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.

equation

Solve for x in the second equation.

equation

Substitute equation for x in the other equation.

equation

Solve this new equation.

equation

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

equation

Check the solution in both original equations.

equation

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.