Forming Simultaneous Equations

Exam code: 4024

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

Forming Simultaneous Equations

How do I form simultaneous equations?

Introduce two letters, x and y, to represent two unknowns
- Make sure you know exactly what they stand for (and any units)
Create two different equations from the words or contexts
- 3 apples and 2 bananas cost $1.80, while 5 apples and 1 banana cost $2.30
  - 3x + 2y = 180 and 5x + y = 230
    x is the price of an apple, in cents
    y is the price of a banana, in cents
  - This question could also be done in dollars, $
Solve the equations simultaneously
Give answers in context (relate them to the story, with units)
- x = 40, y = 30
- In context: an apple costs 40 cents and a banana costs 30 cents
Some questions don't ask you to solve simultaneously, but you still need to
- Two numbers have a sum of 19 and a difference of 5, what is their product?
  - x + y = 19 and x - y = 5
  - Solve simultaneously to get x = 12, y = 7
  - The product is xy = 12 × 7 = 84

Worked example

At a bakery a customer pays £9 in total for six bagels and twelve sausage rolls.

Another customer buys nine bagels and ten sausage rolls, which costs £12.30 in total.

Find the cost of 5 bagels and 15 sausage rolls.

Answer:

The two variables are the price of bagels, $b$ , and the price of sausage rolls, $s$

Write an equation for the first customer's purchases, and label it equation 1

$1 6 b + 12 s = 9$

Write an equation for the second customer's purchases, and label it equation 2

$2 9 b + 10 s = 12.3$

We will choose to eliminate the $b$ terms
Make the $b$ terms equal by multiplying all parts of equation 1 by 3 and all parts of equation 2 by 2
Label these as equations 3 and 4

$\begin{array}{rcl} 1 \times 3 18 b + 36 s & = & 27 3 \\ 2 \times 2 18 b + 20 s & = & 24.6 4 \end{array}$

To eliminate $b$ , subtract equation 4 from equation 3

$3 - 4 16 s = 2.4$

Solve for $s$

$s = \frac{2.4}{16} = 0.15$

Substitute this into either equation to find $b$ , we will use equation 1

$\begin{array}{rcl} 1 6 b + 12 (0.15) & = & 9 \\ 6 b + 1.8 & = & 9 \\ 6 b & = & 7.2 \\ b & = & 1.2 \end{array}$

So sausage rolls cost £0.15 each and bagels cost £1.20 each

Use these values to find the price of 5 bagels and 15 sausage rolls

$(5 \times 1.2) + (15 \times 0.15) = 8.25$

£8.25