Simultaneous equations: step-by-step guide, methods, examples, and practice
Use this page as the main simultaneous-equations hub on CureMath — AI Math Explainer: understand what the system means, compare the main methods, and then choose the right next page or tool.
Start here if you want the short version before reading the full method.
- Simultaneous equations are solved by finding values that make both equations true at the same time.
- The two main routes are elimination and substitution.
- A strong final check is to place the final values back into both original equations, not just one.
What this topic means and what to look for first.
Simultaneous equations are easier when you first decide what the question structure is asking for, rather than jumping straight into a method.
This page is designed to help you spot whether elimination or substitution is cleaner, then give you a dependable check at the end.
One reliable route through the topic.
- 1Write both equations in a clean aligned form so like terms line up vertically.
- 2Decide whether elimination or substitution gives the shortest reliable route.
- 3Solve one variable first, then substitute back to find the second.
- 4Check both values in both original equations before relying on them.
- 5If the arithmetic gets messy, pause and check whether a different method would be cleaner for this system.
Choose the route that fits the quadratic.
Best when the coefficients already line up or can be matched with a short multiplication step.
Best when one equation already isolates a variable or can be rearranged quickly into that form.
Useful for understanding what the solution means: the answer is the intersection point of both lines.
See the method in action.
x + y = 10 and x - y = 2
- Add the equations to eliminate y and get 2x = 12.
- So x = 6.
- Substitute into x + y = 10 to get y = 4.
2x + y = 11 and x - y = 1
- Add the equations directly to eliminate y and get 3x = 12.
- So x = 4.
- Substitute back into x - y = 1 to get y = 3.
y = 2x + 1 and x + y = 10
- Substitute y = 2x + 1 into x + y = 10.
- This gives x + 2x + 1 = 10, so 3x = 9 and x = 3.
- Substitute back to get y = 7.
Things that commonly send the method off track.
- Adding or subtracting the equations in the wrong direction and changing the sign pattern.
- Multiplying one equation to align coefficients but forgetting to multiply every term.
- Finding one variable correctly and then stopping without substituting back for the second.
- Checking the answer in only one equation instead of in both original equations.
Use a short verification pass before moving on.
- Substitute the final x and y values into both original equations and confirm both statements are true.
- If you used elimination, re-read the aligned equations and make sure the coefficients really did cancel the way you intended.
- If you used substitution, check that brackets were handled correctly after replacing one variable with an expression.
Try a few variations before switching to a calculator or solver tool.
- x + y = 11 and x - y = 1
- 2x + y = 7 and x - y = 2
- y = x + 4 and 2x + y = 13
- 3x + 2y = 17 and x + 2y = 9
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Extra algebra revision resources
If you want more printed algebra practice after this page, these broader searches are a sensible next step.
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Algebra workbook and revision book search
Useful if you want more equation, factorising, and worked-example practice in one printed source.
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GCSE algebra practice resources search
A wider GCSE-style search if you want more mixed algebra questions beyond one online guide.
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Short answers worth checking.
They are two equations solved together by finding values that make both of them true at the same time.
Use elimination when the coefficients line up cleanly or can be matched with a short multiplication step.
Substitute both final values into both original equations and confirm that both equations still work.
It is the point where both lines intersect, because that point lies on both equations at the same time.
Continue with the next closely related topic.
Use the public site structure first, then switch into the solver tool only if you need a direct test.
CureMath uses artificial intelligence to suggest how a maths problem could potentially be solved. AI can make mistakes.
Check important answers independently before relying on them.