We have learned so far how the substitution method is effective for finding solutions to pairs of simultaneous equations with two unknown variables. You may also notice that the method is rather tedious. Before finishing our study, we will explore another method that will also give us solutions to the equations. This alternate method is not as intuitive, but it will give us answers with less computational labor. We will show both equations in the general linear form ax + by = c that is already familiar. We will then apply some algebraic “magic” to solve for each of x and y using a general strategy. To begin with, let us see what we can do with two equations that have no multipliers for y, such as ax + y = b, and cx + y = d. This will give us the opportunity to demonstrate the subtraction rule in a way that we have not yet encountered. We will use the subtraction rule to combine or “condense” the two equations into just one for the variable x. How can we do this? We already know that each side of an equation contains an expression that stands for a quantity. We also know that the same quantity can be added or subtracted from both sides of an equation without losing its equality. Let us therefore apply that principle to our two equations, and subtract the two sides of the second equation from both sides of the first. This will give us the result ax – cx = b – d. You may see examples on the following page.