Ana, £204 16/?. 288 ELEMENTARY ALGEBRA. 6. A man bought ten bushels of wheat, on the condition that he should pay 1 cent for the first bushel, 3 for the second, 9 for the third, aiid so on to the last : what did he pay for the last bushel, and for the ten bushels ? j Last bushel, |196 83. ^^' \ Total cost, $295 24. 7. A man plants 4 bushels of barley, which, at the first harvest, produced 32 bushels ; these he also plants, which, in like manner, produce 8 fold ; he again plants all his crop, and again gets 8 fold, and so on for 16 years: what is his last crop, and what the sum of the series ? j Last, 140737488355328 bush. ^^* I Sum, 160842843834660. 910. When the progression is decreasing, we have, r< 1, and Z< a; the above formula, Ir - a for the sum, is then written under the form, a - Ir in order that the two terms of the fraction may be positive4 1. Find the sum of the terms of the progression, 32 : 16 : 8 : 4 : 2 32 - 2 X ^ ^ = 1^ == ? = H = 62. I — r 1 1 2 2 210. What is the formula for the sum of the series of a decreasing progression ? G K u M E T li I C A L P K () G R E 6 8 I O N . iiyU 2. Find the sum of the first twelve terms of the pro- grossion, 1 /IV' 1 64 . 10 : 4 : 1 :-:... : 64(-) , or - 65538 64-—^ X^ 256 ^ - a — lr (15536 4 65536 ,, . 65635 S = " = = 86 + 1 — r 3 3 196008 4 ail. Remark. — We perceive that the principal difficulty consists in obtaining the numerical value of the last term, a tedious operation, even when the number of terms is not very great. 3. Find the sum of six terms of the progression, 612 : 128 : 32 . . . Ans. 682^ 4. Find the sura of seven terms of the progression, 2187 : 729 : 243 . . . Ans. 327ft 5. Fmd the sum of six terms of the progression, 972 : 324 : 108 . . . Ahs. 1456 6. Find the sum of eight terms of the progression, 147456 : 36864 : 9216 . . . Afis. 190605. OP PROGRESSIONS HAVTNO AN ENPrNlTE NXTMBEK OF TERMS 213 Let there be the decreasing progression, a : b : c : d : e : / : , . , il2. When tho progression in decreasing, and the number of terms in- Aoite, what is the eipreKrion for the value of the sun of the eeries? 13 290 ELEMENTARY ALGEBRA. containing an indefinite number of terms. In the formula, substitute for I its value, ar"-\ (Art. 208), and we have, „ a — ar** o = 1 - r ' which expresses the sum of n terms of the progression. This may be put under the form, „ a ar* W^ 1 — r \ — r Now, since the progression is decreasing, r is a proper fraction ; and r" is also a fraction, which diminishes as n increases. Therefore, the greater the number of terms we take, the more will x r" diminish, and consequently, the more will the entire sum of all the terms approximate to an equality with the first part of aS, that is, to • Finally, when n is taken greater than any given number, or w — mfinity, then x r" will be less than any given number, or wdll become equal to ; and the expres- sion, , will then represent the true value of the sum of all the terms of the series. Whence we may conclude, lliat the expression for the sum of the term,s of a decreasing progfession^ in which the number of terms is iiifinite^ ^5, a that is, equal to the first term^ divided by 1 minus t/ie ratio. GEOMETRICAL PROGRESSION. 291 This is, properly speaking, the limit to which the partial sums approach, as we take a greater number of terms in tho progression. The difference between these sums and , nay be made as small as we please, but will only become ixotJiing when the number of terms is infinite. EXAMPLES. 1. Find tlu" sum ot , 111 !..«.. 1 • « • ;: • ^n, * :rr 1 ^o mnnity. 3 9 27 81 ' ^ We have, for the expression of the sum of the terms, .^ = -^— = -^ =1 = n. Ans, 1 - r 1 2 ^ 3 The error committed by taking this expression for the value of the sum of the n first terms, is expressed by 1 ^7 — r 2' 13/ • First take n = = 5; it becomes, 2\3 J 2.3* = 1 162 When n = = 6, we find. 2\3y 162 ^ 3 = 1 486 3 Hence, we see, that the error committed by taking _ for the sum of a certain number of terms, is less in proportion EG this number is greater. 