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4PM1/01R Jun 2023, Question 2: Solving by Substitution

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 8 marks

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The question, in short

Simultaneous equations y = px + 9 and 6x² − xy = 5 have a solution (−1/2, q), with p an integer and q prime. Find p, q, then the other solution.

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Part (a) · 4 marks

Use the given solution point (−1/2, q) in both equations to find the integer p and the prime q.

  1. 1Substitute x = −1/2, y = q into 6x² − xy = 5: 6(1/4) + q/2 = 5
  2. 23/2 + q/2 = 5, so q = 7
  3. 3Substitute into y = px + 9: 7 = −p/2 + 9
  4. 4p = 4

Answerp = 4, q = 7

Part (b) · 4 marks

Using the values of p and q found, solve the equations again to find the second solution point.

  1. 16x² − x(4x + 9) = 5
  2. 22x² − 9x − 5 = 0
  3. 3(2x + 1)(x − 5) = 0, so x = 5 (x = −1/2 is the given solution)
  4. 4y = 4(5) + 9 = 29

Answerx = 5, y = 29

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