Intercepts and Linear Equations

Connect gradient and starting values to equations of the form $y = mx + b$, then use the equation to make predictions.

45 min Algebra Linear relationships Lesson 11 of 13

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Printable worksheet

Open this lesson's worksheet

Use the printable version for identifying intercepts, writing linear equations and making predictions.

Think First

A bike hire costs $12 before riding begins, then $5 per hour. Which number is the starting value, and which number is the rate?

Type the starting value and the rate.

Write the starting value and the rate in your book.

Write your response in your book

Saved

Know

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

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 context meaning.

Can Do

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.

y

Linear Equation Form

$y = mx + b$

$m$ = gradient or rate, $b$ = vertical intercept or starting value

$C = 12 + 5h$

$C$ = total cost, $h$ = hours, 12 = fixed cost, 5 = hourly rate

Core Content

1. The Intercept Is the Starting Value

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

In a cost model, the intercept is often a fixed fee. In a savings model, it is often the amount already saved before regular deposits begin.

Key idea: The intercept is not the final total. It is the starting output.

Worked Example 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.

Starting value or intercept: $12.

Gradient or rate: $5 per hour.

Equation: $C = 12 + 5h$.

For $h = 4$, $C = 12 + 5(4) = 32$, so 4 hours costs $32.

Worked Example 2

Interpret $m$ and $b$

A savings plan is modelled by $S = 75 + 20w$, where $S$ is savings in dollars after $w$ weeks.

The intercept is 75. This means the person starts with $75.

The gradient is 20. This means savings increase by $20 each week.

After 6 weeks: $S = 75 + 20(6) = 195$.

Worked Example 3

Use a table to identify intercept and gradient

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

The intercept is 30 because $d = 30$ when $t = 0$.

The gradient is 12 because distance increases by 12 each minute.

Equation: $d = 30 + 12t$.

Communication habit: Define variables before using the equation. Here, $d$ is distance and $t$ is time in minutes.

2. Do Not Confuse Gradient and Intercept

In $y = mx + b$, the gradient is the repeated change. The intercept is the starting output. In $C = 12 + 5h$, the hourly rate is 5, not 12.

Common error: The intercept is not the amount added each time. It is the amount present when the input is zero.

Activity

Intercept and Equation Practice

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.

Complete the intercept practice in your book.

Revisit the Bike Hire

The bike hire model is $C = 12 + 5h$. The $12 is the intercept or fixed starting cost, and the 5 is the gradient or hourly rate.

Explain the changed equation in your book.

Assessment

MC

Multiple Choice

Random questions from the lesson bank - feedback appears immediately.

SA

Short Answer

Identify gradient and intercept, write equations and make predictions.

1. 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

Answer in your book.

2. In $S = 120 + 25w$, explain the meaning of 120 and 25, then find $S$ when $w = 8$. 4 MARKS

Answer in your book.

3. 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

Answer in your book.

Rate or Start?

Sort each number into gradient or intercept, then write the matching linear equation.

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