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
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
Write down both equations.
- S2
Multiply Equation 1 by -1 to get: -x - y = -5.
- S3
Add this new equation to Equation 2: x 2 + y - x - y = 7 - 5, which simplifies to x 2 - x = 2.
- S4
Solve the quadratic equation: x 2 - x - 2 = 0. Factorize it as (x - 2)(x + 1) = 0, giving x = 2 or x = -1.
- S5
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.
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Write down both equations.
- S2
Multiply Equation 1 by -1 to get: -x - y = -5.
- S3
Add this new equation to Equation 2: x 2 + y - x - y = 7 - 5, which simplifies to x 2 - x = 2.
- S4
Solve the quadratic equation: x 2 - x - 2 = 0. Factorize it as (x - 2)(x + 1) = 0, giving x = 2 or x = -1.
- S5
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.
Final answer: (2, 3) and (-1, 6)
Common mistakes
- Forgetting to check solutions by substituting them back into the original equations. — Always substitute the values of x and y back into both original equations to ensure they satisfy both.
- Incorrectly solving the quadratic equation. — 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. — Always solve the quadratic equation completely and consider both possible values for x.
- Incorrectly substituting values back into equations. — Be careful when substituting values. Ensure you are substituting the correct value for the correct variable.
- Failing to simplify equations before solving. — 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. — 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
- Full worked solution (all marking points)4 marks