..b2x2+x2=3abx, and x= Ex. 11. Given (x3-y2)x(x—y)=3xy, the values of x and y. to find Dividing the second equation by the first, (x2+y2) .(x+y)=15xy; ··· x3+x2y+xy2+y3=15xy ; but from the first, x3-x3y-xy2+y3 = 3xy; 3 ... by addition, 2x3+2y3-18xy, and x3+y3=9xy. But by subtraction, 2x2y+2xy2-12xy, and x+y=6; ... by cubing, 3+3x2y+3xy2+y3=216 ; 3 3 +y3=9xy; .. by subtraction, 3x2y+3xy2=216—9xy, or 3.(x+y).xy=3×6.xy=216-9xy;... 27xy=216, and xy=8. Now x2+2xy + y2=36, - and 4xy = 32; 2 ... by subtraction, x-2xy + y2 = 4; and by extracting the square root, x-y=±2, and x=4, or 2; but x+y= 6, ... by addition, 2x=8, or 4; and by subtraction, 2y=4, or 8; ..y=2, or 4. Ex. 12. Given lues of x. Ex. 13. Given values of x. Ex. 14. Given b' to find the va Ans. x=√(2ab—b3). 18 a1+3x-7=x+2+, to find the Ex. 15. Given x+y:x:: 5: 3, and xy=6, to find the values of x and y. Ans. x=3, and y=±2. Ex. 16. Given x-y:x:: 5: 6 and xy3 =384, to find the values of x and y. Ans. x=24, and y=4. Ex. 17. Given x+y:x:: 7 : 5 and xy+y2=126, to find the values of x and y. Ans. 15, and y=±6. 2 Ex. 18. Given xy2+y=21, and x3y1+y2=333, to find the values of x and Y. 3 Ans. x=2, or ; and y=3, or 18. Ex. 19. Given x2y+xy2180, and x+y=189, to find the values of x and y. Ans. x=5, or 4; and y=4, or 5. Ex. 20. Given x+xy+y=19 and x2+xy+y2= 133, to find the values of x and y. Ans. x=9, or 4; and y=4, or 9. Ex. 21. Given x2y+xy2=6, and x3y+x3y3=12, to find the values of x and y. Ans. x=2, or 1; and y=1, or 2. Ex. 22. Given (x2+y3)x(x+y)=2336, and (x2 —y3).(x—y)=576, to find the values of x and y. Ans. x=11, or 5; and y=5, or 11. Ex. 23. Given 3+y3=(x+y). xy and x2y+xy2 = 4xy, to find the values of x and y. Ans. x=2, and y=2. Ex. 24. Given 2.(x2+y2).(x+y)=15xy and 4 (x4 —y1).(x2+y2)=75x2y2, to find the values of x and y. Ex. 25. Given x : Ans. x=2, and y=1. y: 45 and x2+4y2=181, to find the values of x and y. Ans. x= 19, and y=±5. Ex. 26. Given x2+y3: x2 -y2:: 17: 8, and xy? =45, to find the values of x and Ex. 27. x-Vy=3, and the values of x and y. Ex. 28. Given x+y: y. Aus. x=5, and y=4. x+y=7; to find Ans. x=625, and y=16. x-√y:: 4:1, and x y16, to find the values of x and 3 3 y. Ans. x 25, and y=9. Ex. 29. Given x3 +y3 : x3 —y3 :: 559 : 127, and x2y=294; to find the values of x and y.' Ans. x=7, and y=6. Ex. 30. Given x3+y3=20, and x+y=6; to find the values of x and y. Ans. x=18, or ±√8, and y=32, or 1024. Ex. 31. Given x1+2x2y3 +y1=1296 — 4xy(x2+ xy+y2), and x-y-4; to find the values of x and y. Ans. 5, or 1, and y=1, or —5. Ex. 32. Given (4x+1)+4x=9, to find the √(4x+1)−√4x value of x. -- Ans. x=. Ex. 33. Given xy=a2, and x2+y=s; to find the values of x and y. Ans. x=(s2+2a2)+√(s2 —2a3)], and y(s2+2a2)—√(s2-2a2)]. Ex. 34. Given x2+x/xy2=208, and y2+y3⁄4x2y =1053, to find the values of x and y Ex. 35. Given x+x 3 3 3 y 3 3 Ans. x8, and y=±27 3 and x + xy=y=582193, to find the values of x and y. Ans. 81, or 16; and y=16, or 31. CHAPTER IX. UN THE SOLUTION OF PROBLEMS, 399. In addition to what has been already said, (Arts. 258, 266), with respect to the translation of problems into algebraic equations, it is very proper to observe, that, when two quantities are required which are in the given proportion of m to n, the unknown quantities are represented by mx and nx; then the values of x, found from the equation of the problem by the methods in the preceding chapter, being multiplied by m and n respectively, will give the numbers required. If three quantities are required, which have a given ratio to one another, assume mx, nx, and px, m to n being the ratio of the first to the second, and n to p being that of the second to the third; then proceed as before. 