How to Solve Systems by Elimination

Elimination is a clean way to solve systems when the coefficients line up or can be made to line up.

Quick idea

Add or subtract equations so one variable cancels out, then solve the remaining one-variable equation.

Steps

1. Line up like terms vertically.
2. Multiply one or both equations if needed so one variable has opposite coefficients.
3. Add the equations to eliminate that variable.
4. Solve for the remaining variable and substitute back.

Worked example

Solve 2x plus y equals 7 and x minus y equals 2.

Full solution

Step-by-step walkthrough

Step 1

The y terms are opposites: y and negative y.

Step 2

Add the equations to eliminate y.

Step 3

Solve 3x equals 9, then substitute x equals 3 to find y equals 1.

Common mistake

When you multiply an equation, multiply every term on both sides, not just the variable you want to eliminate.

Practice problems

1. Solve x + y = 8 and x - y = 2.
2. Solve 2x + y = 10 and x - y = 2.
3. Solve 3x + 2y = 12 and 3x - 2y = 6.

Answers

1. x = 5, y = 3
2. x = 4, y = 2
3. x = 3, y = 3/2

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 of Equations by Substitution
• How to Graph a Linear Equation
• How to Solve Equations With Variables on Both Sides