Mathematics • Year 8 • Unit 2 • Lesson 20
Linear Relationships in Context
Build fluency with the whole Unit 2 toolkit: gradient, y-intercept, the equation y = mx + c, sketching, and the choice of substitution vs elimination. One worked context example, one guided example with blanks, then eight independent problems graduated from "find the gradient" to "set up and solve a real-world system".
1. I do — fully worked example
Read every line. The whole unit lives inside y = mx + c — gradient m, intercept c. Translate the words, build the equation, solve.
Problem. A car rental costs $60 per day plus $0.30 per kilometre. (a) Write a linear model for the total cost C in dollars when k kilometres are driven. (b) Use the model to predict the cost for 200 km.
Step 1 — Identify the gradient m and intercept c from the words.
"$0.30 per km" → rate of change = 0.30 → m = 0.30
"$60 per day (fixed)" → cost when k = 0 → c = 60
Reason: gradient = how fast cost grows per extra km. Intercept = what you pay before driving anywhere.
Step 2 — Write the model in y = mx + c form (with C and k).
C = 0.30 k + 60
Step 3 — Substitute k = 200 to predict the cost.
C = 0.30 (200) + 60 = 60 + 60 = 120
Step 4 — Interpret in context.
Driving 200 km in one day costs $120.
Answer: (a) C = 0.30k + 60. (b) $120.
2. We do — fill in the missing steps
Same shape, with blanks. 5 marks
Problem. A taxi charges $5 flagfall plus $2 per kilometre. Write a model C = mk + c and find the cost of a 12 km trip.
Step 1 — Identify m and c from the words:
"$2 per km" → m = ______
"$5 flagfall (fixed)" → c = ______
Step 2 — Write the model:
C = ______ k + ______
Step 3 — Substitute k = 12:
C = ______ (12) + ______ = ______ + ______ = ______
Step 4 — Interpret: A 12 km taxi trip costs $______.
Bonus: What is the gradient of this model, and what does it represent in context?
m = ______; it represents ____________________ per km.
3. You do — independent practice
Show every step. 3.1–3.3 are foundation (read gradient/intercept). 3.4–3.6 are standard (build the equation, use to predict). 3.7–3.8 are extension (set up and solve a context system).
Foundation — read gradient and intercept
3.1 Find the gradient of the line through (2, 3) and (6, 11). 1 mark
3.2 State the gradient and y-intercept of the line y = −4x + 7. 1 mark
3.3 A line crosses the y-axis at (0, −3) and has gradient 2. Write its equation in the form y = mx + c. 1 mark
Standard — model a context with y = mx + c
3.4 A phone plan charges $20 base fee plus $0.15 per text. Write an equation C = mt + c for total cost C when t texts are sent, then use it to find the cost when t = 200 texts. 2 marks
3.5 A pool is being filled. After 2 minutes there are 30 L; after 5 minutes there are 60 L. (a) Find the gradient (rate, in L/min). (b) Write a linear model V = mt + c for volume V at time t minutes. 2 marks
3.6 A line on the Cartesian plane has intercepts (4, 0) and (0, −8). (a) Find its gradient. (b) Write its equation in y = mx + c form. 2 marks
Extension — set up and solve a system
3.7 The sum of two numbers is 18 and their difference is 4. Set up two equations and solve by elimination. State the two numbers. 3 marks
3.8 A rectangle has perimeter 24 cm and the length is twice the width. (a) Let L = length, W = width. Write two equations. (b) Solve by substitution. (c) State the dimensions. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do taxi model
Step 1: m = 2, c = 5.
Step 2: C = 2k + 5.
Step 3: C = 2(12) + 5 = 24 + 5 = 29.
Step 4: $29.
Bonus: m = 2; it represents the extra cost in dollars per km travelled.
3.1 — Gradient through (2, 3) and (6, 11)
m = (11 − 3) / (6 − 2) = 8 / 4 = 2.
3.2 — y = −4x + 7
Gradient m = −4; y-intercept c = 7 (line crosses y-axis at (0, 7)).
3.3 — y-intercept (0, −3), gradient 2
Equation: y = 2x − 3.
3.4 — Phone plan
C = 0.15t + 20. At t = 200: C = 0.15(200) + 20 = 30 + 20 = $50.
3.5 — Pool filling
(a) Gradient = (60 − 30) / (5 − 2) = 30 / 3 = 10 L/min.
(b) Use V = 10t + c with point (2, 30): 30 = 10(2) + c → c = 10. V = 10t + 10. (At t = 0 there were already 10 L in the pool.)
3.6 — Intercepts (4, 0) and (0, −8)
(a) m = (−8 − 0) / (0 − 4) = −8 / −4 = 2.
(b) y-intercept c = −8 → y = 2x − 8. Check: at x = 4, y = 2(4) − 8 = 0 ✓.
3.7 — Sum 18, difference 4
Eq 1: x + y = 18. Eq 2: x − y = 4. Add: 2x = 22 → x = 11. Back-sub: 11 + y = 18 → y = 7. Numbers: 11 and 7. Check: 11 − 7 = 4 ✓.
3.8 — Rectangle, perimeter 24, length twice width
(a) Eq 1: 2L + 2W = 24. Eq 2: L = 2W.
(b) Sub: 2(2W) + 2W = 24 → 6W = 24 → W = 4. Then L = 2(4) = 8.
(c) Length 8 cm, width 4 cm. Check perimeter: 2(8) + 2(4) = 24 ✓.