Linear Relationships

Video coming soon3 worked examples

Theory

Linear models are used to describe situations that involve a constant rate of change. This means that every time the input variable increases by one unit, the output variable changes by the exact same, fixed amount.

1. The Structure of a Linear Model

The formula for a linear model usually takes the form: $y = a + bx$ (or $y = mx + c$ ).
$x$ is the independent variable (the input, often time).
$y$ is the dependent variable (the output).

2. The Intercept and the Gradient

To fully understand and interpret a linear model, you must be able to explain the two key parameters in the context of the real-life situation:

The y-intercept ( $a$ or $c$ ): This is the starting value. It represents the exact value of the dependent variable ( $y$ ) when the independent variable ( $x$ ) is exactly 0.
The Gradient or Slope ( $b$ or $m$ ): This represents the rate of change. It tells us exactly how much the dependent variable changes for every single 1-unit increase in the independent variable.
- A positive gradient means the value is constantly increasing (growth).
- A negative gradient means the value is constantly decreasing (decay).

3. Using the Model

Evaluating: This simply means "plugging in the numbers" to predict a future value. You substitute a given value for $x$ into your formula to calculate $y$ .
Comparing: You will often be asked to create two linear models and compare them to see which is more cost-effective or efficient for a specific value.

Worked examples

Example 1

Example 1: Generating a Model (Positive Gradient)

At the start of the year, a new online streaming service launches with 250 founding subscribers. The platform steadily gains 65 new subscribers at the end of every week.

(a) Write down a formula that models the relationship between the number of weeks since launch, $w$ , and the total number of subscribers, $S$ .

(b) State the gradient of your linear model and explain what information it gives us in this context.

Solutions:

(a) $S = 250 + 65w$
(b) The gradient is 65. This tells us that the total number of subscribers increases by exactly 65 for every 1 week that passes.
(c) $S = 250 + 65(12) = 250 + 780 = 1030$ subscribers.

Example 2

Example 2: Interpreting a Model (Negative Gradient)

A large 800-litre water tank develops a leak and begins to drain at a steady rate. The relationship between the time the tank has been leaking in minutes, $t$ , and the volume of water remaining in litres, $V$ , is modelled by the formula: $V = 800 - 24t$

(a) State the y-intercept of this formula and explain what it tells us in context.

(b) State the gradient and explain what it tells us.

Solutions:

(a) The y-intercept is 800. This tells us that before the leak started ( $t = 0$ ), the initial volume of water in the tank was 800 litres.
(b) The gradient is -24. This tells us that the volume of water in the tank decreases by 24 litres for every 1 minute that passes.
(c) The tank is completely empty when $V = 0$ .
$0 = 800 - 24t$
$24t = 800$
$t = 800 \div 24 = 33.33 \text{ minutes}$

Example 3

Example 3: Comparing Two Linear Models

Two local garage mechanics charge different rates for labour.

Mechanic A charges a fixed inspection fee of £45, plus £30 per hour of labour.
Mechanic B charges a fixed inspection fee of £80, but only £15 per hour of labour.

(a) Form an equation for both mechanics to model the total cost, $C$ , of a repair taking $h$ hours.

(b) Determine which mechanic would be cheaper to hire for a repair job that takes exactly 3 hours.

Solutions:

(a) Mechanic A: $C = 45 + 30h$ . Mechanic B: $C = 80 + 15h$ .
(b) Cost for Mechanic A: $C = 45 + 30(3) = 45 + 90 = £135$ . Cost for Mechanic B: $C = 80 + 15(3) = 80 + 45 = £125$ .
Conclusion: Mechanic B is the cheaper option for a 3-hour repair job.

Ready to practise?

9 Linear Relationships questions with answers and video solutions