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GCSE Maths / Edexcel

Simultaneous equations

Solve two linear equations by eliminating one unknown, then substitute back to find the other unknown.

AlgebraFoundation and HigherGrades 5 to 7Skill

Curriculum path: GCSE Maths > Edexcel > Algebra > Simultaneous equations

Pearson Edexcel GCSE Maths algebra A19: solve two simultaneous equations in two variables.

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Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

Simultaneous equations are two equations that must be true at the same time.

The answer is a pair of values, usually one value for x and one value for y. Both values must work in both original equations.

Elimination means removing one unknown so you can solve for the other unknown.

If the signs of the matching terms are different, add the equations. If the signs are the same, subtract the equations.

After finding the first unknown, substitute it back into either original equation to find the second unknown.

For Edexcel method marks, write the elimination step clearly and state both final values.

Key ruleEliminate one unknown first, then substitute back to find the other.

Worked examples

Add to eliminate

Solve x + y = 9 and x - y = 3.

  1. Add the equations: 2x = 12.
  2. x = 6.
  3. Substitute into x + y = 9: 6 + y = 9.

Answer: x = 6, y = 3

Subtract to eliminate

Solve 2x + y = 11 and x + y = 7.

  1. Both equations contain +y, so subtract one equation from the other to eliminate y.
  2. (2x + y) - (x + y) = x.
  3. 11 - 7 = 4, so x = 4.
  4. Substitute into x + y = 7: 4 + y = 7.

Answer: x = 4, y = 3

Make coefficients match

Solve 3x + y = 13 and x + y = 7.

  1. Subtract the second equation from the first: 2x = 6.
  2. x = 3.
  3. Substitute into x + y = 7.

Answer: x = 3, y = 4

Common mistakes

  • Adding when the equations need subtracting, or subtracting in the wrong order.
  • Finding one unknown but not substituting back to find the other.
  • Changing only one term when multiplying an equation.
  • Not checking the final pair of values in both original equations.

Quick exercise

Try these before moving to the exam-style questions.

  1. Solve x + y = 8 and x - y = 2.
  2. Solve x + y = 10 and x - y = 4.
  3. Solve 2x + y = 12 and x + y = 8.
  4. Solve 3x + y = 14 and x + y = 8.
  5. Solve x + 2y = 11 and x + y = 7.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleFoundation and HigherNon-calculator3 marks

Solve x + y = 8 and x - y = 2.

simultaneous equationseliminationfoundation algebra
Standard exam styleFoundation and HigherNon-calculator4 marks

Solve 2x + y = 10 and x + y = 7.

eliminationsubstitution checkmethod marks
ChallengeHigherEither5 marks

Adult tickets cost a pounds and child tickets cost c pounds. 2 adult tickets and 3 child tickets cost 31 pounds. 1 adult ticket and 3 child tickets cost 22 pounds. Find a and c.

forming simultaneous equationscontext problemhigher reasoning