A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Simultaneous Equations: Linear and Quadratic — mark scheme explained
The short answer
Simultaneous equations are a set of two or more equations that must be solved together. In AQA A-Level Mathematics, you will encounter problems involving one linear equation and one quadratic equation. This topic is crucial for understanding how to find the points where these types of equations intersect.
The question
Solve the simultaneous equations: x + y = 5 (Equation 1) and x 2 + y = 7 (Equation 2).
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
Marks are typically awarded for each step in solving the equations, including setting up the equations, performing algebraic manipulations, and finding the correct solutions. Partial marks may be given for correct working even if the final answer is incorrect.
What the command words demand
- Solve
- Find the values of the variables that satisfy the given equations.
- Verify
- Check your solutions by substituting them back into the original equations.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Aim to spend about 5-7 minutes on a typical question involving simultaneous equations with one linear and one quadratic equation. This allows time for careful calculation and verification of solutions.
- Write down both equations.0 marks
- Multiply Equation 1 by -1 to get: -x - y = -5.1 mark
- Add this new equation to Equation 2: x 2 + y - x - y = 7 - 5, which simplifies to x 2 - x = 2.1 mark
- Solve the quadratic equation: x 2 - x - 2 = 0. Factorize it as (x - 2)(x + 1) = 0, giving x = 2 or x = -1.1 mark
- Substitute these values back into Equation 1 to find y: For x = 2, 2 + y = 5 → y = 3; for x = -1, -1 + y = 5 → y = 6.1 mark
Final answer: (2, 3) and (-1, 6)
Work through every step correctly and you earn all 4 marks.
Another worked example
Solve the simultaneous equations: 2x + y = 8 (Equation 1) and x 2 + y = 10 (Equation 2).
- Write down both equations.0 marks
- Multiply Equation 1 by -1 to get: -2x - y = -8.1 mark
- Add this new equation to Equation 2: x 2 + y - 2x - y = 10 - 8, which simplifies to x 2 - 2x = 2.1 mark
- Solve the quadratic equation: x 2 - 2x - 2 = 0. Use the quadratic formula x = (2 ± √(4 + 8)) / 2, giving x = 1 + √3 or x = 1 - √3.2 marks
- Substitute these values back into Equation 1 to find y: For x = 1 + √3, 2(1 + √3) + y = 8 → y = 6 - 2√3; for x = 1 - √3, 2(1 - √3) + y = 8 → y = 6 + 2√3.1 mark
Final answer: (1 + √3, 6 - 2√3) and (1 - √3, 6 + 2√3)
Work through every step correctly and you earn all 5 marks.
Common mistakes
Forgetting to check solutions by substituting them back into the original equations.
Why it happens: Students often solve for the variables but forget to verify their solutions, leading to incorrect answers.
Fix: Always substitute the values of x and y back into both original equations to ensure they satisfy both.
Incorrectly solving the quadratic equation.
Why it happens: Students may make algebraic errors when factoring or using the quadratic formula, leading to incorrect solutions.
Fix: Double-check your factorization and use the quadratic formula carefully. Verify each step of the calculation.
Forgetting that a quadratic equation can have two solutions.
Why it happens: Students may only find one solution to the quadratic equation, missing the second solution.
Fix: Always solve the quadratic equation completely and consider both possible values for x.
Incorrectly substituting values back into equations.
Why it happens: Students may substitute values incorrectly, leading to incorrect solutions.
Fix: Be careful when substituting values. Ensure you are substituting the correct value for the correct variable.
Failing to simplify equations before solving.
Why it happens: Students may skip simplifying steps, making the problem more complex and prone to errors.
Fix: Simplify equations as much as possible before solving. This can make the problem easier to handle.
Choosing the wrong method (elimination or substitution) for the given problem.
Why it happens: Students may choose a more complicated method when a simpler one is available, leading to unnecessary work and potential errors.
Fix: Assess the problem carefully and choose the method that seems easier. Practice both methods to become proficient in choosing the best approach.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Simultaneous Equations | Solve a linear and quadratic pair of equations by elimination and factorising. | 4 |
| Solve Simultaneous Equations | Solve a linear and quadratic pair of simultaneous equations for both x and y. | 5 |
| Simultaneous Equations | Solve one linear and one quadratic equation by substitution to find x and y. | 5 |
| Simultaneous Equations | Solve a linear and quadratic equation pair using substitution to find x and y. | 6 |
| Total across these question types | 20 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.