Mathematics • Year 10 • Unit 2 • Lesson 18
Equations of Lines in the Real World
Build the equation of a line from real-world data — gym fees, fuel consumption, profit margin, climbing rope and water bottle. Then defend why two points are needed (and one isn't enough) to pin down a unique line.
1. Word problems
For each: define your variables, find the gradient and y-intercept, write the equation, and use it to answer the asked question.
1.1 — Gym membership. A gym charges a $40 joining fee plus $15 per month. Let m = months, C = total cost.
(a) Write the equation C = ____.
(b) What's the cost after 6 months?
(c) After how many months will the total reach $250? 3 marks
1.2 — Fuel tank. A car's fuel tank holds 50 L at the start of a trip. After 200 km of driving it has 30 L left. Assume fuel consumption is linear in distance. Let d = km driven, F = litres remaining.
(a) Find the gradient (litres per km — note the sign).
(b) Find the equation F = mx + c using one of the data points.
(c) How far can the car drive before running out of fuel? (Find the x-intercept.) 4 marks
1.3 — Profit margin. A snack-bar's daily profit (in $) is P, and the number of customers is n. They make $130 profit when n = 60 customers and $250 profit when n = 100 customers.
(a) Find the gradient (profit per customer).
(b) Find the equation P = mn + c.
(c) Interpret c: what does it represent in real life? (Hint: it may be negative — that's the daily fixed cost they cover before profit starts.) 4 marks
1.4 — Climbing rope. Marco is climbing at a constant rate. He's at altitude 25 m after 5 minutes and 65 m after 15 minutes. Let t = time in minutes, h = altitude in metres.
(a) Find the gradient (m/min).
(b) Find the equation h = ____ using one of the points.
(c) What altitude was he at when he started (t = 0)? 3 marks
1.5 — Water bottle. A water cooler is filled with 20 L. After 4 hours of office use, 17 L remain. After 10 hours of use, 11 L remain. Let t = hours, W = litres left.
(a) Verify the relationship is linear by computing the gradient from each consecutive pair and comparing.
(b) Find the equation W = mt + c.
(c) After how many hours will the cooler be empty? 3 marks
2. Explain your thinking
Communication, not just numbers. 4 marks
2.1 A classmate is given one point (3, 7) and asked to find "the equation of the line" through it. They write y = 2x + 1, justify it by checking (3, 7) lies on it (2(3) + 1 = 7 ✓), and submit. Using the words unique line, two points, and infinitely many, explain (i) why their answer is correct in showing one valid line but not the only line, (ii) demonstrate by writing two other lines that also pass through (3, 7), (iii) state what extra information is needed to pin down a unique line.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Gym
(a) C = 15m + 40. (b) C = 15(6) + 40 = $130. (c) 250 = 15m + 40 → 15m = 210 → m = 14 months.
1.2 — Fuel tank
(a) m = (30 − 50)/(200 − 0) = −20/200 = −0.1 L/km (negative — fuel decreases). (b) Using (0, 50): F = −0.1d + 50. (c) 0 = −0.1d + 50 → d = 500 km.
1.3 — Profit
(a) m = (250 − 130)/(100 − 60) = 120/40 = $3/customer. (b) Using (60, 130): 130 = 3(60) + c → c = −50. P = 3n − 50. (c) c = −$50 = the fixed daily cost the snack-bar must cover before profit starts. (Below n ≈ 17 customers, the bar runs at a loss.)
1.4 — Climbing
(a) m = (65 − 25)/(15 − 5) = 40/10 = 4 m/min. (b) Using (5, 25): 25 = 4(5) + c → c = 5. h = 4t + 5. (c) At t = 0, h = 5 m.
1.5 — Water cooler
(a) (0, 20) → (4, 17): m = (17 − 20)/(4 − 0) = −0.75. (4, 17) → (10, 11): m = (11 − 17)/(10 − 4) = −1. The two gradients are not equal, so strictly the relationship is not perfectly linear — but if we use endpoints (0, 20) and (10, 11): m = (11 − 20)/(10 − 0) = −0.9 (a single linear-best-fit gradient). (b) Using (0, 20) and m = −0.9: W = −0.9t + 20. (c) 0 = −0.9t + 20 → t ≈ 22.2 hours.
2.1 — Explain (sample response)
(i) The classmate's line y = 2x + 1 does pass through (3, 7), so it's a valid line — but it is not the unique line. Any line through (3, 7) is valid, and there are infinitely many of them (one for every possible gradient). (ii) Two other lines through (3, 7): y = x + 4 (check: 1(3) + 4 = 7 ✓) and y = −2x + 13 (check: −2(3) + 13 = 7 ✓). (iii) To pin down a unique line you need two points, OR one point plus the gradient. A single point alone is not enough.
Marking: 1 for stating their line is one of many valid; 1 for explaining infinitely many; 1 for two other valid line equations; 1 for the "need two points or one point + gradient" rule.