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.