Question 1Warm-upFoundation
Solve the simultaneous equations
The y terms have different signs, so add the equations.
Worked solution
- Add the equations: , so .
- .
- Substitute into : , so .
- Check: ✓
Answer:
GCSE & IGCSE maths practice
Simultaneous equations questions and answers
Simultaneous equations are two equations with two unknowns that are true at the same time. Solving them means finding the values that make both work. The questions start with equations you can add or subtract straight away, move on to ones where you multiply first, then word problems, and finish with the Higher-tier pairs where one equation is a quadratic.
11 questions · Foundation and Higher tier · Every answer is checked as you type it, with a full worked solution. Updated .
Where simultaneous equations comes up
| GCSE (AQA, Edexcel, OCR) | Two linear equations: Foundation and Higher. One linear and one quadratic: Higher only. |
|---|---|
| Cambridge IGCSE (0580) | Two linear equations: Core and Extended. One linear and one non-linear: Extended only. |
| Edexcel International GCSE (4MA1) | Two linear equations: Foundation and Higher. One linear and one quadratic: Higher only. |
The method
- Line the equations up so the x terms, the y terms and the numbers sit under each other.
- Make the number in front of one letter the same in both equations (multiply one or both equations if you need to).
- Same signs: subtract the equations. Different signs: add them. One letter disappears.
- Solve for the letter that's left, then substitute it into either original equation to find the other.
- Check both values in the equation you didn't use.
- Linear and quadratic (Higher): rearrange the linear equation to make x or y the subject, substitute it into the quadratic, solve, and find the matching value for each solution.
Simultaneous equations questions
Try each one before looking at the hint. If you're stuck, the AI tutor can talk you through it step by step.
Warm-up
Question 2Warm-upFoundation
Solve
Both equations have +y, so subtract one from the other.
Worked solution
- Subtract: , so .
- Substitute into : , so .
- Check: ✓
Answer:
Question 3Warm-upFoundation
Solve
Both have +2y. Subtract to get rid of y.
Worked solution
- Subtract: , so .
- Substitute into : , so and .
- Check: ✓
Answer:
Exam-style
Question 4Exam-styleFoundation
Solve
The y terms are +3y and −3y: adding removes them.
Worked solution
- Add: , so .
- Substitute into : , so and .
- Check: ✓
Answer:
Question 5Exam-styleFoundation
Solve
Multiply the first equation by 3 and the second by 4 so the y terms become +12y and −12y.
Worked solution
- First × 3: . Second × 4: .
- Add: , so .
- Substitute into : , so and .
- Check: ✓
Answer:
Question 6Exam-styleFoundation
3 adult tickets and 2 child tickets cost £36. 1 adult ticket and 4 child tickets cost £32. Find the cost of an adult ticket () and a child ticket (), in pounds.
Write two equations: 3a + 2c = 36 and a + 4c = 32.
Worked solution
- and .
- Multiply the second by 3: .
- Subtract the first: , so .
- Then , so .
- Check: ✓
Answer: Adult £8, child £6
Question 7Exam-styleFoundation
Solve
Make the y terms −10y and +10y: multiply the first by 2 and the second by 5.
Worked solution
- First × 2: . Second × 5: .
- Add: , so .
- Substitute into : , so .
- Check: ✓
Answer:
Question 8Exam-styleFoundation
The lines and meet at the point P. Find the coordinates of P.
Substitute y = 3x − 5 into the second equation.
Worked solution
- , so .
- .
- Check: ✓
Answer:
Question 9Exam-styleHigher
Solve the simultaneous equations
Solution 1
Solution 2
Substitute y = x + 1 into the circle equation and solve the quadratic.
Worked solution
- , so .
- Divide by 2: , so .
- gives ; gives .
- Check: ✓ and ✓
Answer: and
Challenge
Question 10ChallengeHigher
Solve
Solution 1
Solution 2
Both equations give y, so set them equal to each other.
Worked solution
- , so .
- gives ; gives .
- Check in : ✓ and ✓
Answer: and
Question 11ChallengeHigher
Solve
Solution 1
Solution 2
Make x the subject of the linear equation: x = 5 − 2y.
Worked solution
- , so .
- , so .
- Divide by 4: , so .
- gives ; gives .
- Check: ✓ and ✓
Answer: and
Common mistakes
- Subtracting when the signs are different (or adding when they're the same), so the letter doesn't disappear.
- Multiplying only the letter terms by the scale factor and forgetting the number on the right-hand side.
- Sign slips when subtracting a negative: 3y − (−2y) is 5y.
- With a quadratic, finding both x values but pairing them with the wrong y values, or giving only one solution.
Stuck on one of these?
The TutorAura AI tutor works through it with you step by step, with the reason for each step, then gives you one like it to try. It checks your working too.
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Sources
The questions are our own. The “where it comes up” notes are from each board's published specification, checked on 26 October 2026:
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