RULE III. 249. Find the value of the same unknown quantity in terms of the other and known quantities, in each of the equations; then, let the two values, thus found, be put equal to each other; an equation arises involving only one unknown quantity; the value of which may be found, and therefore, that of the other unknown quantity, as in the preceding rules. This rule depends upon the well known axiom, (Art 47); and the two preceding methods are founded on principles which are equally simple and obvious. From the first equation, x=100-3y, and from the second, x= 100-y 2 Multiplying by 2, 100-y=200—by, by transposition, 6y-y-200-100, or, 5y=100; .. by division, y=20. whence, x100-3y=100-3x20; .. x=40. Here, two values of y might have been found, which would have given an equation involving only x; and from the solution of this new equation, a value of x, and therefore of y, might be found. Ex. 2. Given x+y=7, and 1x+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 3x-13y=1, to find the values of x and y. From the first equation, x= 47-9y 4 And multiplying both sides of this equation, by 8, 94-18y=1+13y; Ex. 8. Given 3x-7y_2x+y+1, and 8 3 5 6, to find the values of x and y. y 2 -y Ans. x=13, and y=3. Ex. 9. Given x+y=10, and 2x-3y=5, to find Ex. 10. Given 3x-5y-13, and 2x+7y=81, to the values of x and y. Ans. x=7, and y=3. find the values of x and y. Ans. x16, and y=7. Ex. 11. Given *+2+8y=31, and 3 192, to find the values of x and Ex. 12. Given Ans. 19, and y = 3 251. EXAMPLES in which the preceding Rules are applied, in the Solution of Simple Equations, Involving two unknown Quantities. the values of x and y. Multiplying the first equation by 20, .. by transposition, 48y-17x=155. Multiplying the second equation by 6, by transposition, 2y+30x160 ... (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, 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. and x 3y4:7, to find the values of x and y. Reducing the first equation to lower terms, 4x-1 4+ y x-y 9 18 1. 4+y+ 3 6 and therefore, (Art. 147), multiplying by 18, 2y-4x+1=18-24-6y+3x-3y; .. by transposition, 7=7x-11y. But from the second equation, 7x=12y. Substituting therefore this value in the preceding equation, it becomes 12y-11y=7, or y=7. 12y_84 and. x =12. 7 7 to find the values of x and y. Multiplying the first equation by 33, 33x-9y+6-3x=33+15x+. 4y ; 3 multiplying again by 3, and transposing, we shall have 45x-31y=81. Multiplying the second equation by 12, ... by transposition, 19y-5x=131. Multiplying this by 9, 171y-45x=1179; but 45x-31y= 81; .. by addition, 140y=1260; and by division, y=9. Now, 5x=19y-131-171-131—40; .. by division, x=8. |