Forming pairs of equations

Some word problems can be solved by using the information given to form equations.

If there are two sets of information these can form pairs of simultaneous equations.

At the cinema, one family bought 2 drinks and 4 bags of popcorn and the total cost was £15.00.

Another family bought 3 drinks and 2 bags of popcorn and the total cost was £10.50.

What is the cost of one drink (d) and one bag of popcorn (p)?

This time, we need to create each equation from the information given above.

\(2d + 4p = 15\,(Equation\,1)\)

\(3d + 2p = 10.5\,(Equation\,2)\)

Mulitiply equation 1 by -1

\(- 2d - 4p = - 15\,(Equation\,3)\)

Mulitiply equation 2 by 2

\(6d + 4p = 21\,(Equation\,4)\)

Adding equations 3 and 4 together:

\(( - 2d - 4p = - 15) + (6d + 4p = 21)\)

\(-2d-4p=-15\)

\(6d+4p=21\)

Adding these equations we get:

\(4d = 6\)

\(d = \frac{6}{4}\)

\(d = 1.5\)

Substitute \(d\) into equation 1: \(2d + 4p = 15\)

When \(d = 1.5\)

\(2 \times 1.5 + 4p = 15\)

\(3 + 4p = 15\)

\(4p = 15 - 3\)

\(4p = 12\)

\(p = \frac{{12}}{4} = 3\)

Therefore since \(d = 1.5\) and \(p = 3\), then the cost of one drink is £1.50 and the cost of one bag of popcorn is £3.00.

Working with simultaneous equationsForming pairs of equations