Simultaneous equations

Foundation and Higher

The idea

Two equations with two unknowns usually share one solution: the point where their lines cross. Eliminate one letter to find the other.

Make the coefficients of one letter match by multiplying an equation.
Add (different signs) or subtract (same signs) to eliminate it, then solve.
Substitute back to find the other letter, and check in the other equation.

Watch out Find both values: an answer of just $x$ is half the answer.

See it worked

Worked exampleNon-calculator3 marks

Solve the simultaneous equations $x + y = 10$ and $x - y = 4$ .

Find the values of $x$ and $y$ . Give your answer as $(x, y)$ .

Add the equations to eliminate $y$ . $2x = 14 \Rightarrow x = 7$
Substitute into $x + y = 10$ . $7 + y = 10 \Rightarrow y = 3$

Answer $x = 7$ , $y = 3$

Your turn

Question 1Non-calculator3 marks

Solve the simultaneous equations 2x + y = 13 and x $-$ y = 2. Give your answer as (x, y).

Question 2Non-calculator3 marks

Solve the simultaneous equations $3x + 2y = 16$ and $5x - 2y = 8$ .

Find the values of $x$ and $y$ . Give your answer as $(x, y)$ . You must show your working.

Question 3Calculator allowed3 marks

Four pens and three notebooks cost £17.40. Two pens and five notebooks cost £21.30. Each pen has the same price and each notebook has the same price. Find the price of one notebook.

In past papers

65 official questions in 63 papers since 2017. The latest:

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