10 4+2 23-30 7 5 2x-5 1 5 10. 2– 10 3 2.x-5 -3 4.x - 3 11. + 11b 5 2x – 15 4(x+3) 8x +37 7x – 29 12. 9 18 5.0 - 12 ) 2x+5 13. 14. 6x (x+4) 5x +3 4 2 5 27 15. 3+3 2x + 6 2x+2 7 60 101 8 16. X -4 5x – 30 3.x - 12 3 5 + 2x+1 =0. 5x + 2 17. 18. 5+ 19. + 14+ 20. 21. 22. 25 3 16x +4} 23 + x+1 48 x+1 X 8 -9 1 - 2 1 0 6 x -7 x+5 6 . 4 8 - 15 2 – 16 13 10-15 x-17, x +3 X - 6 4 + X-1 - 6 23. x-7 24. + x-2 2.0 - 3 •4.x – 6 -4 27. 28. =56. •3.x - 4 *06x – .07 *0625 29. •083 (x – 625)= '09 (x – +59375). 30. (2x+1.5) (3x – 2.25)=(2x – 1•125) (3x +1.25). •3x – 1 •5 + 1.2.x 1-1:40 •7 (x - 1) 32. •5x – '4 2.x - 1 2 + 30 1- *5x 3x ) 1 3x ). 2.x - 1 6 31. . 33. LITERAL EQUATIONS. 151. In the equations we have discussed hitherto the coefficients have been numerical quantities, but equations often involve literal coefficients. [Art. 6.] These are supposed to be known, and will appear in the solution. Example 1. Solve (x+a) (x+b) - c (a + c)=(x – c) (x + c)+ab. x2 + ax + bx + ab – ac – c2=x2 – c2 + ab; whence ax + bx =ac, (a+b)x=ac; Simplifying the left side, we have a (oc – B) – 6 (2c — a) a - 6 : 1 (3 -- a) (x - 1) Multiplying across, ac? — cx= x2 — ax – bx + ab, ax+ bx – cx=ab, ab X-C EXAMPLES XVII. b. +1 1) a 1. ax – 2b=5bx - 3a. 2. a? (x – a)+b+ (x – b)=abx. 3. 12+a=(-x) 4. (x-a) (x+6)=(x-a+b). 5. a (x - 2)+2x=6+a. 6. m2 (m – x) – mnx=n2 (n +x). 7. (a+x) (6+x)=x (x -c). 8. (a - b)(x-a)=(a - c) (x – 6). 2x +3a 2 (3x + 2a) 2 (0-6) 2x + 6 9. 10. xta 3x +a 3x -c "3(x -c) 1 1 1 2 3 / x 11. 12. 6 3 4 la 6 9a 3.x 46 2x 13. =c(a-6) + 14. b b (x — b)2 15. 16. 2 2x - a 1 x ta 17. 4 2 3 18. (a+1)x2 - a (bx+a)=bx (-a)+ ax (x - b). bc2 19. b(a + x)-(a+x) (6 – x)= 22+ a al a a X - a a-X a CHAPTER XVIII. PROBLEMS LEADING TO FRACTIONAL AND LITERAL EQUATIONS. 152. We here give some problems which lead to equations with fractional and literal coefficients. Example 1. Find two numbers which differ by 4, and such that one-half of the greater exceeds one-sixth of the less by 8. Let x be the smaller number, then 3+4 is the greater. 1 One-half of the greater is represented by a (x+4), and one-sixth 2 1 of the less by a x. Example 2. A has $180, and B has $84; after B has won from A a certain sum, A has then five-sixths of what B has ; how much did B win ? Suppose that B wins x dollars, A has then 180— dollars, and B has 84 +3 dollars. 5 Hence 180–2=(84+2); 1080—6x=420+5x, 11x=660; .. X=60. Therefore B wins $60. EXAMPLES XVIII. 1. Find a number such that the sum of its sixth and ninth parts may be equal to 15. 2. What is the number whose eighth, sixth, and fourth parts together make up 13? 3. There is a number whose fifth part is less than its fourth part by 3 : find it. 4. Find a number such that six-sevenths of it shall exceed fourfifths of it by 2. 5. The fifth, fifteenth, and twenty-fifth parts of a number together make up 23: find the number. 6. Two consecutive numbers are such that one-fourth of the less exceeds one-fifth of the greater by 1: find the numbers. 7. Two numbers differ by 28, and one is eight-ninths of the other; find them. 8. There are two consecutive numbe such that one-fifth of the greater exceeds one-seventh of the less by 3: find them. 9. Find three consecutive numbers such that if they be divided by 10, 17, and 26 respectively, the sum of the quotients will be 10. 10. A and B begin to play with equal sums, and when B has lost five-elevenths of what he had to begin with, A has gained $6 more than half of what B has left: what had they at first ? 11. From a certain number 3 is taken, and the remainder is divided by 4; the quotient is then increased by 4 and divided by 5 and the result is 2 : find the number. 12. In a cellar one-fifth of the wine is port and one-third claret: besides this it contains 15 dozen of sherry and 30 bottles of hock. How much port and claret does it contain? 13. Two-fifths of A's money is equal to B's, and seven-ninths of B's is equal to C's: in all they have $770, what have they each? 14. A, B, and C have $1285 between then : A's share is greater than five-sixths of B's by $25, and C's is four-fifteenths of B's: find the share of each. 15. A man sold a horse for $35 and half as much as he gave for it, and gained thereby $10: what did he pay for the horse ? 16. The width of a room is two-thirds of its length. If the width had been 3 feet more, and the length 3 feet less, the room would have been square: find its dimensions. 17. What is the property of a person whose income is $430, when he has two-thirds of it invested at 4 per cent., one-fourth at 3 per cent., and the remainder at 2 per cent. ? 18. I bought a certain number of apples at three for a cent, and five-sixths of that number at four for a cent; by selling them at sixteen for six cents, I gained 3] cents ; how many apples did I buy ? |