14. Given x9x+27x2-27-2197, to find x. Result, x=4. 15. Given x2-17=130-2x2, to find x. Result, 2=7. 16. Given x3-100-412-7x3, to find x. Result, x=4. SECTION V. SIMPLE EQUATIONS. Containing two unknown quantities. RULE 1. 55. Multiply or divide one or both equations, by such numbers or quantities as will make the co-efficients of one of the unknown quantities the same in both equations. Then, take the sum or difference of these equations, according as the signs of the corresponding terms are different, or the same; the result will be a new equation, containing but one unknown quantity. The value of the remaining unknown quantity may then be determined by the methods in the last section. EXAMPLES. Given 4x+y=34, and 4y+x=16, to find the values of x and y. First, to eliminate x, we multiply the second equation by 4; whence, 16y+4x=64. From this take the first equation and 15y=30; whence y=2. The value of y being now obtained, we find x=16— When, by this or any other process, a quantity or letter is caused to vanish from the equations, it is said to be eliminated. E 4y=16-8-8; or to eliminate x, divide the first equa 1 34 tion by 4, and we shall have x+y=, which sub 15 30 tracted from the second equation, leaves 4y=4 whence y=2, as before. 2. Given ax+by=m, cx-dy=n, to determine the values of x and y. To eliminate x, multiply the first equation by c, and the second by a, whence acx+bey=cm, and acx-ady= *an; by subtraction bcy+ady=cm-an; and by dividing cm-an by bc+ad, y=bc+ad Or, to eliminate y, multiplying the equations by d and b respectively, adx+bdy=dm,bcx-bdy-bn; adding these dm+bn equations, adx+bcx=dm+bn; whence x= 3. Given S5x+7y=50 ad+bc 8x+3y=39, to find the values of x and y. Result, x=3, y=5. 18x-17y=58, to find the values of x and y. 4. Given { 15x+34y=241, 6x-4y=34, 5. Given and y. Result, x=7, y=4. {9y-2x=27. Required the values of x 6. Given {11x+3y=100, Ans. x=9, y=5. y. Result, x=8, y=4. 4x-7y=4, to find the values of x and 56. Find, from each equation, the value of one unknown quantity in terms of the other quantities. Assume the values thus obtained, as the different members of a new equation. The value of the unknown quantity contained in this equation may then be determined as in the last section. 92-614, to find the values of x and y. 8 From the first equation y=357-49x. From the second, y=72x-490, Hence, 72x-490-357-49x, 72x+49x=357+490, 847 121 =2. Quere the values of x and y? 4. Given 30 y 2x 3y ·+ =22, Ans. x=20, y=18. 3x 2y 67 = 07. Quere the values of x and y? 5 3 3 6. Given -2y-d { Ans. x=15, y=20. xy::a:b. Quere the values of x and y? 57. Find, in either equation, the value of one unknown quantity, and substitute the value thus found in the other equation; whence will arise a new equation, containing but one unknown quantity, whose value may be determined as already taught. From the first equation x=12-. This value being 2 substituted for x in the second equation, we have Multiplying by -2, 7y+6y-168=-116. 2. Given 10x+15y=825. Quere the values of x and y? Ans. x=45, y=25. 58. Multiply one of the equations by an indefinite quantity, and to the equation thus formed, add the other given equation. Assume the sum of the co-efficients of one required quantity equal to 0, and thence determine the value of the assumed multiplier. The new equation will then contain but one unknown quantity, which may be found as before. |