2 Problem 1. There are two numbers in the proportion of 4 to 5, the difference of whose squares is 81. What are those numbers? Let 4x and 5x=the numbers; then (25x-16x2) 9x2=81; .. x2=9, and x= +3. Consequently the numbers are +12 and 15. Prob. 2. It is required to divide 18 into two such parts, that the squares of those parts may be in the proportion of 25 to 16. Let x=the greater part; then 18-x-the less; ..x:(18-x): 25: 16, and 16x225(18-x)2; .. extracting the square root, 4x=5(18-x), and 9x=90; .. x=10, and the parts are 10 and 8. Prob. 3. What two numbers are those whose difference, multiplied by the greater, produces 40, and by the less 15? Let x=the greater, and y=the less; .. x2-xy=40, and xy-y=15; ... by subtraction, x2-2xy+y2=25, and x-y-±5. ... from the 1st equation, x(x-y)=+5x=40, and x=18. From the 2d, y(x-y)=±5y=15; ..±3. Prob. 4. What two numbers are those whose difference, multiplied by the less, produces 42, and by their sum 133? Let x=the greater, and y=the less; .'. (x—y).y=42, and (x-y). (x+y)=133; by subtracting twice the first from the second, but x=y±7; ...x=±6±7=±13. Prob. 5. What two numbers are those, which being both multiplied by 27, the first product is a square, and the second the root of that square; but being both multiplied by 3, the first product is a cube, and the second the root of that cube? Let x and y be the numbers; then 27x=27y, and .. x=27y2; .*. also 3x=3y; and .. x=9y3; whence 9y327y', and y=3; .. x=9×27=243; ... the numbers are 243, and 3. Prob. 6. Two travellers, A and B, set out to meet each other; A leaving the town C at the same time that B left D. They travelled the direct road, C D; and, on meeting, it appeared that A had travelled 18 miles more than B; and that A could have gone B's journey in 15 3-4th days, but B would have been 28 days in performing A's journey. What was the distance between C and D? Let x=the number of miles A has travelled; ... x 18 the number B has travelled; and x 18 x 153: the number of days A travelled, 63x 4.(x-18) ; also xx-18:: 28: to the number of days B travelled= 63x 4(x —18); or 16.(x—18)2 = 9x3 ; .• •. 4. (x — 18)=±3x. and x=72, or 102; whence A travelled 72, and B 54 miles; and, the whole distance, C D 126 miles. Prob. 7. Two partners, A and B, dividing their gain (601.), B took 201. A's money continued in trade 4 months; and if the number 50 be divided by A's money, the quotient will give the number of months that B's money, which was 1001., continued in trade. What was A's money, and how long did B's money continue in trade? 50 Suppose A's money was x pounds; .. the number of months B's money was in trade; and since B gained 201. A gained 401. 4x2=10000, and x2=2500; .'. x=±50. A's money was 50l., and B's money was one month in trade. Prob. 8. A detachment from an army was marching in regular column, with 5 men more in depth than in front; but, upon the enemy coming in sight, the front was increased by 845 men; and by this movement the detachment was drawn up in five lines. Required the number of men. Let x=the number in front; ..x+5=the number in depth, |
