Forming linear equations from situations | Year 12

01 / LEARN

The idea, step by step

Choose the changing quantities

Define the input quantity and its units before writing a formula. In a linear model, equal increases in input produce equal changes in output. The gradient measures that constant rate; the intercept is the output at input zero. An intercept may be mathematically defined even when input zero is outside a model's sensible practical range.

Combine rate and fixed amount

A booking fee gives a constant term. A cost per item is multiplied by the number of items. State practical restrictions, such as non-negative distance or whole-number tickets, and check the units of each term. Substituting a known value provides a quick check that the model describes the original information.

C = m x + b

A common trap

A negative gradient makes the model wrong.

Instead: A negative gradient correctly describes a quantity decreasing at a constant rate.

02 / WORKED EXAMPLES

See the thinking

Example 1

A delivery fee

A courier charges AUD 18 plus AUD 2.40 per kilometre. Write its charge for $d$ kilometres and find the charge for 15 kilometres.

Let $C$ be dollars and $d$ be kilometres, with $d \geq 0$ .
The model is $C = 18 + 2.40 d$ .
At $d = 15$ , $C = 18 + 36 = 54$ .

C = 18 + 2.40 d

; AUD 54.

Example 2

A draining tank

A tank holds 950 litres and drains at 35 litres per minute. Write its volume after $t$ minutes and its useful domain.

A loss gives a negative rate: $V = 950 - 35 t$ .
Require $V \geq 0$ and $t \geq 0$ .
At empty, $t = 950 / 35 = 190 / 7$ , about 27.14 minutes.

V = 950 - 35 t

for

0 \leq t \leq 190 / 7

minutes.

Check your understanding

A room costs AUD 90 plus AUD 7 per participant. Write its total cost.

Show answer

$C = 90 + 7 n$ for whole-number $n \geq 0$ .

The room cost is fixed and each participant adds 7.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A printer charges AUD 12 setup and AUD 0.08 per page. Write $C$ for $p$ pages.
2 marks
A 24 cm candle burns down at 1.5 cm per hour. Write its height and useful time domain.
2 marks
Temperature rises from 8 degrees Celsius at 6 am to 20 degrees at noon at a constant rate. Let $t$ be hours after 6 am. Find $T (t)$ .
2 marks
An account starts with AUD 250 and receives AUD 40 each week, with no interest. Find its balance after 9 weeks.
2 marks
A linear cost is AUD 46 for 3 units and AUD 70 for 7 units. Find its rule.
2 marks
Interpret $M = 180 - 4 t$ , where mass is in grams and time in minutes. When does it become physically impossible?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A printer charges AUD 12 setup and AUD 0.08 per page. Write $C$ for $p$ pages.
2 marks
The setup is constant; multiply the page rate by the number of pages.
$C = 12 + 0.08 p$ .
A 24 cm candle burns down at 1.5 cm per hour. Write its height and useful time domain.
2 marks
Subtract the amount burnt.
Set height to zero for the 16-hour endpoint.
$h = 24 - 1.5 t$ , $0 \leq t \leq 16$ .
Temperature rises from 8 degrees Celsius at 6 am to 20 degrees at noon at a constant rate. Let $t$ be hours after 6 am. Find $T (t)$ .
2 marks
The rate is $(20 - 8) / 6 = 2$ degrees per hour.
$T = 8 + 2 t$ for $0 \leq t \leq 6$ .
An account starts with AUD 250 and receives AUD 40 each week, with no interest. Find its balance after 9 weeks.
2 marks
$B = 250 + 40 w$ .
$B (9) = 250 + 360 = 610$ .
AUD 610.
A linear cost is AUD 46 for 3 units and AUD 70 for 7 units. Find its rule.
2 marks
$m = (70 - 46) / (7 - 3) = 6$ .
$46 = 6 (3) + b$ gives $b = 28$ .
$C = 6 n + 28$ .
Interpret $M = 180 - 4 t$ , where mass is in grams and time in minutes. When does it become physically impossible?
2 marks
Solve $180 - 4 t = 0$ .
For $t > 45$ the rule predicts negative mass.
Initial mass 180 g, loss 4 g/min; stop at 45 minutes.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics: Modelling with Linear Relationships, a compulsory school-assessed topic. The external examination scope differs from the full course.

Official curriculum source ↗

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