The elimination method (also called the addition method) removes one variable by adding or subtracting the two equations. It works best when both equations are in standard form, Ax + By = C.
The method
- Line up the equations so the x terms, y terms and numbers are in columns.
- Make one variable’s coefficients the same size, multiplying one or both equations if needed.
- Add the equations if those coefficients have opposite signs, or subtract if they have the same sign. That variable disappears.
- Solve for the remaining variable, then substitute it back to find the other.
- Check in both original equations.
Worked examples
Example 1: add straight away
Solve x + y = 10 and x − y = 4.
The y terms are +y and −y, opposite signs, so add the equations: 2x = 14, and x = 7.
Substitute into x + y = 10: 7 + y = 10, so y = 3. The solution is (7, 3).
Example 2: opposite signs again
Solve 2x + 3y = 12 and 4x − 3y = 6.
Add: 6x = 18, so x = 3.
Substitute: 2 × 3 + 3y = 12, so 3y = 6 and y = 2. The solution is (3, 2).
Check: 4 × 3 − 3 × 2 = 6 ✓.
Example 3: multiply both equations
Solve 3x + 2y = 16 and 2x + 5y = 18.
Neither variable matches yet. To make the x terms both 6x, multiply the first equation by 2 and the second by 3:
6x + 4y = 32 and 6x + 15y = 54.
The x terms have the same sign, so subtract the first from the second: 11y = 22, and y = 2.
Substitute: 3x + 2 × 2 = 16, so 3x = 12 and x = 4. The solution is (4, 2).
Check: 2 × 4 + 5 × 2 = 18 ✓.
See Example 3 in the Intersection Finder →
Example 4: a fraction in the answer
Solve 5x − 2y = 3 and 3x + 4y = 20.
Multiply the first equation by 2 to get −4y: 10x − 4y = 6.
The y terms are −4y and +4y, so add: 13x = 26, and x = 2.
Substitute: 5 × 2 − 2y = 3, so 2y = 7 and y = 7/2. The solution is (2, 7/2).
Check: 3 × 2 + 4 × 7/2 = 6 + 14 = 20 ✓.
Add or subtract?
- Opposite signs (such as +3y and −3y): add the equations.
- Same signs (such as 6x and 6x): subtract one equation from the other.
- When you subtract, subtract every term, including the number on the right. Most mistakes happen here.
Elimination or substitution?
Use substitution when one equation already gives x or y on its own, or a variable has a coefficient of 1. Use elimination when both equations are in standard form with larger coefficients. Both methods always give the same answer, and the Intersection Finder shows either one.