GMAT Quantitative Review

3.0 Math Review

3.2 Algebra

4. Solving Two Linear Equations with Two Unknowns

For two linear equations with two unknowns, if the equations are equivalent, then there are infinitely many solutions to the equations, as illustrated at the end of section 3.2.2 (“Equations”). If the equations are not equivalent, then they have either one unique solution or no solution. The latter case is illustrated by the two equations:


Note that image implies image, which contradicts the second equation. Thus, no values of x and y can simultaneously satisfy both equations.

There are several methods of solving two linear equations with two unknowns. With any method, if a contradiction is reached, then the equations have no solution; if a trivial equation such as image is reached, then the equations are equivalent and have infinitely many solutions. Otherwise, a unique solution can be found.

One way to solve for the two unknowns is to express one of the unknowns in terms of the other using one of the equations, and then substitute the expression into the remaining equation to obtain an equation with one unknown. This equation can be solved and the value of the unknown substituted into either of the original equations to find the value of the other unknown. For example, the following two equations can be solved for x and y.


In equation (2), image. Substitute image in equation (1) for x:


If image, then image and image.

There is another way to solve for x and y by eliminating one of the unknowns. This can be done by making the coefficients of one of the unknowns the same (disregarding the sign) in both equations and either adding the equations or subtracting one equation from the other. For example, to solve the equations


by this method, multiply equation (1) by 3 and equation (2) by 5 to get


Adding the two equations eliminates y, yielding image, or image. Finally, substituting image for x in one of the equations gives image. These answers can be checked by substituting both values into both of the original equations.