Mathematics • Year 8 • Unit 2 • Lesson 11
y = mx + c in the Real World
Use gradient-intercept form where it actually shows up: pool filling, gym memberships, savings plans and trip distances. Then explain what m and c mean in plain English.
1. Word problems
Each problem can be modelled by an equation of the form y = mx + c. Identify what m and c stand for, write the equation, then use it.
1.1 — Pool filling. A pool has 50 L of water in it. A hose adds 20 L every minute.
(a) Write an equation for V (litres) after t minutes, in the form V = mt + c.
(b) How much water is in the pool after 6 minutes? 3 marks
1.2 — Gym membership. A gym charges a $40 sign-up fee plus $15 per visit.
(a) Write an equation for total cost C in dollars after x visits.
(b) State m and c in words — what does each one represent in this situation? 3 marks
1.3 — Savings plan. Mia starts with $25 in her savings account and adds $10 each week.
(a) Write S = mw + c for savings after w weeks.
(b) After how many weeks will she have $115? (Set S = 115 and solve for w.) 3 marks
1.4 — Road trip. A family is 30 km outside the city when their journey begins. They then drive at 80 km/h.
(a) Write d = mt + c for distance d (km) from the city after t hours of driving.
(b) How far from the city are they after 2.5 hours? 3 marks
1.5 — Phone battery drain. Your phone starts a long flight at 100% battery and drops 8% every hour you watch movies.
(a) Write B = mt + c for the battery percentage B after t hours.
(b) Find B after 6 hours. (Why is the gradient negative here?) 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate writes the equation for a taxi as C = 2.50d, where d is distance in km. The taxi actually has a $4 flag-fall fee that you pay just for stepping in, plus $2.50 per km. In your own words explain (i) which letter (m or c) is missing from your classmate's equation, (ii) what the correct equation should be in the form C = md + c, and (iii) how the missing piece affects the cost for a 0 km trip. Use the phrase "y-intercept" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Pool filling
(a) m = 20 (litres per minute), c = 50 (litres already there). V = 20t + 50.
(b) V = 20(6) + 50 = 120 + 50 = 170 L.
1.2 — Gym membership
(a) C = 15x + 40.
(b) m = 15 is the cost per visit (gradient — how fast cost grows with each visit). c = 40 is the one-off sign-up fee, paid even if you visit 0 times.
1.3 — Savings plan
(a) S = 10w + 25.
(b) 115 = 10w + 25 → 90 = 10w → w = 9 weeks.
1.4 — Road trip
(a) d = 80t + 30.
(b) d = 80(2.5) + 30 = 200 + 30 = 230 km.
1.5 — Phone battery drain
(a) B = −8t + 100. The gradient is negative because the battery DECREASES as time passes — y goes down as x goes up.
(b) B = −8(6) + 100 = −48 + 100 = 52%.
2.1 — Explain your thinking (sample response)
The classmate's equation C = 2.50d is missing c, the y-intercept. The correct equation is C = 2.50d + 4, where m = 2.50 (the cost per km) and c = 4 (the flag-fall paid just for getting into the taxi). With the missing piece, a 0 km trip in the classmate's version would cost $0, but the real taxi would still charge $4 — the y-intercept is what you pay before you have travelled anywhere. Forgetting c always makes the model wrong at x = 0.
Marking: 1 mark for identifying c as missing; 1 mark for the correct equation C = 2.50d + 4; 1 mark for explaining what happens at d = 0; 1 mark for a clear full-sentence answer using "y-intercept".