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Module 1 · L11 of 13 ~45 min ⚡ +75 XP available

Intercepts and Linear Equations

Connect gradient and starting values to equations of the form $y = mx + b$, then use the equation to make predictions. In this lesson you'll learn how to identify the y-intercept as a starting value, write equations from practical situations, and interpret every part of a linear model.

Today's hook — A bike hire costs $12 before you ride a single metre, then $5 per hour. Which number is the starting value, and which is the rate? And if the price changed, exactly which part of the equation would change?

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Worksheets

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Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question

Recall — your gut answer first

+5 XP warm-up

A bike hire costs $12 before riding begins, then $5 per hour.

Without writing an equation yet — which number is the starting value, and which number is the rate? What do you think each number represents in the real world?

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The equation you need to own

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Every linear relationship in Maths Standard can be described using one equation. Lock this in — everything in Module 1 builds from it.

$m$ is the gradient — it tells you how much $y$ changes each time $x$ increases by 1. In a practical context this is a rate (e.g. cost per hour). $b$ is the vertical intercept — it is the output when $x = 0$. In a practical context this is often a fixed fee or starting amount.

$y = mx + b$ — gradient $\times$ input + starting value

$m$ is the gradient or rate

The amount $y$ increases each time $x$ goes up by 1. In $C = 5h + 12$, the rate is $5 per hour.

$b$ is the vertical intercept

The output when the input is zero. In $C = 5h + 12$, the fixed cost is $12 (when $h = 0$).

Define your variables

Always state what $C$, $h$, $S$ etc. mean before using an equation. Marks are lost without definitions.

What you'll master

Know

Key facts

In $y = mx + b$, $m$ is the gradient or rate of change.
In $y = mx + b$, $b$ is the vertical intercept or starting value.
The intercept often represents a fixed cost or initial amount.

Understand

Concepts

Gradient and intercept together describe a linear relationship.
The intercept is the output when the input is zero.
Variables must be defined so the equation has contextual meaning.

Can do

Skills

Identify gradient and intercept from a practical equation.
Write a linear equation from a rate and starting value.
Use a linear equation to make predictions.

Key terms

Gradient ($m$)The rate of change; how much $y$ increases each time $x$ increases by 1.

Vertical intercept ($b$)The value of $y$ when $x = 0$; the starting output of the relationship.

Linear equationAn equation of the form $y = mx + b$ whose graph is a straight line.

Fixed costAn amount paid regardless of usage; corresponds to the intercept in a cost model.

RateA comparison of two quantities; in $y = mx + b$ this is $m$ (e.g. dollars per hour).

PredictionUsing an equation to find an output for a given input, including inputs not in the original data.

The intercept is the starting value

core concept

The vertical intercept is the output when the input is zero.

In a cost model, the intercept is often a fixed fee — something you pay before any usage begins. In a savings model, it is often the amount already saved before regular deposits start.

Key idea — the intercept is not the final total. It is the starting output. The gradient drives the total up from that starting point. Confusing the two is the most common error on exam questions about linear models.

A line crosses the y-axis at the y-intercept (green, starting value) and the x-axis at the x-intercept (red)

What to write in your book

In $y = mx + b$: $m$ is the gradient (rate of change), $b$ is the vertical intercept (starting value).
The intercept is the output when input = 0. It is NOT the amount added each step.
Example: $C = 12 + 5h$ — the $12 is the fixed cost (intercept), the $5 is the hourly rate (gradient).
Always define variables before using the equation in context.

Did you get this? True or false: in the equation $C = 12 + 5h$, the vertical intercept is $5.

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · WRITE A HIRE-COST EQUATION

A bike hire costs $12 plus $5 per hour. Write an equation for total cost $C$ after $h$ hours.

Identify the starting value (intercept): $b = 12$

The $12 is paid regardless of time ridden. When $h = 0$, cost is still $12.

PROBLEM 2 · INTERPRET $m$ AND $b$

A savings plan is modelled by $S = 75 + 20w$, where $S$ is savings in dollars after $w$ weeks. Interpret $m$ and $b$, then find savings after 6 weeks.

Identify the intercept: $b = 75$

When $w = 0$, $S = 75$. This means the person starts with $75 already saved.

PROBLEM 3 · TABLE TO EQUATION

Use the table below to identify the intercept and gradient, then write the equation.

Minutes, $t$	0	1	2	3
Distance, $d$	30	42	54	66

Intercept: $b = 30$ (when $t = 0$, $d = 30$)

Read directly from the table: the output when input is zero is the intercept.