292 ELEMENTARY ALGEBRA. 2. Again, take the progression, 1 111 11 o We Lave, JS = - — ~ = = 2. Am. 3. What is the sum of the progression, '» iV li' li' r^o' &«-'^^i"fi^i^y- JS= -^ = -i- =. il. ^M.. 1 — r 1 9 10 213. In the several questions of geometrical progres sion, there are five numbers to be considered : 1st. The first t^'rm, . . a. 2d. The ratio, . . . . r. 3d. The number of terms, n. 4th. Tlie last term, . . l. 5th. The sum of the terms, S. 214. We shall terminate this subject by solving this problem : To find a mean proportional between any two numbers, as m and n. Denote the required mean by x. We shah then have (Art. 197), x^ = rn X n : and hence, x = ^m x n. 218. How many numbers are considered in a geometrical progrefisioii ? What are they ? 214. How do you find a mean proportional between two cumbers? GEOMETRICAL PROGKES6ION. 293 That is : Multiply the two numbers togetJier^ and extract the square root of the product. 1. What is the geometrical mean between the numbers 2 and 8? Mean = -/S x 2 = -/iS = 4. Ans, 2. What is the mean between 4 antl 16 ? Ari8, ^ 3. What is the mean between 3 and 27 ? Ans, 9 4. Wliut is the mean between 2 and 72 ? An9, 12, 5. What is the mean between 4 and 64 ? Aii^. 16, // y- /. ^y*^ ^ 294 ELEMENTARY ALGEBEA. CHAl^ER X. OF LOGARITHMS. 215. The nature and properties of the logaritliins in common use, will be readily understood by considering attentively the different powers of the number 10. They are, IQO = 1 10^ = 10 102 = 100 103 = 1000 10* = 10000 10* = 100000 &c.. &c. It ia plain that the exponents 0, 1, 2, 3, 4, 5, 10, and the exponcut by m, what will represent the corresponding number? What is the logarithm of a number ? 217. To what is the sum of the logarithms of any two numbers equal ? Tx) what, then, will the addition cf logarithms corre^poud ? 296 ELEMENTARY ALGEBRA. The sum of the logarithms of atiy two numbers is equal to the logarithm, of their product. Therefore, the addition of logarithms corresponds to the multiplication of their numbers. 218. If we divide Equation ( 1 ) by Equation ( 2 ), mora- ler by member, we liave, 10« M but since 10 is the base of the system, m — 10 is the loga- ritlim of -^ix.; hence, If one number be divided by another.^ the logarithm of the quotient will he equal to the logarithm of the dividend^ diminished by that of the divisor. Therefore, the subtraction of logarithms corresponds to the division of their nuinbers. 219. Let U3 examine further the equations, 10° = 1 101 — 10 102 = 100 103 — 1000 &c., &c. It is plain that the logarithm of 1 is 0, and that the loga- rithra of any number between 1 and 10, is greater than 218. If one number be divided by another, what will the logarithm of the quotient be equal to ? To what, then, will the subtraction of loga ithms correspond ? 219. What is the logarithm of 1? Between what limits are the loga- rithms of ax numbers between 1 and 10? How are they generally ex- pressed ? OF LOOABITHMS. 297 and less than 1. Tlie logarithm is generally expressed by decimal fractions ; thus, log 2 = 0.301030. Tlie logarithm of any nmnber greater than 10 and lese than 100, is greater than 1 and less than 2, and is expressed by 1 and a decimal fraction ; thus, log 60 = 1.098970. The part of the logarithm which stands at the left of the decimal point, is called the characteristic of the logarithm. The characteristic is always 07ie less than the number of places ofjigicres in the number whose logarithm is taken. Thus, in the first case, for numbers between 1 and 10, there is but one place of figures, and the characteristic is 0. For numbers between 10 and 100, there are two places of figures, and the charactQristic is 1 ; and similarly for other numbers. TABLE OP LOGARmnrs. 