How do you remove ten percent GST to find the pre GST price?

The GST inclusive price is one point one times the pre GST price, so divide the inclusive price by one point one rather than subtracting ten percent.

How do you remove ten percent GST to find the pre GST price?

The GST inclusive price is one point one times the pre GST price, so divide the inclusive price by one point one rather than subtracting ten percent.

Year 11 Methods (Unit 1 & 2) Coordinate Geometry And Linear Relations

Linear Models

Q: What do the gradient and intercept mean in a linear model?

The gradient is the constant rate of change per unit of input, and the intercept is the initial or fixed value when the input is zero.

Q: How do you form a linear model from two data points?

Find the rate as the change in output over the change in input, then substitute one point into y equals m x plus c to find the constant.

20 practice questions 1 video lesson Theory + worked examples

Building linear models is a foundational skill assumed for Queensland Year 11 Mathematical Methods (QCAA). A linear model describes a quantity changing by a constant amount each step, where the gradient is the rate of change and the constant term the starting value.

You will build a model from a worded situation, evaluate it for a given input, solve it to reach a target output, and attach the correct units.

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Theory

In Year 11 Mathematical Methods (QCAA), a linear model describes a real situation that changes at a constant rate using $y=mx+c$. The gradient $m$ is the rate (with units), and the constant $c$ is the initial or fixed value. This page shows how to form a model, interpret its parts, evaluate it, and solve it — including break-even and reverse-percentage problems.

A linear model represents a quantity that changes by the same amount each step. Written as $y=mx+c$, the gradient $m$ is the constant rate of change (for example dollars per hour or litres per minute) and the constant term $c$ is the initial value when the input is $0$ (a fixed fee, a starting amount).

To evaluate the model, substitute a value of the input and compute the output. To solve it, set the output to a target value and solve for the input. Always attach the correct metric units to your answers.

Gradient = rate, intercept = starting value. Identify these two numbers from the words, and the model \(y=mx+c\) writes itself.

In a cost model $C=90+70h$, $c=90$ is the fixed fee and $m=70$ the rate.

A decreasing model $h=10-2t$: solve $h=0$ to find when it runs out.

The general linear model, with rate $m$ and initial value $c$:

\[y=mx+c\]

y = m x + c

The rate from two data points $(x_1,y_1)$ and $(x_2,y_2)$:

\[m=\dfrac{y_2-y_1}{x_2-x_1}\]

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

Break-even: set two models equal, \(m_1x+c_1=m_2x+c_2\), and solve for \(x\) to find where they give the same output.

How to build and use a linear model

Identify the rate $m$ and the initial value $c$ from the words (or find $m$ from two data points).
Write the model $y=mx+c$, naming the variables and their units.
Evaluate or solve: substitute an input to predict an output, or set the output and solve for the input; then interpret in context.

Example 1 — Interpreting and evaluating a model

A plumber charges $C=90+70h$ dollars for $h$ hours of work. State the call-out fee and the hourly rate, then find the cost of a $3$-hour job.

Solution

Match to $y=mx+c$: the constant is the fixed fee, the coefficient of $h$ is the rate:

$c$	$=$	$90$
$m$	$=$	$70$

So the call-out fee is $\$90$ and the hourly rate is $\$70$ per hour.

Evaluate at $h=3$:

$C$	$=$	$90+70(3)$
	$=$	$90+210$
	$=$	$300$

Call-out $\$90$, rate $\$70$/h; a $3$-hour job costs $\$300$.

C = 300

Example 2 — Solving a decreasing model

A candle's height is $h=10-2t$ centimetres after $t$ minutes. After how long does the candle burn out?

Solution

The candle is gone when its height is $0$; set $h=0$:

$0$

$=$

$10-2t$

Solve for $t$:

$2t$	$=$	$10$
$t$	$=$	$5$

The candle burns out after $5$ minutes.

t = 5

Example 3 — Solving a cost model

A van hire costs $C=50+0.4d$ dollars for $d$ kilometres driven. How far can you drive for $\$90$?

Solution

Set the cost to $90$:

$90$

$=$

$50+0.4d$

Solve for $d$:

$0.4d$	$=$	$40$
$d$	$=$	$\dfrac{40}{0.4}$
	$=$	$100$

You can drive $100$ km for $\$90$.

d = 100

Example 4 — Building a model from two data points

A tank holds $250$ L after $2$ minutes and $350$ L after $6$ minutes, filling at a constant rate. Form a model $V=mt+c$ and find when the tank holds $450$ L.

Solution

Rate — change in volume over change in time:

$m$	$=$	$\dfrac{350-250}{6-2}$
	$=$	$\dfrac{100}{4}$
	$=$	$25$

Initial value — substitute $(2,250)$:

$250$	$=$	$25(2)+c$
$250$	$=$	$50+c$
$c$	$=$	$200$

So the model is $V=25t+200$ (litres after $t$ minutes).

Solve $V=450$:

$450$	$=$	$25t+200$
$25t$	$=$	$250$
$t$	$=$	$10$

Model $V=25t+200$; the tank holds $450$ L after $10$ minutes.

t = 10

Common pitfalls

Swapping the rate and the starting value. The gradient $m$ is the rate (per unit of input); the constant $c$ is the value when the input is $0$. Read the words carefully.

Dropping the units. A model answer is a quantity: $\$300$, $100$ km, $5$ minutes. State the units and interpret the number in context.

Confusing evaluate with solve. Given the input, substitute to find the output. Given the output, set the model equal to it and solve for the input.

Frequently asked questions

What do the gradient and intercept mean in a linear model?

The gradient $m$ is the constant rate of change (per unit of input) and the intercept $c$ is the initial or fixed value when the input is $0$.

How do you form a linear model from two data points?

Find the rate $m=\dfrac{y_2-y_1}{x_2-x_1}$, then substitute one point into $y=mx+c$ to find $c$, giving $y=mx+c$.

How do you solve a linear model?

Set the output equal to the target value and solve the resulting linear equation for the input; then state the answer with units.

What does break-even mean?

The input value at which two linear models give the same output. Set the two models equal, $m_1x+c_1=m_2x+c_2$, and solve for $x$.

How do you remove 10% GST to find the pre-GST price?

The GST-inclusive price is $1.1$ times the pre-GST price, so divide the inclusive price by $1.1$ (not subtract $10\%$).

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More in Coordinate Geometry And Linear Relations

\(c\)	\(=\)	\(90\)
\(m\)	\(=\)	\(70\)

\(C\)	\(=\)	\(90+70(3)\)
	\(=\)	\(90+210\)
	\(=\)	\(300\)

\(2t\)	\(=\)	\(10\)
\(t\)	\(=\)	\(5\)

\(0.4d\)	\(=\)	\(40\)
\(d\)	\(=\)	\(\dfrac{40}{0.4}\)
	\(=\)	\(100\)

\(m\)	\(=\)	\(\dfrac{350-250}{6-2}\)
	\(=\)	\(\dfrac{100}{4}\)
	\(=\)	\(25\)

\(250\)	\(=\)	\(25(2)+c\)
\(250\)	\(=\)	\(50+c\)
\(c\)	\(=\)	\(200\)

Linear Models

📖 Theory

How to build and use a linear model

Common pitfalls

Frequently asked questions

Theory