Finding an expression for the value of x from both (1) and (2), we have (3) and (4). Placing these two values of x equal to each other (Art. 13, Ax. 8), we form (5), which contains but one unknown quantity. Reducing (5) we obtain (7), or y = 5. Substituting this value of y in (3), we have (8), or x — 16. Hence, RULE. Find an expression for the value of the same unknown quantity from each equation, and put these expressions equal to each other. By this method of elimination find the values of x and y in the following equations: + 1. Given CASE III. 114. Elimination by combination. (3x-2y=7, to find x and y. If we multiply (1) by 2, and (2) by 3, we have (3) and (4), in which the coefficients of x are equal; subtracting (4) from (3), we have (5), which contains but one unknown quantity. Reducing (5), we have (6), or y 1; substituting this value of y in (2), we obtain (7), which reduced gives (8), or x = 3. = If we multiply (1) by 2, we have (3), an equation in which y has the same coefficient as in (2); since the signs of y are different in (2) and (3), if we add these two equations together, we have (4), which contains but one unknown quantity. Reducing (4), we have (5), or x 18. Substituting this value of x in (1), we have (6), which reduced gives (7), or y =12. Hence, RULE. Multiply or divide the equations so that the coefficients of the quantity to be eliminated shall become equal; then, if the signs of this quantity are alike in both, subtract one equation from the other; if unlike, add the two equations together. NOTE.—The least multiplier for each equation will be that which will make the coefficient of the quantity to be eliminated the least common multiple of the two coefficients of this quantity in the given equations. It is always best to eliminate that quantity whose coefficients can most easily be made equal. By this method of elimination find the values of x and y in the following equations: 4 115. Find the values of x and y in the following EXAMPLES. NOTE. -Which of the three methods of elimination should be used depends upon the relations of the coefficients to each other. That one which will eliminate the quantity desired with the least work is the best. 819 Ans. x= 9. y = 1. x=11. y= 5. |