220. A table of logarithms is a table in which are writ- ten the logarithms of all numbers between 1 and some other given number. A table showing the logarithms of the numbers between 1 and 100 is annexed. The numbers are written in the column designated by the letter N, and the logarithms in the column designated by Log. How is it with the logarithms of Lumbers between 10 and 100? What \n that part of the logarithm called which stands at the loft of the char arterjptic? What is the value of the characteristic? 220. What is a table of logarithms ? Explain the manner of finding the logarithns of numbers between 1 and 100? 13* — r 'IF 298 ELEMENTARY ALQEBEA. TABLE. IT 1 Log. "N. Log. IT 51 Log. r'NT" Log. 0.000000 26 1.414973 1.707570 76 1.880814 2 0,301030 27 1.431364 52 1.716003 77 1.886491 3 0.477121 28 1.447158 53 1.724276 78 1.892095 i 0.602060 29 1.462398 54 1.732394 79 1.897627 5 0.698970 30 1.477121 55 1.740363 80 1.903090 6 0.778151 31 1.491362 56 1.748188 81 1.908485 7 0.845098 32 1.505150 57 1.755875 82 1.913814 8 0.903090 33 1.518514 58 1.763428 83 1.919078 9 0.954243 34 1.531479 69 1.770852 84 1.924279 10 1.000000 35 1.544068 60 1.778151 85 1.929419 11 1.041393 36 1.556303 61 1.785330 86 1.934498 12 1.079181 37 1.568202 62 1.792392 87 1.939519 13 1.113943 38 1.579784 63 1.799341 88 1.944483 14 1.146128 39 1.591065 64 1.806180 89 1.949390 15 1.176091 40 1.602060 65 1.812913 90 1.954243 16 1.204120 41 1.621784 66 1.819544 91 1.959041 17 1.230449 42 1.623249 67 1.826075 92 1.963788 18 1.255273 43 1.633468 68 1.832509 93 1.968483 19 1.278754 44 1.643453 69 1.838849 94 1.973128 20 1 . 301030 45 1.653213 70 1.845098 95 1.977724 21 1.322219 46 1.662758 71 1.851258 96 1.982271 22 1.342423 47 1.672098 72 1.857333 97 1.986772 23 1.361728 48 1.681241 73 1.863323 98 1.991226 24 1.380211 49 1.690196 74 1.869232 99 1.995635 25 1.397940 50 1.698970 75 1.875061 100 2.000000 EXAMPLES. 1. Let it be required to multiply 8 by 9, by means of logarithms. We have seen, Art. 216, that the sum of the logarithms is equal to the logarithm of the product. There- fore, find the logarithm of 8 from the table, which is 0.903090, and then the logarithm of 9, which is 0.954243 ; and their sum, which is 1.857333, will be the logarithm of the product. In searching along in the table, we find that 72 stands oppo- site this logarithm ; hence, 72 is the product of 8 by 9, OF LOG A KI TH M S. 2U9 2. Wliat is the product of 7 by 12? Logarithm of 7 is, . Logarithm of 12 is, Logarithm of their product, and the corresponding number is 34. 3. What is the product of 9 b} 11? Logarithm of 9 is, . Logarithm of 1 1 is, Logarithm of their product, and the corresponding number is 99. 0.845098 1.079181 1.924279 0.954243 1.041393 1.99563C 4. Let it be required to divide 84 by 3. We have seen in Art. 218, that the subtraction of Logarithms corresponds to th<; division of their numbers. Hence, if we find the logarithm of 84, and then subtract from it the logarithm of 8, the remainder will be the logarithm of the quotient. The logarithm of 84 is, . . . 1.924279 llie logarithm of 3 is, . . . 0.477121 Their difference is, . . .' . 1.447158 and the corresponding number is 28. 6. Wliat is the product of 6 by 7 ? Logarithm of 6 is, . . . .0.778151 Logarithm of 7 is, . . 0.845098 Their sum Is, .... 1.623249 and the corresponding number of the table, 42. The JVational Series of Standard School- T^ooK's, MATHEMATICS. DAVIES' NATIONAL COURSE, ARITHMETIC. 1. Daviea' Primary Arithmetic, $ 25 $ ^2 2. Davies' Intellectual Arithmetic, 40 48 3. Davies' Elements of Written Arithmetic, .... 50 60 4. Davies' Practical Arithmetic, 00 1 00 Key to Practical Arithmetic, 00 5. Davies* University Arithmetic, 1 40 1 50 Key to University Arithmetic, *1 40 ALGEBRA. 1. Davies' New Elementary Algebra, *1 25 1 85 Key to Elementary Algebra, *1 25 2. Davies' University Algebra, 1 50 1 CO Key to University Algebra, *1 50 3. Davies' New Bourdon's Algebra, 2 25 2 88 Key to Bourdon's Algebra, *2 25 GEOMETRY. 1. Davies* Elementary Geometry and Trigonometry, 1 40 1 50 2. Davies' Legendre's Geometry, 2 25 2 38 3. Davies' Analytical Geometry and Calculus, ... 2 50 2 63 4. Davies' Descriptive Geometry, 2 75 2 88 6. Davies' New Calculus, 2 00 MENSURATION. 