168. Definition. When two or more equations are
satisfied by the same values of the unknown quantities, they are called simultaneous equations.
169. In the example already worked, we have used the method of solution which best illustrates the meaning of the term simultaneous equations; but in practice it will be found that this is rarely the readiest mode of solution. It must be borne in mind that since the two equations are simultaneously true, any equation formed by combining them will be satisfied by the values of x and y which satisfy the original equations. Our object will always be to obtain an equation which involves one only of the unknown quantities.
170. The process by which we cause either of the unknown quantities to disappear is called elimination. It may be effected in different ways, but three methods are in general use : (1) by Addition or Subtraction ; (2) by Substitution ; and (3) by Comparison.
ELIMINATION BY ADDITION OR SUBTRACTION.
171. Ex.1. Solve 7x + 2y=47 (1), 5x-4y = 1 (2).
Here it will be more convenient to eliminate y.
Multiplying (1) by 2, 14x + 4y = 94, and from (2) 5x - 4 y = 1; adding, 19 x = 95 ; x = 5.
To find y, substitute this value of x in either of the given equations.
Thus from (1) 35 + 2y = 47 ; y = 6, and x = 5.
In this solution we eliminated y by addition.
Ex.2. Solve 3x+7y = 21 (1), 5x + 2y = 10 (2).
