| When solving more than one equation at a time, we are interested in finding the set of all solutions that will satisfy both equations. An easy way to find this set of solutions for the linear equations 3x + 4y = 10 and 2x + y = 5 is to express one variable in terms of the other. In this case, the variable y is isolated in the second equation, which may be rewritten y = 5 - 2x. If we substitute this new expression of y into the first equation, we get: 3x + 4(5 - 2x) = 10. Now there is only one variable and the equation may be solved. The solution, x = 2, may then be substituted into both equations, which yields a value of y = 1. Thus, the set of solutions which satisfies both equations is (2,1). Graphically, any values that satisfy both equations will result in an intersection of the lines (see graph). |
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