GCSE Maths Simultaneous Equations Notes
Simultaneous equations are two equations with two unknowns solved together. Use elimination or substitution to find the pair of values that satisfies both.
Key concepts
- Simultaneous equations
- Two (or more) equations that share the same unknowns, e.g. 2x + y = 7 and x − y = 2. The solution works in both at once.
- Elimination
- Add or subtract the equations so one unknown cancels. Match coefficients first if needed by multiplying one equation.
- Substitution
- Rearrange one equation for a single letter, then substitute it into the other. Best when a coefficient is already 1.
Key formulas
Elimination setup
Make one pair of coefficients equal, then add or subtract
Same signs → subtract; different signs → add.
Worked example
Solve 2x + y = 7 and x − y = 2.
- 1The y-terms have opposite signs, so add the equations: (2x + y) + (x − y) = 7 + 2.
- 2This gives 3x = 9, so x = 3.
- 3Substitute x = 3 into x − y = 2: 3 − y = 2.
- 4Solve: y = 1.
Answer: x = 3, y = 1
Common mistakes
✗ Adding when you should subtract (or vice versa) during elimination.
✓ Same signs on the matched term → subtract; different signs → add.
✗ Only finding one unknown.
✓ Substitute back to find the second value, then state both.
✗ Forgetting to multiply every term when matching coefficients.
✓ Multiply the whole equation, both sides, by the same number.
Quick check
Try these, then reveal the answers.
Solve x + y = 10 and x − y = 4.Reveal answer
x = 7, y = 3
Solve 3x + y = 11 and y = 2.Reveal answer
x = 3, y = 2
In 2x + y = 7 and x − y = 2, what cancels when you add?Reveal answer
The y-terms (+y and −y).
Exam tips
- Check your pair works in both original equations before writing the final answer.
- Label your equations (1) and (2) so the examiner can follow your elimination.
- If one equation is already 'y = …', substitution is usually fastest.
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Frequently asked questions
Should I use elimination or substitution?+
Use substitution when one equation is already solved for a letter; otherwise elimination is usually quicker for GCSE.
Can simultaneous equations have no solution?+
Yes — if the lines are parallel they never meet, so there is no pair of values that works.
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