AQA GCSE Mathematics Paper 3 (Foundation), June 2025: Question 24

2 marks · Medium difficulty · Short Answer

Find the equation of a straight line parallel to a given line and identify the equation of a line with a given gradient passing through a given point.

Practise this question

Question

Question 24(a) asks to write down the equation of a straight line parallel to y - 2x = 9 for 1 mark. Question 24(b) states that a straight line has gradient 5 and passes through the point (3, 7), and asks to circle the correct equation from four options: y = 3x - 2, y = 3x + 7, y = 5x, and y = 5x - 8 for 1 mark.

Mark scheme

Show the mark scheme Mark scheme for Question 24: Part (a) awards B1 for y = 2x + c or y - 2x = c (oe where c ≠ 9), with additional guidance that leaving 'c' as a letter gives B0, y - 2x + 1 = 9 gives B1, and y - 2x + 1 = 10 gives B0. Part (b) awards B1 for selecting y = 5x - 8.

How to answer it

Straight Line Graphs: Parallel Lines & Equations

📌 What this question tests

This question assesses your ability to find equations of straight lines using y = mx + c . Specifically, it tests:

  • Identifying the gradient of a line from an implicit form and writing a parallel equation.
  • Understanding that parallel lines have identical gradients ( m ) but different y-intercepts ( c ).
  • Finding the exact equation of a straight line given its gradient and a coordinate point on the line.

Question 24 (a)

Parallel Lines [1 mark]

✅ Correct Answers [1 Mark]

Any equation of the form y = 2x + c (or y - 2x = c ) where c ≠ 9.

Examples of full-mark answers:

  • y = 2x
  • y = 2x + 1
  • y = 2x - 5
  • y - 2x = 0

💡 Key Knowledge

  • The standard form is y = mx + c , where m is the gradient and c is the y-intercept.
  • Rearrange the given line into standard form:
    y - 2x = 9 → y = 2x + 9
  • The gradient is m = 2.
  • Parallel lines have identical gradients, so your line must also have a gradient of 2.

📐 Step-by-Step Method

  1. Rearrange to find the gradient:
    y - 2x = 9 ⇒ add 2x to both sides ⇒ y = 2x + 9 .
    The gradient is 2 .
  2. Choose a new intercept:
    Keep m = 2 , but pick any number other than 9 for c (e.g., c = 3 ).
  3. Write the final equation:
    y = 2x + 3 .

❌ Common Errors to Avoid

  • Writing literally y = 2x + c : You must choose an actual numerical value for the intercept. Writing the algebraic letter c scores 0 marks.
  • Writing the same line: If you write y = 2x + 9 or y - 2x + 1 = 10 (which simplifies to y - 2x = 9 ), you score 0 marks because it is the exact same line, not a parallel line.
  • Sign mistake on rearrangement: Thinking the gradient is -2 by incorrectly moving the 2x term.
Mark Scheme Breakdown: B1 for any correct linear equation with gradient 2 and a different y-intercept ( c ≠ 9 ).

Question 24 (b)

Finding the Equation of a Line [1 mark]

✅ Correct Answer [1 Mark]

Circle:

y = 5x - 8

🧠 Exam Technique: Elimination

  • The question states the line has gradient 5.
  • In y = mx + c , the number in front of x must be 5.
  • Instantly eliminate options with gradient 3:
    y = 3x - 2 and y = 3x + 7 .
  • Only two options remain: y = 5x or y = 5x - 8 .

📐 Step-by-Step Calculation

  1. Identify given information:
    Gradient m = 5
    Point (x, y) = (3, 7)
  2. Substitute into y = mx + c :
    7 = 5(3) + c
  3. Solve for c :
    7 = 15 + c
    c = 7 - 15 = -8
  4. State the full equation:
    y = 5x - 8

❌ Common Errors to Avoid

  • Confusing the y-intercept with the y-coordinate: Some students mistakenly think that because the point has y = 7 , the equation must end in + 7 (choosing y = 3x + 7 ). The y-intercept is only equal to y when x = 0 !
  • Swapping x and y: Calculating 3 = 5(7) + c instead of 7 = 5(3) + c .
  • Guessing without testing: You can always test the point: 5(3) - 8 = 15 - 8 = 7 . This confirms the point (3, 7) lies on the line.
Mark Scheme Breakdown: B1 for selecting/circling y = 5x - 8 only.

Topics

Algebra · 3.2.2 Graphs

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 3 (Foundation), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.