1. Davies' Practical Mathematics and Mensuration, . 1 50 1 60 2. Davies' Elements of Surveying, 2 50 2 C:{ 3. Davies' Shades, Shadows, and Perspective,. . . 3 75 3 88 MATHEMATICAL SCIENCE. Davies' Grammar of Arithmetic, * 50 Davies* Outlines of Mathematical Science, *1 00 Davies' Nature and Utility of Mathematics, 8vo, +2 00, 12mo, *l 50 Davies' Metric System, *l •'>0 Davies & Peck's Dictionary of Mathematics, *5 00 Davies' Foundations Mathematical Science, * 25 17 The jVationa t Series of Standard School- ^ooJhs. MATHEMATICS-Continued. ARITHMETICAL EXAMPLES. Reuck's Examples in Denominate Numbers $ no Reuck's Examples in Arithmetic i oo These volumes differ from the ordinary arithmetic in their peculiarly practical character. They are composed mainly of examples, and afford the moat severe and thorough discipline for the mind. While a book which should contain a complete treatise of theory and practice would be too cumbersome for every-day use, the iasufficieucy oi in-actical examples has been a source of complaint. HIGHER MATHEMATICS. Church's Elements of Calculus 2 50 Church's Analytical Geometry 2 50 Church's Descriptive Geometry, with Shades, ShadoY/s, and PerspectiYe 4 00 These volumes constitute the " West Point Course" in their several departments. Courtenay's Elements of Calculus . . - - 3 oo A wwrk especially popular at the South. Hackley's Trigonometry 2 53 With applications to navigation and surveying, nautical and practical geometry and geodesy. Peck's Analytical Geometry l 75 Peck's Practical Calculus i 75 APPLIED MATHEMATICS. Peck's Ganot's Popular Physics i 75 Peck's Elements of Mechanics 2 oo Peck's Practical Calculus i 75 Peck's Analytical Geometry, i 75 Prof. W. G. Peck, of Columbia College, has designed the first of these works for the ordinary wants of schools in the department of "Natural Philosophy. The other volumes are the briefest treatises on those subjects now published. Their nsethods are purely practical, and unembarrassed by the details which rather coa- lube than simplify science. SLATED ARITHMETICS. This consists of the application of an artificially slated surface to the inner cover of a book, with flap of the same opening outward, so that students may refer to the book and use the slate at one and the same time, and as though the slate were detached. When folded up, the slate preserves examples and memoranda til' needed. The material used is as durable as the stone slate. The additional cost •jf books thus improved is trifling. 20 One-third or one-half shorter than any other similar course containing the same amount of Knowl' edge, and thoroughly Scientific, PECK'S BRIEF COURSE IN ARITHMETIC, By W. G. PECK, LL.D., Professor of Mathematics and Astronomy in Columbia College, N. Y. Author of " Analytical Geometry," " Practical Calculus," *' Elementary Mechanics," " Ganot's Physics." The Theory of this concise as well as comprehensive Course of Arithmetic to meet the wants of all classes is as follows : I.— FIRST LESSONS IN NUMBERS. 18mo, half bound, ; slated, This book beg^ with the simplest Elementary Combinations, illu* trating the processes by suitable cuts, but not by pictures culled from the primary readers and children's magazines. The true idea of illustration is to have a leading picture at the head of each part as Counting, Ada . tion. Subtraction, Multiplication, Division, and Fractions. The indivi- dual stops should be illustrated by diagrams neatly engraved and grouped, and aiding in developing the arithmetical ideas desired. This style of treatment, covering 50 or 60 pages, precedes the subject of Mental Arithmetic, The book thus formed should be all the Arithmetic needed to enter upon either the Manual or the Complete Arithmetic. Its place in all schools would be in dasees of pupils younger than about twelve years. 1 deck's jirlt?imetlcat Course, II.— MANUAL OF PRACTICAL ARITHMETIC. 