IGCSE Maths Simultaneous Equations Notes
Simultaneous equations are solved together to find values that satisfy both. Use elimination, substitution, or graphs.
Key concepts
- Elimination
- Add or subtract the equations to remove one variable, then solve for the other.
- Substitution
- Rearrange one equation and substitute it into the other — best when one is linear and one quadratic.
- Two solutions
- A linear/quadratic pair often gives two pairs of solutions.
Key formulas
Linear pair
ax + by = c and dx + ey = f
Match coefficients, then eliminate.
Worked example
Solve 3x + y = 11 and x − y = 1.
- 1Add the equations to eliminate y: 4x = 12.
- 2Solve: x = 3.
- 3Substitute back: 3(3) + y = 11, so y = 2.
Answer: x = 3, y = 2
Common mistakes
✗ Adding when you should subtract (or vice versa).
✓ Add when the matching coefficients have opposite signs; subtract when they are the same.
✗ Only finding one variable.
✓ Substitute back to find the second variable and state both.
✗ Losing a solution in linear/quadratic pairs.
✓ A quadratic can give two answers — find both x and both y values.
Quick check
Try these, then reveal the answers.
Solve x + y = 5, x − y = 1.Reveal answer
x = 3, y = 2
Best method for one linear, one quadratic?Reveal answer
Substitution
Solve 2x = 8 (from elimination). x = ?Reveal answer
4
Exam tips
- Number your equations (1) and (2) and show which you add or subtract.
- Scale equations first so one variable's coefficients match.
- Substitute your answers into the original equations to check.
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Frequently asked questions
When is substitution better than elimination?+
Substitution is best when one equation is already rearranged for a variable, or when one equation is quadratic.
Can simultaneous equations have no solution?+
Yes — if the lines are parallel there is no solution; if they are identical there are infinitely many.
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