CHAPTER XI. SIMULTANEOUS SIMPLE EQUATIONS. 184. If we have two unknown numbers and but one relation between them, we can find an unlimited number of pairs of values for which the given relation will hold true. Thus, if x and y are unknown, and we have given only the one relation x + y = 10, we can assume any value for x, and then from the relation x + y = 10 find the corresponding value of y. For from x + y = 10 we find y = 10 - х. If a stands for 1, y stands for 9; if x stands for 2, y stands for 8; if x stands for 2, y stands for 12; and so on without end. 185. We may, however, have two equations that express different relations between the two unknowns. Such equations are called independent equations. Thus, x + y = 10 and x y = 2 are independent equations, for they evidently express different relations between x and y. 186. Independent equations involving the same unknowns are called simultaneous equations. If we have two unknowns, and have given two independent equations involving them, there is but one pair of values which will hold true for both equations. Thus, if in § 184, besides the relation x + y = 10, we have also the relation x - y = 2, the only pair of values for which both equations will hold true is the pair x = 6, y = 4. Observe that in this problem z stands for the same number in both equations; so also does y. 187. Simultaneous equations are solved by combining the equations so as to obtain a single equation with one unknown number; this process is called elimination. There are three methods of elimination in general use: I. By Addition or Subtraction. II. By Substitution. III. By Comparison. 189. To Eliminate by Addition or Subtraction, therefore, Multiply the equations by such numbers as will make the coefficients of one of the unknown numbers equal in the resulting equations. Add the resulting equations, or subtract one from the other, according as these equal coefficients have unlike or like signs. NOTE. It is generally best to select the letter to be eliminated that requires the smallest multipliers to make its coefficients equal; and the smallest multiplier for each equation is found by dividing the L. C. M. of the coefficients of this letter by the given coefficient in that equation. Thus, in example 2, the L. C. M. of 6 and 8 (the coefficients of x) is 24, and hence the smallest multipliers of the two equations are 4 and 3, respectively. Sometimes the solution is simplified by first adding the given equations or by subtracting one from the other. |