GMAT Exam: A Simple Guide To Algebra - Systems of Linear Equations

A pair of two linear equations, each with the same two variables, is a system when the two equations are considered simultaneously.

Solving a System of Equations by Substitution
To solve a system of two equations by substitution: Solve one of the equations for one of the variables in terms of the other variable and then use substitution to solve for the other variable. Plug the obtained value into one of the original equations, and solve for the unknown variable.

Here is an example.

Solve the system.

Begin by solving the first equation, –x + y = –7, for y in terms of x.

Substitute into the second equation and solve for x.

Tip: Enclose substituted expressions in parentheses to avoid errors.

Substitute 2 for x in the first equation and solve for y.

Tip: You can substitute the value for x in either equation. Just pick the one you think would be easier to work with.

The ordered pair (2, –5) satisfies the system.

Solving a System of Equations by Elimination
To solve a system of two equations by elimination: Multiply one or both equations by constants to produce opposite coefficients of one variable so that it can be eliminated by adding the two equations. Solve the resulting equation for the remaining variable. Plug the obtained value into one of the original equations, and solve for the unknown variable.

Here is an example.

Solve the system.

First, to eliminate x, multiply the first equation by 3.

Add the resulting two equations, and then solve for y.

Substitute –5 for y in the first equation and solve for x.

Tip: You can substitute the value for x in either equation.

Just pick the one you think would be easier to work with.

The ordered pair (2, –5) satisfies the system.