Teach Me Maths

Simultaneous Equations (Three Unknowns)

This section is in development. More questions and features coming shortly.

AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 12 marks

a, b and c satisfy the simultaneous equations

4a+b+2c=13 a+3b+3c=8 2a+3b+2c=7

Work out the value of \;a+b+c

You do not need to find a, b and c separately.

Hint

Add all three equations together and look at the coefficient of each letter.

Worked solution
  1. Add the three equations.
  2. a: 4+1+2=7, b: 1+3+3=7, c: 2+3+2=7
  3. So 7a+7b+7c=13+8+7=28
  4. Divide by 7: \;a+b+c=4
  5. (Check: a=2, b=-1, c=3 satisfies all three, and 2-1+3=4.)

Question 23 marks

A shop sells pens, rulers and erasers.

  • A pen, a ruler and an eraser cost £1.15 altogether.
  • Two pens and a ruler cost £1.60
  • A pen costs four times as much as an eraser.

Work out the cost of a ruler.

Give your answer in pence.

Hint

Let a pen cost p pence, a ruler r pence and an eraser e pence, and write one equation for each fact.

Worked solution
  1. Let the costs in pence be p, r and e.
  2. p+r+e=115, 2p+r=160, p=4e
  3. Substitute p=4e: \;5e+r=115 and 8e+r=160
  4. Subtract: \;3e=45, so e=15
  5. r=115-5(15)=40
  6. (Check: p=60; 60+40+15=115 and 120+40=160.)
  7. A ruler costs 40p

Question 3Challenge5 marks

Solve the simultaneous equations

4x-y+2z=-5 2x+3y+6z=4 6x+2y+2z=8

Do not use trial and improvement.

Give any answers that are not integers as fractions in their simplest form.

x=

2 marks

y=

2 marks

z=

1 mark

Hint

Label the equations, then remove the same letter from two different pairs. Here y is the easiest to eliminate because equation (1) has -y.

Worked solution

x=

  1. Label the equations (1), (2), (3).
  2. Eliminate y: 3\times(1)+(2): \;14x+12z=-11 (4)
  3. Eliminate y: 2\times(1)+(3): \;14x+6z=-2 (5)
  4. (4) - (5): \;6z=-9, so z=-\dfrac32
  5. Substitute into (5): \;14x-9=-2, so 14x=7 and x=\dfrac12

y=

  1. Using x=\dfrac12 and z=-\dfrac32:
  2. Substitute into (1): \;2-y-3=-5
  3. y=4

z=

  1. From (4) - (5): \;6z=-9, so z=-\dfrac32
  2. Check in (2): \;1+12-9=4 ✓ and (3): \;3+8-3=8 ✓

More on this topic: Simultaneous Equations (Three Unknowns) worksheet with full solutions

All AQA Level 2 Further Maths practice questions

Unofficial practice questions written by Teach Me Maths. Not produced or endorsed by AQA.