Linear Models (HSC SSCE Mathematics Standard): Revision Notes

Linear Models

What is a linear model?

A linear model is a mathematical way to represent real-world situations using a straight-line equation. When we create a linear model, we're describing a practical situation using a linear function. This allows us to make predictions, understand relationships, and solve problems mathematically.

Linear models are particularly useful because many everyday situations involve constant rates of change. For example, when a catering company charges a fixed base fee plus an amount per person, or when water flows into a tank at a steady rate, these situations can be modelled using straight-line graphs.

The gradient-intercept form

The most common way to write a linear model is using the gradient-intercept form:

$y = mx + b$

where:

• $y$ is the dependent variable (the value we're calculating)
• $x$ is the independent variable (the value we know or choose)
• $m$ is the gradient (the rate of change)
• $b$ is the $y$-intercept (the starting value when $x = 0$)

Components of a linear model

Every linear model has two key components that tell us important information about the situation.

The gradient (m)

The gradient represents the rate of change. It tells us how much $y$ changes for every unit increase in $x$. In practical situations, the gradient often represents:

• A cost per item
• A speed or velocity
• A rate of flow
• A rate of change over time

To calculate the gradient using two points $(x_1, y_1)$ and $(x_2, y_2)$:

$m = \frac{\text{Rise}}{\text{Run}} = \frac{y_2 - y_1}{x_2 - x_1}$

The intercept (b)

The intercept represents the starting value when $x = 0$. In practical situations, the intercept often represents:

• A base fee or fixed cost
• An initial amount
• A starting value before any changes occur

Example: catering costs

A catering company charges a base amount of $100 plus a rate of $25 per guest. We can model this situation using a linear equation.

Let $c$ be the cost of the event (in dollars) and $n$ be the number of guests. The equation becomes:

$c = 25n + 100$

In this model:

• The gradient is 25, which represents the rate per guest ($25 per person)
• The $c$-intercept is 100, which represents the base amount ($100 fixed cost)
• The number of guests $n$ must be greater than zero and a whole number

The graph shows how the cost increases linearly with the number of guests. The steepness of the line is determined by the gradient ($25 per guest), and the line crosses the vertical axis at $100 (the base amount).

Reading values from linear graphs

Linear graphs are useful tools for making conversions and reading values. You can read information from a graph by tracing horizontal and vertical lines from known values to find unknown values.

Worked example: currency conversion

Currency exchange rates can be represented using linear graphs, allowing us to convert between different currencies quickly.

The graph below converts Australian dollars (AUD) to euros (EUR).

Finding equations from graphs

When you're given a linear graph, you can find its equation by identifying key information from the graph and using the gradient-intercept form.

Worked example: water tank problem

Water is pumped into a partially full tank. The graph shows the volume of water $V$ (in litres) after $t$ minutes.