Substitution Method Calculator

What Is the Substitution Method?

The substitution method solves a system of linear equations by isolating one variable in one equation and substituting that expression into the other equation. This reduces a 2-variable system to a single equation in one unknown.

For the system:

$$\begin{aligned} a_1 x + b_1 y &= c_1 \\ a_2 x + b_2 y &= c_2 \end{aligned}$$

choose an equation and variable, solve for that variable, substitute into the remaining equation, and back-substitute to find the second variable.

Substitution Method Steps

1. Write both equations in standard form with variables on the left and constants on the right.
2. Solve one equation for $x$ or $y$.
3. Substitute the resulting expression into the other equation.
4. Solve the new one-variable equation.
5. Back-substitute to find the remaining variable.
6. Check the solution in both original equations.

Special Cases

• No solution: Substitution produces a false statement such as $0 = 1$. The lines are parallel.
• Infinitely many solutions: Substitution produces a true identity such as $0 = 0$. The equations are dependent.

Compare with the Elimination Method Calculator, Linear Equation Solver, and Cramer's Rule Calculator.