Simultaneous Equations — GCSE Mathematics Revision
Revise Simultaneous Equations for GCSE Mathematics. Step-by-step explanation, worked examples, common mistakes and exam-style practice aligned to AQA, Edexcel, OCR, WJEC, Eduqas, CCEA, Cambridge International (CIE), Pearson Edexcel International, OxfordAQA International, SQA, IB, AP.
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Go to SequencesWhat is Simultaneous Equations?
Simultaneous equations are two or more equations that share the same unknowns. You solve them to find values of x and y that satisfy both equations at the same time. There are two main methods: elimination (adding or subtracting equations to remove one variable) and substitution (rearranging one equation and substituting into the other). At Higher tier, you also need to solve one linear and one quadratic simultaneously.
Board notes: Linear-quadratic simultaneous equations are Higher tier only on all boards. AQA and Edexcel often set these as 4-5 mark questions.
Step-by-step explanationWorked examples
Worked example 1: Core method
Solve: 2x + 3y = 12 and 5x - 3y = 9. Add the equations: 7x = 21, so x = 3. Substitute back: 2(3) + 3y = 12, 3y = 6, y = 2. Solution: x = 3, y = 2.
Worked example 2: Exam variation
Now change one detail in the question and keep the same structure: name the Simultaneous Equations idea being tested, show the method or evidence, then explain why it answers the command word. This helps GCSE Mathematics students avoid memorising one surface pattern.
Worked example 3: Mark-scheme check
Finish by checking the answer against marks: one point for the correct Simultaneous Equations idea, one for accurate working or evidence, and one for a precise final statement. If any step is vague, rewrite it before moving to timed practice.
Mini lesson for Simultaneous Equations
1. Understand the core idea
Simultaneous equations are two or more equations that share the same unknowns. You solve them to find values of x and y that satisfy both equations at the same time.
Can you explain Simultaneous Equations without copying the notes?
2. Turn it into marks
Solve: 2x + 3y = 12 and 5x - 3y = 9. Add the equations: 7x = 21, so x = 3.
Underline the method, evidence, or command-word move that would earn credit in GCSE Algebra.
3. Fix the likely mark leak
Watch for this mistake: Forgetting to multiply the ENTIRE equation (both sides) when scaling up for elimination.
Write one correction rule before doing another practice question.
Practise this topic
Jump into adaptive, exam-style questions for Simultaneous Equations. Free to start; sign in to save progress.
Mini quiz: Simultaneous Equations
Three quick checks for revision practice. They are original StudyVector prompts, not official exam-board questions.
Question 1
In one GCSE sentence, explain what Simultaneous Equations is testing.
Answer: Simultaneous equations are two or more equations that share the same unknowns. You solve them to find values of x and y that satisfy both equations at the same time.
Mark focus: Precise definition and topic focus.
Question 2
A student sees a Simultaneous Equations question but is not sure how to start. What should the first method line establish?
Answer: It should identify the rule, equation, diagram feature, or transformation before any calculation. That protects method marks and makes later checking easier.
Mark focus: Method selection and command-word control.
Question 3
A student makes this mistake: "Forgetting to multiply the ENTIRE equation (both sides) when scaling up for elimination." What should their next repair task be?
Answer: Do one Simultaneous Equations question and review the mistake type.
Mark focus: Error correction and next-step practice.
Simultaneous Equations flashcards
Core idea
What is the main idea in Simultaneous Equations?
Simultaneous equations are two or more equations that share the same unknowns. You solve them to find values of x and y that satisfy both equations at the same time.
Common mistake
What mistake should you avoid in Simultaneous Equations?
Forgetting to multiply the ENTIRE equation (both sides) when scaling up for elimination.
Practice
What is one useful practice task for Simultaneous Equations?
Answer one Simultaneous Equations question and review the mistake type.
Exam board
How should you use board notes for Simultaneous Equations?
Linear-quadratic simultaneous equations are Higher tier only on all boards. AQA and Edexcel often set these as 4-5 mark questions.
Common mistakes
- 1Forgetting to multiply the ENTIRE equation (both sides) when scaling up for elimination.
- 2Sign errors when subtracting equations — especially when subtracting a negative term (e.g. 3y - (-2y) = 5y, not y).
- 3Only finding x and forgetting to substitute back to find y.
- 4In linear-quadratic pairs, not expanding brackets correctly after substitution.
Simultaneous Equations exam questions
Exam-style questions for Simultaneous Equations with mark-scheme style solutions and timing practice. Aligned to AQA, Edexcel, OCR, WJEC, Eduqas, CCEA, Cambridge International (CIE), Pearson Edexcel International, OxfordAQA International, SQA, IB, AP specifications.
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Step-by-step method
Step-by-step explanation
4 steps · Worked method for Simultaneous Equations
Core concept
Simultaneous equations are two or more equations that share the same unknowns. You solve them to find values of x and y that satisfy both equations at the same time. There are two main methods: elimin…
Frequently asked questions
When should I use elimination vs substitution?
Use elimination when the coefficients of one variable are the same (or easy to make the same). Use substitution when one equation is already in the form y = ... or x = ..., or when one equation is quadratic.
How do I solve simultaneous equations with a quadratic?
Rearrange the linear equation to make one variable the subject, substitute into the quadratic, expand and simplify to get a quadratic in one variable, then solve using factorising or the quadratic formula.