How to Solve Systems of Equations by Substitution

A system of equations has more than one equation at the same time. The solution is the point that makes both equations true.

Quick idea

Substitution works by replacing a variable with an equal expression from the other equation.

Steps

1. Solve one equation for one variable if it is not already solved.
2. Substitute that expression into the other equation.
3. Solve the one-variable equation.
4. Substitute back to find the other variable.

Worked example

Solve the system y equals 2x and x plus y equals 9.

Full solution

Step-by-step walkthrough

Step 1

Because y equals 2x, replace y in the second equation with 2x.

Step 2

Solve x plus 2x equals 9 to get x equals 3.

Step 3

Substitute x equals 3 into y equals 2x to get y equals 6.

Common mistake

After finding one variable, do not stop. A system solution usually needs both x and y.

Practice problems

1. Solve y = x + 1 and x + y = 7.
2. Solve y = 3x and x + y = 12.
3. Solve x = y + 2 and x + y = 10.

Answers

1. x = 3, y = 4
2. x = 3, y = 9
3. x = 6, y = 4

Ask Pennpaper to explain it live

If the steps make sense but you still feel stuck, start a Pennpaper lesson and ask the tutor to draw the problem on the whiteboard. Seeing the symbols move step by step is often what makes the concept click.

Related guides

• How to Solve Systems by Elimination
• Slope-Intercept Form Explained
• How to Solve Equations With Variables on Both Sides