## Linear Equations: Solutions Using Substitution with Two Variables

To solve systems using substitution, follow this procedure:

**Select one equation and solve it for one of its variables.**
**In the other equation, substitute for the variable just solved.**
**Solve the new equation.**
**Substitute the value found into any equation involving both variables and solve for the other variable.**
**Check the solution in both original equations.**

Usually, when using the substitution method, one equation and one of the variables leads to a quick solution more readily than the other. That's illustrated by the selection of *x* and the second equation in the following example.

Example 1

Solve this system of equations by using substitution.

Solve for *x* in the second equation.

Substitute for *x* in the other equation.

Solve this new equation.

Substitute the value found for *y* into any equation involving both variables.

Check the solution in both original equations.

The solution is *x* = 1, *y* = –2.

If the substitution method produces a sentence that is always true, such as 0 = 0, then the system is dependent, and either original equation is a solution. If the substitution method produces a sentence that is always false, such as 0 = 5, then the system is inconsistent, and there is no solution.