Linear equations solve problems involving a relationship between two variables, usually represented by x and y. These problems involve finding one variable's value given the other variable's value. In real-life situations, linear equations can help us make decisions by modelling scenarios with different variables.

Here are some examples of linear equations word problems:

Example 1: A mobile phone company offers a pay-as-you-go plan that charges 10p per minute for calls and a monthly contract that costs £20 per month and includes 500 free minutes. Which plan is better value for someone who makes 300 minutes of calls per month?

Let x be the total cost of a pay-as-you-go plan.
Let y be the total cost of the monthly contract.

For the pay-as-you-go plan, the cost is proportional to the number of minutes used: x = 10p × 300 = £30.

The monthly contract costs are fixed at £20, regardless of the number of minutes used.

To determine which plan is better value, we need to compare the total cost for each plan: x > y or x < y.

y = £20 (because 500 minutes are included in the monthly contract and 300 minutes are being used)

x > y because £30 > £20, therefore the pay-as-you-go plan is better value.

Example 2: A school is planning a trip to a theme park. The cost of transportation is £200, and each student must pay for their admission ticket. If the total cost of the trip is to be shared equally among 30 students, how much should each student pay if the admission ticket costs £25?