Ex. 2. Given 4x+y=7, and x+y=8, to find the values of x and y. Multiplying both equations by 6, and we shall have 3x+2y=42, and 2x+3y=47, 42-2y From the first of these equations, x=- 3 Multiplying each member by 6, we shall have 84-4y=144-9y; by transposition, 9y-4y-144-84, or 5y=60;.. y=12. And, by substituting this value of y, in one of the values of x, the first, for instance, we shall have Ex. 3. Given 8x+18y=94, and 8x-13y=1, to find the values of x and y.* From the first equation, x=; 47-9y 4 4 8 And multiplying both sides of this equation, by 8, 94-18y=1+13y; by transposition, 18y-13y=—94+1; Changing the signs, or what amounts to the same thing, multiplying both sides by -1, and we shall 18y+13y=94-1, or 31y=93; have 1+13y 1+39 40 93 y= 3 31 whence x 5. Ex. 4. Given x+y=a, to find the values bx+cy=de, of x and y. From the first equation, x=a—y ; de-cy and from the second, x= b ab-by-de-cy; by transposition, cy-by-de-ab ; by collecting the coefficients, (c—b) y=de-ab; de-ab .. by division, y= whence x= ca-ab-de-ab ca -de that is, x C c-b 250. If in the above equations, there existed, between the coefficients, these relations, cb, and ca> or <de; then, And therefore, (Art. 233), the two proposed equations would be contradictory. In order to give a numerical example, let c=b 4, a=3, and de-10; then, by substituting these values, we shall have Where the values of x and y are both infinite, and therefore, under these relations, there can be no finite values of x and y, which would fulfil both equations at once; this is what will still appear more evident, if we substitute these values in the proposed equations; for then, we shall have, x+y =3, and 4x+4y-10; which are evidently contradictory; since, if we multiply the first by 4, and subtract the second from the result, we should have 0=2. 0 0' Again, if cb=4, a 3, and de=12; then a 0 and y; therefore, under these relations, the 0 two proposed equations would be indeterminate ; and, in fact, this appears evident by inspection only; for the second furnishes no condition, but what is contained in the first, since the two proposed equations, in this case, would become x+y=3, and 4x+4y-12. Ex. 5. Given 3x+7y=79, and 2y-1-9, to find the values of x and y. Ans. 10, and y=7. Ex. 6. Given ++1=6, and "="+3=4, to 3 7 find the values of x and y. Ans. 11, and y=4 Ex. 8. Given Ans. x= = 8, and y X 3x—7y_2x+y+1, and 8—*—y 3 5 =6, to find the values of x and y. 5 Ans. x 13, and y=3, Ex. 9. Given x+y=10, and 2x-3y=5, to find the values of x and y. Ans. x7, and y=3. Ans. x=16, and y=7. Ex. 10. Given 3x-5y=13, and 2x+7y=81, to find the values of x and y. Ex. 11. Given *+2+8y=31, and +5 3 192, to find the values of x and y. 4 +10x= Ans. x=19, and y= Ex. 12. Given 2x+14=18, and 2y+x 2 to find the values of x and y. and 7y=32=11+y, √ y. Ans. x 6, and y=8. 251. EXAMPLES in which the preceding Rules are applied, in the Solution of Simple Equations, Involv. ing two unknown Quantities. the values of x and y. Multiplying the first equation by 20, 40y-5x-15=140+12x-3y; .. by transposition, 48y-17x=155. Multiplying the second equation by 6, ... by transposition, 2y+30x=160. . . (A). Multiplying this by 24, we have 48y+720x=3840; but 48y- 17x= 155; .. by subtraction, 737x=3685, and by division, x=5. From equation (A), 2y=160-30x ; ... by substitution, 2y-160-150, 10 by division, y==;•'•y=5. The values of x and y might be found by any of the methods given in the preceding part of this Section; but in solving this example, it appears, that Rule I, is the most expeditious method which <we could apply. ~ |