Simultaneous equations

The method

1. Label the two equations and decide whether to eliminate or substitute.
2. To eliminate, multiply one or both equations so a pair of coefficients matches.
3. Add or subtract the equations to remove one unknown, then solve for the other.
4. Substitute that value back into the simpler equation to find the second unknown.
5. For a linear and a quadratic pair, expect two answers and check both in both equations.

Mistakes that lose marks

• Adding the equations when the matching coefficients have opposite signs, or subtracting when they match.
• Finding one unknown and forgetting to substitute back for the second.
• Giving only one pair of values when a linear and a quadratic equation meet twice.
• Multiplying only part of an equation when scaling it.
• Leaving the answer inside a square root or fraction instead of simplifying it.

7 exam-style questions

These are our own questions written in the style of the exam, not past-paper questions. The mark codes are M = method, A = accuracy and B = independent mark. Try each one, then open the solution.

1. Exam-style core Solve the simultaneous equations 2x + 3y = 12 and x − y = 1. [3 marks]
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   1. Rearrange the second equation: x = y + 1 M1
   2. Substitute into the first: 2(y + 1) + 3y = 12, so 5y = 10 and y = 2 M1
   3. Then x = 2 + 1 = 3 A1

   Answer: x = 3, y = 2

2. Exam-style core Solve the simultaneous equations 3x + 2y = 16 and x + 2y = 12. [3 marks]
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   1. The y terms match, so subtract the second equation from the first: 2x = 4 M1
   2. x = 2 M1
   3. Substitute back: 2 + 2y = 12, so y = 5 A1

   Answer: x = 2, y = 5

3. Exam-style core Three coffees and two teas cost £13. Two coffees and three teas cost £12. Find the cost of one coffee and the cost of one tea. [4 marks]
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   1. Let a coffee cost c and a tea cost t: 3c + 2t = 13 and 2c + 3t = 12 M1
   2. Multiply the first equation by 3 and the second by 2: 9c + 6t = 39 and 4c + 6t = 24 M1
   3. Subtract: 5c = 15, so c = 3 M1
   4. Then 6 + 3t = 12, so t = 2 A1

   Answer: A coffee costs £3 and a tea costs £2

4. Exam-style extended Solve the simultaneous equations 4x + 3y = 25 and 3x + 2y = 18. [4 marks]
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   1. Multiply the first equation by 2 and the second by 3: 8x + 6y = 50 and 9x + 6y = 54 M1
   2. Subtract the first from the second: x = 4 M1
   3. Substitute back: 16 + 3y = 25, so 3y = 9 M1
   4. y = 3 A1

   Answer: x = 4, y = 3

5. Exam-style extended Solve the simultaneous equations 2x + 3y = 13 and 3x − 2y = 0. [4 marks]
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   1. From the second equation y = 3x ÷ 2 M1
   2. Substitute into the first: 2x + 3(3x ÷ 2) = 13, so 13x ÷ 2 = 13 M1
   3. x = 2 M1
   4. Then y = 6 ÷ 2 = 3 A1

   Answer: x = 2, y = 3

6. Exam-style extended Solve the simultaneous equations y = x + 1 and x² + y² = 25. [5 marks]
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   1. Substitute y = x + 1 into the second equation: x² + (x + 1)² = 25 M1
   2. Expand: 2x² + 2x + 1 = 25, so x² + x − 12 = 0 M1
   3. Factorise: (x + 4)(x − 3) = 0 M1
   4. x = 3 gives y = 4 A1
   5. x = −4 gives y = −3 A1

   Answer: x = 3, y = 4 and x = −4, y = −3

7. Exam-style higher A rectangle has a perimeter of 26 cm and an area of 40 cm². Find its length and width. [5 marks]
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   1. Let the sides be x cm and y cm: x + y = 13 and xy = 40 M1
   2. Substitute y = 13 − x into the area equation: x(13 − x) = 40 M1
   3. Rearrange and factorise: x² − 13x + 40 = 0, so (x − 5)(x − 8) = 0 M1
   4. x = 5 or x = 8 A1
   5. The sides are 5 cm and 8 cm, whichever way round they are taken A1

   Answer: 5 cm by 8 cm

Where this is in the syllabus

Simultaneous equations appears in Cambridge IGCSE Mathematics 0580 and Pearson Edexcel International GCSE Mathematics A 4MA1. For the official specifications and past papers, use the board's own website: Cambridge IGCSE Mathematics 0580 and Pearson Edexcel International GCSE Mathematics A 4MA1.

Questions students ask