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Home/Resources/Maths/Simultaneous Equations
📘 GCSE Maths · Notes & worksheet

Simultaneous Equations.

GCSE Higher (Grades 6-9)board-neutral (AQA / Edexcel / OCR)Worksheet: 23 marks

Notes

Linear pairs: elimination

  1. Multiply one or both equations so a variable has the same coefficient.
  2. Add (if signs differ) or subtract (if signs match) to eliminate it.
  3. Solve for the remaining variable, then substitute back.
  4. Check in the other original equation.

Example: 2x + 3y = 13 and 5x − 2y = 4. Multiply the first by 2 and the second by 3 to get 4x + 6y = 26 and 15x − 6y = 12. Add: 19x = 38, so x = 2 and y = 3.

Linear and quadratic pairs: substitution

  1. Rearrange the linear equation to y = … (or x = …).
  2. Substitute into the quadratic or circle equation.
  3. Expand and collect into a quadratic equal to 0 and solve.
  4. Substitute each solution back into the linear equation to find the matching partner. Always give answers as coordinate pairs.

Example: y = 2x − 1 and x² + y² = 29 gives x² + (2x − 1)² = 29, so 5x² − 4x − 28 = 0 and x = −2 or x = 14/5.

What the solutions mean

  • Solutions are where the graphs intersect.
  • Two solutions: the line cuts the curve twice (a chord).
  • One repeated solution: the line is a tangent.
  • No real solutions (discriminant < 0): they never meet.

Forming equations from a context

Define variables clearly (for example "let p = price of a pen in £"), build two equations from the two statements, then solve.

Common mistakes

  • Squaring (2x − 1) as 4x² − 1. It must be 4x² − 4x + 1.
  • Finding both x values but pairing them with the wrong y values.
  • Giving only the x values when the question asks for the coordinates.

Exam tips

  • Pair your answers explicitly: (x₁, y₁) and (x₂, y₂).
  • Use the discriminant to answer "show that the line does not intersect the circle".

Worksheet

Prints the questions only, without answers. Choose “Save as PDF” in the print window to download.

Total: 23 marks.

  1. Solve 2x + 3y = 13 and 5x − 2y = 4. (3)
  2. Solve y = 2x − 1 and x² + y² = 29. (5)
  3. The curve y = x² − 3x and the line y = 2x − 6 intersect at two points. Find the coordinates of both points. (4)
  4. Solve x² + y² = 25 and y = x + 1. (4)
  5. Three pens and 2 notebooks cost £7.80. Two pens and 5 notebooks cost £12.90. Find the cost of one pen and the cost of one notebook. (4)
  6. Show that the line y = 2x + 5 does not intersect the circle x² + y² = 4. (3)
Show mark scheme and answers (tutor copy)

Try the worksheet first, then check your work here.

  1. Eliminate y: 4x + 6y = 26 and 15x − 6y = 12 (M1). Add: 19x = 38 (M1), so x = 2, y = 3. (A1) Check: 5(2) − 2(3) = 4 ✓.
  2. x² + (2x − 1)² = 29 (M1). 5x² − 4x − 28 = 0 (A1), which factorises as (x + 2)(5x − 14) = 0 (M1), so x = −2 or x = 2.8 (A1). (−2, −5) and (2.8, 4.6), that is (14/5, 23/5). (A1)
  3. x² − 3x = 2x − 6 (M1), so x² − 5x + 6 = 0 (A1), so x = 2 or 3 (A1). (2, −2) and (3, 0). (A1)
  4. x² + (x + 1)² = 25 (M1), so 2x² + 2x − 24 = 0 and x² + x − 12 = 0 (A1), so (x + 4)(x − 3) = 0 (A1). (−4, −3) and (3, 4). (A1)
  5. 3p + 2n = 7.80 and 2p + 5n = 12.90 (M1 for both equations). Multiply to eliminate p: 6p + 4n = 15.60 and 6p + 15n = 38.70 (M1). Subtract: 11n = 23.10, so n = 2.10 (A1). Pen £1.20, notebook £2.10. (A1) Check: 3(1.20) + 2(2.10) = 7.80 ✓.
  6. x² + (2x + 5)² = 4 (M1). 5x² + 20x + 21 = 0 (A1). b² − 4ac = 400 − 420 = −20 < 0, so there are no real solutions and the line does not meet the circle. (A1)
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