Solving simultaneous equations by elimination
Use this page when elimination is the route your worksheet, revision page, or teacher has chosen and you want to see the process line by line.
Start here if you want the short version before reading the full method.
- Elimination works by removing one variable so the other can be solved directly.
- It is often the cleanest method when the coefficients already match or can be matched with one short multiplication step.
What this topic means and what to look for first.
Elimination is often the first method taught because it turns two equations into one simpler equation by cancelling one variable deliberately.
The main judgement call is whether you should add or subtract the equations after aligning them.
One reliable route through the topic.
- 1Write the equations so like terms are vertically aligned.
- 2Choose the variable that is easiest to eliminate.
- 3Multiply one or both equations if needed so the chosen coefficients match.
- 4Add or subtract the equations to cancel the chosen variable.
- 5Solve the remaining equation, then substitute back for the second variable.
See the method in action.
x + y = 10 and x - y = 2
- Add the equations so y cancels and 2x = 12.
- So x = 6.
- Substitute into x + y = 10 to get y = 4.
2x + y = 11 and x - y = 1
- Here the y coefficients already cancel by addition, so add the equations directly to get 3x = 12.
- So x = 4.
- Substitute into x - y = 1 to get y = 3.
3x + 2y = 16 and x + 2y = 8
- Subtract the second equation from the first so 2y cancels.
- This gives 2x = 8, so x = 4.
- Substitute into x + 2y = 8 to get y = 2.
Things that commonly send the method off track.
- Multiplying one equation to align coefficients but forgetting to multiply every term.
- Adding when subtraction was needed, or subtracting when addition would have cancelled the variable.
- Cancelling the wrong variable because the equations were not written in a clean aligned form.
Use a short verification pass before moving on.
- Substitute the final x and y values into both original equations.
- Re-read the aligned equations and confirm the variable really did cancel the way you intended.
Try a few variations before switching to a calculator or solver tool.
- x + y = 9 and x - y = 3
- 2x + y = 12 and x - y = 0
- 4x + 3y = 23 and 2x + 3y = 11
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Short answers worth checking.
It is often better when the coefficients already match or can be matched quickly, so one variable disappears cleanly.
No. You add or subtract depending on which operation actually cancels the chosen variable.
Put both final values back into both original equations and confirm both statements are true.
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