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.

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