Mathematics • Year 8 • Unit 2 • Lesson 15
Linear Modelling
Build fluency turning real situations into the equation y = mx + c. Identify the rate (m), the fixed amount (c), write the equation, then substitute. One worked example, one guided fill-in, then eight independent problems.
1. I do — fully worked example
Read every line. The same four-step recipe works for every linear model.
Problem. A plumber charges $80 call-out plus $50 per hour. Find the total cost C for 3 hours of work.
Step 1 — Identify the fixed amount (c) and the rate (m).
Fixed call-out fee: c = 80 (paid even for 0 hours).
Rate per hour: m = 50.
Reason: a "fee" or "starting fixed cost" is always c (y-intercept). A "per hour" / "per km" / "per text" is the gradient m.
Step 2 — Write the equation in y = mx + c form.
C = 50x + 80, where x is hours of work.
Step 3 — Substitute the value asked for.
C = 50 × 3 + 80 = 150 + 80 = 230.
Step 4 — State the answer in context with units.
The total cost for 3 hours is $230.
Answer: $230.
2. We do — fill in the missing steps
A cyclist starts 5 km from home and rides at 15 km/h. Find the distance from home after 2 hours. Fill in each blank. 4 marks
Step 1 — Identify variables:
Fixed starting distance: c = ______ km (y-intercept).
Rate (speed): m = ______ km/h (gradient).
Step 2 — Write the equation:
d = ______ × t + ______
Step 3 — Substitute t = 2:
d = ______ × 2 + ______ = ______ + ______ = ______
Step 4 — Answer in context:
After 2 hours the cyclist is ______ km from home.
3. You do — independent practice
For each, write the equation in y = mx + c form, then answer the question with units. Show your substitution.
Foundation — write the equation
3.1 A phone plan costs $30 per month plus $0.10 per text. Write C in terms of x texts. 1 mark
3.2 Water costs $1.50 per kilolitre plus a $15 service fee. Write C in terms of k kilolitres. 1 mark
3.3 A taxi has a $4 flag-fall and charges $2.50 per km. Write C in terms of d km. 1 mark
3.4 A car travels at 80 km/h starting from a point 20 km outside the city. Write d in terms of t hours. 1 mark
Standard — write AND evaluate
3.5 Water bill (from 3.2): find C when k = 20 kL. Show every step. 2 marks
3.6 Taxi (from 3.3): find C for a 12 km trip. 2 marks
Extension — solve for x
3.7 Using the phone plan from 3.1, how many texts will make the total bill exactly $42? 2 marks
3.8 A printing service charges $20 setup plus $0.40 per page. How many pages can you print for exactly $60? 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (cyclist)
Step 1: c = 5, m = 15.
Step 2: d = 15 × t + 5.
Step 3: d = 15 × 2 + 5 = 30 + 5 = 35.
Step 4: after 2 hours the cyclist is 35 km from home.
3.1 — Phone plan
m = 0.10 (per text), c = 30 (monthly). C = 0.10x + 30.
3.2 — Water bill
m = 1.50 (per kL), c = 15 (service fee). C = 1.50k + 15.
3.3 — Taxi
m = 2.50 (per km), c = 4 (flag-fall). C = 2.50d + 4.
3.4 — Car
m = 80 (speed), c = 20 (starting distance). d = 80t + 20.
3.5 — Water bill at k = 20
C = 1.50 × 20 + 15 = 30 + 15 = $45.
3.6 — Taxi for 12 km
C = 2.50 × 12 + 4 = 30 + 4 = $34.
3.7 — Phone bill = $42
42 = 0.10x + 30 → 12 = 0.10x → x = 120. So 120 texts.
3.8 — Print bill = $60
60 = 0.40p + 20 → 40 = 0.40p → p = 100. So 100 pages.