What to write in your book

When $x = 0$ in a table, the output is the intercept directly — no calculation needed.
Check the gradient is constant by subtracting consecutive outputs. If not constant, the relationship is not linear.
$d = 30 + 12t$: intercept 30 = starting distance; gradient 12 = 12 metres per minute.
Communication habit: define all variables before writing the equation ($d$ is distance in metres, $t$ is time in minutes).

Quick check: In the equation $C = 12 + 5h$, which value is the vertical intercept?

Common errors · 3 traps that cost marks

Trap 01

Swapping gradient and intercept

In $C = 12 + 5h$, the hourly rate is 5, not 12. The 12 is the fixed starting cost. Always ask: "what happens when the input is zero?" — that value is the intercept.

Trap 02

Saying the intercept is "added each time"

The intercept is the amount present when $x = 0$. It is the starting output, not the repeated addition. The gradient is what gets added each time $x$ increases by 1.

Trap 03

Forgetting to define variables

Writing $C = 12 + 5h$ without stating that $C$ is total cost in dollars and $h$ is time in hours loses communication marks in extended responses. Always define every variable.

What to write in your book

Always check: which number is the fixed amount (intercept) and which grows with the input (gradient)?
$b$ = output when input = 0. $m$ = amount added each time input increases by 1.
Defining variables is a communication skill worth marks in HSC extended responses.
Trap check: if someone says "the intercept is what gets added each time" — they mean the gradient.

Fill the gap: In $S = 75 + 20w$, the starting savings are $ and the weekly deposit rate is $.

Quick-fire practice · 4 calculations

A taxi fare is $8 plus $2.50 per kilometre. Write an equation for cost $C$ after $k$ kilometres.

In $P = 40 + 18w$, explain what 40 and 18 mean if $P$ is pay after $w$ weeks.

A table has outputs 10, 16, 22, 28 for inputs 0, 1, 2, 3. Write the equation.

Use your equation from Question 3 to predict the output when the input is 7.

Match it: In $d = 30 + 12t$, which is correct?

Revisit your thinking

Earlier you predicted which number was the starting value and which was the rate in the bike hire problem. Let's confirm:

The bike hire model is $C = 12 + 5h$. The $12 is the intercept (fixed starting cost — paid before riding) and the $5 is the gradient (hourly rate). If the hourly rate rose to $6, only the gradient changes: $C = 12 + 6h$. The fixed cost stays the same.

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Multiple choice

+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.

Short answer

ApplyBand 34 marks

Q1. A delivery cost is $15 plus $4 per suburb zone. Write an equation for total cost $C$ for $z$ zones and interpret the intercept. (4 marks)

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ApplyBand 34 marks

Q2. In $S = 120 + 25w$, explain the meaning of 120 and 25, then find $S$ when $w = 8$. (4 marks)

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AnalyseBand 44 marks

Q3. A table has outputs 18, 25, 32, 39 for inputs 0, 1, 2, 3. Write the equation and predict the output for input 6. (4 marks)

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📖 Comprehensive answers (click to reveal)

Drill 1: $C = 8 + 2.50k$ · 2: 40 = starting pay (intercept), 18 = weekly pay rate (gradient) · 3: $y = 10 + 6x$ · 4: $y = 10 + 6(7) = 52$

Q1 (4 marks): $C = 15 + 4z$ [1] where $C$ = total cost ($), $z$ = number of zones [1]. The intercept 15 means a fixed delivery fee of $15 applies regardless of the number of zones [2].

Q2 (4 marks): 120 = starting savings / initial amount ($120 saved at the start) [1]. 25 = rate of saving / $25 per week deposited [1]. $S = 120 + 25(8) = 120 + 200 = \$320$ [2].

Q3 (4 marks): Intercept = 18 (output when input = 0) [1]. Gradient = $25 - 18 = 7$ [1]. Equation: $y = 18 + 7x$ [1]. When $x = 6$: $y = 18 + 7(6) = 18 + 42 = 60$ [1].

Boss battle · Rate or Start?

earn bronze · silver · gold

Sort each number into gradient or intercept, then write the matching linear equation. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.

⚔ Enter the arena

Science Jump · platform challenge

Climb platforms by answering intercept and gradient questions. Pool: lesson 11.

Mark lesson as complete

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← Lesson 10 · Gradient as Rate Lesson 12 · Direct Variation →

Module overview · Maths Standard Year 11