Linear Equations: Solutions Using Substitution

To solve systems using substitution, follow this procedure.

1. Select one equation and solve for one of its variables.

2. In the other equation, substitute for the variable just solved.

3. Solve the new equation.

4. Substitute the value found into any equation involving both variables and solve for the other variable.

5. Check the solution in both original equations.

Example 1: Solve this system of equations by using substitution.

Solve for x.

Substitute for x in the other equation.

Solve this new equation.

Substitute 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.

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