208 pp., 18mo, half bound, 50/^ ; slated, 60)^. Tliis book has tlie definitions clearly laid down (just as tliej are to stand throughout the course) ; the rules too are laid down exactly as they are to stand in all the after course of mathematics. There is a carefully illustrated example after each rule (illustrated, that is, by being wrought out and explained), and then follows a sufficient number of graded ex- amples to impress the rule on the minds of the pupils. The place of this book would be in the ordinary district schools where the pupils are simply fitting themselves for the farm and the workshop, or in graded schools as a good practice before entering on the study of the Complete Arithmetic. It is adapted to children twelve to fourteen years of age, and contains enough of practical arithmetic for common life. As this course of books is chiefly intended for live teachers, and not so much for lazy ones, such questions are omitted as, "If one cow has two horns, how many horns have two cows ? " The live teacher, after having taught the First Lessons, can form enough of these examples from the objects around him, and will do so. III.— THE COMPLETE ARITHMETIC. 318 pp., 12mo, half bound, 90^ ; slated, $1.00. This book contains everything necessary to a complete arithmetician. Every step is explained scientifically. Every principle is laid down in clear language. Every rule is demonstrated. A suitable number of illustrative examples are given. In this book pupils of intelligence are addressed, such as are our children of fourteen years in our average schools. The book is made consecutive, logical, scientific, concise, simple. A student who follows this course in the order indicated will be an Arithmetician capable of making any application of his principles, and able to give a reason for the faith that is in him. 3 iPecJb^s Ai'Uhmetical Course, Such a course requires for its full development a live teacher — but in the end the fruits will be worthy of his labors. An Arithmetical course should be progressive, and, as fiir as possible, repetitions should be avoided. The place for such questions as a recent author uses to usher in his subjects, is in the Primary and Mental. To introduce them into either of the higher books would be a needless repe- tition, and one of our ablest teachers assures us that such ques- tions are always passed over by all good teachers. No course in Arithmetic can be studied and mastered with- out much labor on the part of both pupil and teachers, and we have yet to learn of any plan by which the subject can be made so easy that children will cry for it. With respect to the outcry of keeping up to the spirit of the age, we will say that the continually-widening circle of knowl- edge demands that each subject should be made ever more and more concise, more and more abbreviated. By abbreviations emasculation is not meant, but rather elimination of all trash and superfluous matter. The repeti- tion of primary principles in an advanced work, for instance, and the introduction of pictures from Chatterbox, are not in the direction of what we may consider the spirit of the age. How well these ideas have been carried out in this course wiU be determined by the popular verdict from the great mass of intelligent teachers of the country, and their name is le- gion. We will send specimen pages free, or copies for exam- ination to teachers at one-half the retail price, or a full set, WITH SLATES ATTACHED, for $1.00. Address A. S. BARNES & CO., Publishers, NEW YORK AND CHICAGO. 3 deck's A.rithnietical Course. f^edk'^ Si'ief doui'^e MATHEM ATICS and ME CHANICS. GANOT'S POPULAR PHYSICS For the use of schools and academies. 12ino, 504 pp., half roan. Elegantly illustrated. Price $1.75. PECK'S TREATISE ON MECHANICS, With Calculus. For the use of Colleges, Academies, and Iligli Schools, where the Calculus is not separately taught. 12mo, 344 pp., half roan, marbled edges. Price $2.00. PECK'S ELEMENTARY MECHANICS, Without Calculus. For the use of Colleges and Schools of Science. 