Modelling a problem using a system

Why model?

Many real-world problems can naturally be expressed as a system of two equations with two unknowns: the ages of two people, the prices of two items, a mixture of solutions, etc.

General method

1. Identify the two unknowns and label them clearly (for example, x and y).
2. Translate each sentence of the problem statement into an equation.
3. Solve the resulting system (using substitution or elimination).
4. Check that the solution makes sense in the context of the problem (for example, an age cannot be negative).
5. Answer the question asked in a complete sentence.

Practical example

A cinema sells adult tickets for 10 euros and children’s tickets for 6 euros. A group of 8 people paid 68 euros in total. How many adults and how many children are there?

Let x be the number of adults and y the number of children.

{ x + y = 8
{ 10x + 6y = 68

From the first equation: y = 8 - x

We substitute: 10x + 6(8 - x) = 68 -> 10x + 48 - 6x = 68 -> 4x = 20 -> x = 5

Therefore, y = 8 – 5 = 3.

There are 5 adults and 3 children. Check: 5 + 3 = 8 people, and 10(5) + 6(3) = 50 + 18 = 68 euros.

Common pitfall

The main pitfall is choosing or defining the unknowns incorrectly (swapping x and y around compared to the question), or forgetting to answer the final question using the correct units (euros, people, years, etc.). Always re-read the question to ensure that the solution you have found makes sense.