Linear Models Revision Notes for HSC SSCE Mathematics Standard

Linear Models

What are linear models?

Linear models are mathematical representations of real-world situations using linear functions. When we observe a relationship between two quantities that changes at a constant rate, we can describe this relationship using a linear equation. This allows us to make predictions, solve problems, and understand patterns in everyday contexts.

The most common form for expressing a linear model is the gradient-intercept form:

$y = mx + c$

infoNote

Understanding the gradient-intercept form:

In the equation $y = mx + c$ :

$y$ represents the dependent variable (the output or result)
$x$ represents the independent variable (the input or what we control)
$m$ represents the gradient (the rate of change)
$c$ represents the vertical intercept (the starting or initial value)

Real-world example: iTunes collection

Consider Logan's music collection. He starts with $100$ songs and adds $15$ new songs every month. We can create a linear model to represent this situation.

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Worked Example: Modelling a Music Collection

Let $N$ be the number of songs and $m$ be the number of months. The equation becomes:

$N = 15m + 100$

In this model:

The gradient is $15$ (songs added per month)
The vertical intercept is $100$ (initial number of songs)
The number of months must be greater than zero and a whole number

Key features of linear models

Every linear model has two important characteristics that help us understand the situation it represents.

The gradient (rate of change)

The gradient tells us how quickly one quantity changes in relation to another. It represents the rate of change and is calculated using:

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Calculating the Gradient:

$m = \frac{\text{Rise}}{\text{Run}}$

This formula tells us how much the dependent variable changes (rise) for each unit change in the independent variable (run).

In Logan's iTunes collection:

The gradient is $15$
This means the collection grows by 15 songs per month
We can also express this as $\frac{300}{20}$ songs (from the graph), which simplifies to $15$ songs per month

infoNote

Exam tip: The gradient always includes units. In linear models, clearly state what the gradient represents (e.g., "songs per month" or "dollars per policy").

The vertical intercept (starting value)

The vertical intercept is the value of $y$ when $x = 0$ . It represents the initial or starting value in most real-world contexts.

In Logan's iTunes collection:

The vertical intercept is $100$
This represents the initial number of songs he owned before adding any new ones
On a graph, this is where the line crosses the vertical axis

Working with linear graphs

Reading values from graphs

Linear graphs allow us to quickly find corresponding values between two quantities. We can read values in both directions.

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Worked Example: Currency Conversion

The graph below converts Australian dollars (AUD) to New Zealand dollars (NZD).

Part a: Convert $40 AUD to NZD

Locate $40$ on the horizontal (AUD) axis
Draw a vertical line up to meet the graph
Draw a horizontal line across to the vertical (NZD) axis
Read the value: $50 NZD

Part b: Convert $25 NZD to AUD

Locate $25$ on the vertical (NZD) axis
Draw a horizontal line across to meet the graph
Draw a vertical line down to the horizontal (AUD) axis
Read the value: $20 AUD

Finding equations from graphs

To create an equation from a linear graph, we need to determine two things: the gradient and the vertical intercept.

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Worked Example: Weekly Income with Commission

Grace sells insurance policies. She earns a base salary plus commission on each policy sold. The graph shows her weekly income ( $I$ ) against the number of policies sold ( $n$ ).

Part a: What is Grace's base salary?

The base salary is the amount Grace earns when she sells zero policies. Looking at the graph when $n = 0$ , we find $I = 400$ .

Grace's base salary is $400 per week

Part b: What is Grace's salary when she sells 8 new policies?

Locate $8$ on the horizontal axis
Read up to the line and across to find the income
When $n = 8$ , we find $I = 800$

Grace earns $800 for the week

Part c: How many policies must Grace sell to earn $700?

Locate $700$ on the vertical axis
Read across to the line and down to find the number of policies
When $I = 700$ , we find $n = 6$

Grace needs to sell 6 new policies

Part d: Find the equation of the line in terms of $I$ and $n$

To find the equation, we need the gradient and the intercept.

Step 1: Choose two clear points from the graph

Point 1: $(0, 400)$
Point 2: $(8, 800)$

Step 2: Calculate the gradient using the formula

m = \frac{\text{Rise}}{\text{Run}}

m = \frac{800 - 400}{8 - 0}

m = \frac{400}{8}

m = 50

Step 3: Identify the vertical intercept

From the graph, when $n = 0$ , $I = 400$

Therefore, $c = 400$

Step 4: Write the equation using $y = mx + c$

Using the appropriate variables:

$I = 50n + 400$

Part e: Use the equation to calculate weekly income when Grace sells 3 new policies

Substitute $n = 3$ into the equation:

I = 50n + 400

I = 50 \times 3 + 400

I = 150 + 400

I = 550

Grace earns $550 when she sells 3 policies

We can verify this answer by checking the graph at $n = 3$ .

Part f: How much does Grace earn for each new policy?

The gradient of the graph represents the commission per policy. We calculated the gradient as $50$ .

Grace earns $50 commission for each new policy

Understanding context in linear models

When working with linear models, always consider what the mathematical features mean in the real-world context.

chatImportant

Interpreting Linear Models in Context:

Gradient: Represents the rate of change. Ask yourself: "How much does the dependent variable change for each unit increase in the independent variable?"
Vertical intercept: Represents the starting value or baseline. Ask yourself: "What is the value when the independent variable is zero?"
Domain restrictions: Some linear models have constraints. For example, Logan's months must be whole numbers greater than zero (you cannot have negative months or half a month).

infoNote

Exam tip: When interpreting linear models, always include appropriate units and context in your answers. Don't just state numbers – explain what they represent in the situation.

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Key Points to Remember:

Linear models represent real-world situations using linear functions of the form $y = mx + c$
The gradient ( $m$ ) represents the rate of change and is calculated using $m = \frac{\text{Rise}}{\text{Run}}$
The vertical intercept ( $c$ ) represents the initial or starting value when the independent variable equals zero
To find an equation from a graph, identify two clear points, calculate the gradient, determine the intercept, and substitute into $y = mx + c$
Always interpret your answers in context – explain what the numbers mean in the real-world situation

Linear Models (HSC SSCE Mathematics Standard): Revision Notes

Linear Models

What are linear models?

Real-world example: iTunes collection

Key features of linear models

The gradient (rate of change)

The vertical intercept (starting value)

Working with linear graphs

Reading values from graphs

Finding equations from graphs

Understanding context in linear models

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