12mo, 300 pp., half roan, marbled edges. Price $2.00. This work is a rewritten edition of the Treatise ok Mechanics. Its principal difference is in the omission of the Calculus, which is published separately in fuller and perfected form. PECK'S PRACTICAL CALCULUS. 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It includes a life of the author, notes, dissertations on each poem, a faultless text, and is the only edition of Milton with a complete verbal Index. 51 "F9UR1EEN WEEKS" lyUTURIlL SCIENCE RIEB^ TRE^^nSE IN EA.CII J3RA, J. QOHMAH STEELE, A.M. 14 . NATURAL PHILOSOPHY, WMLU 1 ASTRONOMY. COURSES CHEMISTRY, GEOLOGY. The-e volumes constitute the most RTallable, practical, and attractive text-books on the Scieuoes ever pabUehed. Each volume may be completed in a single term of stodj. THE FAMOUS PRACTICAL QUESTIONS devleed by this anthor are alone eufflclent to place his books In every Academy and Grammar School of the land. These are questions as to the nature and cause or com- mon phenomena, and are not directly answered in the text, the design being to test and promote an intelligent use of the student's knowledge of the foregoing prmciples. TO MAKE SCIENCE POPULAR Ib a prime object of these books. To this end each subject is invested with a charm- inf interest oy the peculiarly happy use of language and illustration in which thia author excels. THEIR HEA VY PREDECESSORS demand as much of the student's time for the acquisition of the principles of a single branch as these for the whole course. PUBLIC APPRECIA TION. The author's great success in meeting an urgent, popular need. Is Indicated by the fiujt (probably unparalleled in the history of scientific text-books), that although the first volume was issued in 1867, the yearly sale is already at the rate of PHYSIOLOGY AND HEALTH^ By EDWARD JARVIS, M.D. ELFJIEMS OF PHYSIOLOGY, PHYSIOLOGY AIVD LAWS OF HEALTH. The only books extant which approach this subject ■with a proper view of the true object of teaching Physiology In schools, viz., that scholars may know how to take care of their own health. The child instructed from these works will be always ^'^onaidc^ the lilies." BOTANY. WOOD'S AMERICAN BOTANIST AND FLORIST. This new and eagerly expected work Is the result of the author's experience and life-long labors in CLASSIFYING THE SCIENCE OF BOTANY. He has at lensrth attained the realization of his hopes br a wonderfully ingenious pro- cess of condensation and arrangement, and presents to the world in this single modern ate-siised volume a COMPL.ETE MANUAL. In 870 duodecimo pages be has actually recorded and defined NEA RL Y 4,ooo SPECIES. The treatises on Descriptive and Structural Botanv are models of concise statement, whicn leave nothing to be said. Of entirely new features, the most notable are the Synoptical Tables for the blackboard^ and the distinction of species and rarleties by yariation in the type. FroC Wood, by this work, establishes • Just claim to his title of the great AMERICAN EXPONENT OF BOTANY. JlRwIb fS® WBSf l^@liri ©@WSi And Only Thorough and Complete Mathematical Series lisr TKCIiEB I>uA.R,TS- /. COMMON SCHOOL COURSE. Davies' Primary Arithmetic- -—The fundamental, principles displa Object Lessons. Davies' Intellectual Arithmetic— Heferrtng all operations to the ui the only tangible basis for logical development. Davies' Slements of Written Arithmeticr— A practical introdnc the whole subject. Theory subordinated to Practice. Davies' Practical Arithmetic-*— The most successful combination of ' and Practice, clear, exact, brief, and comprehensive. //. ACADEMIC COURSE. Davies' University Arithmetic-*— Treating the subject exhausth a science, in a logical series of connected propositions. Davies' Elementary Algebra-*— A connecting link, conducting the easily from arithmetical processes to abstract analysis. Davies' University Alg-ebra-* — For institutions desiring a more co but not the fullest course in pure Algebra. 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