Solving Simultaneous Equations from Context –

Solving simultaneous equations from context is a common GCSE Maths skill where you translate a real-world situation into two equations and then solve them. These questions test both your ability to form equations and your algebraic solving skills.

What Are Simultaneous Equations from Context?

In many real-life situations, two unknown quantities are linked by two pieces of information. For example, if 3 teas and 2 coffees cost £8.50, and 1 tea and 4 coffees cost £9.50, you can set up two equations and solve them to find the price of each drink.

The challenge is translating words into algebra. You must choose sensible letters for the unknowns, write an equation for each piece of information, and then solve the pair of simultaneous equations using elimination or substitution.

These questions appear on both Foundation and Higher tier papers. Foundation questions usually involve straightforward addition and subtraction contexts, while Higher questions may include more complex setups or require you to interpret the answer in context.

Key Formulas

Step-by-Step Method

1. Read the problem carefully and identify the two unknowns. Choose letters to represent them.
2. Write the first equation from the first piece of information.
3. Write the second equation from the second piece of information.
4. Solve the pair of equations using elimination (make coefficients the same and add/subtract) or substitution.
5. Interpret the solution in context and check it makes sense.

Worked Example 1 — Foundation Level

Question: 2 sandwiches and 3 drinks cost £9.50. 4 sandwiches and 1 drink cost £11.50. Find the cost of one sandwich and one drink.

Working:

Step 1 — Let s = cost of a sandwich and d = cost of a drink.

Step 2 — Equation 1: 2s + 3d = 9.50.

Step 3 — Equation 2: 4s + d = 11.50.

Step 4 — Multiply Equation 2 by 3: 12s + 3d = 34.50.

Step 5 — Subtract Equation 1: 12s + 3d - 2s - 3d = 34.50 - 9.50, so 10s = 25, giving s = 2.50.

Step 6 — Substitute into Equation 2: 4(2.50) + d = 11.50, so 10 + d = 11.50, giving d = 1.50.

Step 7 — Check in Equation 1: 2(2.50) + 3(1.50) = 5 + 4.50 = 9.50. Correct.

Answer: A sandwich costs £2.50 and a drink costs £1.50.

Worked Example 2 — Higher Level

Question: A father is 4 times as old as his daughter. In 6 years, he will be 3 times as old. Find their current ages.

Working:

Step 1 — Let d = daughter's current age and f = father's current age.

Step 2 — Equation 1: f = 4d.

Step 3 — Equation 2 (in 6 years): f + 6 = 3(d + 6).

Step 4 — Expand Equation 2: f + 6 = 3d + 18, so f = 3d + 12.

Step 5 — Substitute Equation 1 into this: 4d = 3d + 12, so d = 12.

Step 6 — f = 4 × 12 = 48.

Step 7 — Check: in 6 years, father is 54, daughter is 18. 54 = 3 × 18. Correct.

Answer: The daughter is 12 years old and the father is 48 years old.

Worked Example 3 — Exam Style

Question: At a cinema, 5 adult tickets and 3 child tickets cost £44. 2 adult tickets and 6 child tickets cost £32. Find the cost of one adult ticket and one child ticket. (5 marks)

Working:

Step 1 — Let a = cost of an adult ticket and c = cost of a child ticket.

Step 2 — Equation 1: 5a + 3c = 44.

Step 3 — Equation 2: 2a + 6c = 32.

Step 4 — Multiply Equation 1 by 2: 10a + 6c = 88.

Step 5 — Subtract Equation 2: 10a + 6c - 2a - 6c = 88 - 32, so 8a = 56, giving a = 7.

Step 6 — Substitute into Equation 2: 2(7) + 6c = 32, so 14 + 6c = 32, 6c = 18, c = 3.

Step 7 — Check in Equation 1: 5(7) + 3(3) = 35 + 9 = 44. Correct.

Answer: An adult ticket costs £7 and a child ticket costs £3.

Common Mistakes

• Using the same letter for both unknowns. Choose two different letters and define them clearly at the start.
• Setting up the wrong equations. Read each sentence carefully. "3 teas and 2 coffees cost £8.50" means 3t + 2c = 8.50, not 3t × 2c.
• Forgetting to check the answer in context. A negative price or a child older than a parent should alert you to an error.

Exam Tips

• Always start by writing "let x = ... and y = ..." — this earns a mark and keeps your work organised.
• Label each equation (e.g. Equation 1, Equation 2) so the examiner can follow your working.
• After solving, substitute back into both original equations to verify your answer.

Practice Questions

Q1 (Foundation): 3 pens and 2 rulers cost £3.80. 1 pen and 2 rulers cost £2.20. Find the cost of a pen and a ruler.

Q2 (Foundation): 4 apples and 3 bananas cost £2.50. 2 apples and 5 bananas cost £2.30. Find the cost of each fruit.

Q3 (Higher): The perimeter of a rectangle is 34 cm. The length is 5 cm more than the width. Find the dimensions.

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Related Topics

• Simultaneous Equations: Elimination
• Simultaneous Equations: Substitution
• Forming and Solving Equations

Summary

• Read the problem carefully and define two variables with clear meanings.
• Write one equation from each piece of information given in the question.
• Solve by elimination or substitution, whichever is more convenient.
• Interpret the solution in context and check it makes sense.
• Always substitute back into both original equations to verify your answer.