Mathematics Standard • Year 11 • Module 1 • Lesson 11
Intercepts and Linear Equations
Apply y = mx + b to real Australian contexts — phone plans, hire costs, savings, taxi fares and tank refills.
Problem 1 — Bondi bike hire
A bike-hire kiosk at Bondi Beach charges $12 to begin, plus $5 for each hour of hire.
Set up: What are we solving for?
(i) Write an equation for total cost C after h hours. State what each number means. 2 marks
(ii) Use the equation to find the cost of hiring the bike for 3 hours and for 6.5 hours. 2 marks
(iii) A tourist has $40. Use the equation to find the largest whole number of hours the tourist can afford. 2 marks
Stuck on (iii)? Solve 12 + 5h = 40 for h, then take the integer part.Problem 2 — Saving for a laptop
Jordan starts a savings plan with $75 already saved, then adds $20 each week. Let S be total savings in dollars after w weeks.
Set up: What are we solving for?
(i) Write the linear equation for S in terms of w. 1 mark
(ii) Calculate S after 12 weeks. 1 mark
(iii) A laptop costs $475. Solve the equation to find the number of weeks needed to reach this amount. 2 marks
Stuck? Revisit lesson § Worked Example 2 — Interpret m and b.Problem 3 — Sydney taxi fare
A Sydney taxi charges a $8 flagfall (booking fee) plus $2.50 per kilometre travelled.
Set up: What are we solving for?
(i) Write the equation for cost C after k kilometres. State the gradient and y-intercept. 2 marks
(ii) Find the cost of a 6 km taxi ride. 1 mark
(iii) A passenger is charged $33. How many kilometres did the ride cover? 2 marks
Stuck on (iii)? Set 8 + 2.50k = 33, subtract 8, then divide by 2.50.Problem 4 — Suburban delivery zones
A grocery delivery service charges a $15 base fee plus $4 for each suburb zone the truck crosses. Let z be the number of zones.
Set up: What are we solving for?
(i) Write the equation for total cost C after z zones. 1 mark
(ii) Complete the table of values, then describe the pattern in one sentence. 2 marks
z 0 1 2 3 4 5
C __ __ __ __ __ __
(iii) A customer is quoted $43 for delivery. How many zones did the order cross? 2 marks
Stuck? Revisit lesson § Worked Example 3 — Use a table to identify intercept and gradient.Problem 5 — Rainwater tank during a dry week
A 240 L rainwater tank is being used for the garden. The tank loses 3 L every minute as it drains. Let V be the volume remaining (L) after t minutes.
Set up: What are we solving for?
(i) Write the linear equation for V in terms of t. Explain why the gradient is negative. 2 marks
(ii) Calculate the volume remaining after 25 minutes. 1 mark
(iii) How long does it take for the tank to empty? Solve V = 0 to find the answer in minutes. 2 marks
Stuck on (iii)? Set 240 − 3t = 0 and solve for t.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Bondi bike hire
Set up. Build C = 12 + 5h, then substitute or rearrange depending on the part.
(i) C = 12 + 5h. The 12 is the fixed kiosk fee paid before riding; the 5 is the cost per hour.
(ii) 3 hours: C = 12 + 5(3) = $27. 6.5 hours: C = 12 + 5(6.5) = 12 + 32.50 = $44.50.
(iii) Set 12 + 5h = 40: 5h = 28, h = 5.6. The largest whole number of hours is 5 hours (cost $37, leaving $3 unspent).
Problem 2 — Saving for a laptop
Set up. Intercept 75, gradient 20.
(i) S = 75 + 20w.
(ii) S = 75 + 20(12) = 75 + 240 = $315.
(iii) 75 + 20w = 475 ⇒ 20w = 400 ⇒ w = 20 weeks.
Problem 3 — Sydney taxi fare
Set up. Build C = 8 + 2.50k from the flagfall and per-km rate.
(i) C = 8 + 2.50k. Gradient = 2.50 ($ per km). y-intercept = 8 ($ flagfall).
(ii) C = 8 + 2.50(6) = 8 + 15 = $23.00.
(iii) 8 + 2.50k = 33 ⇒ 2.50k = 25 ⇒ k = 10 km.
Problem 4 — Suburban delivery zones
Set up. Build C = 15 + 4z, then read off the table and rearrange where needed.
(i) C = 15 + 4z.
(ii) Table: z = 0, 1, 2, 3, 4, 5 gives C = 15, 19, 23, 27, 31, 35. The cost increases by $4 for each extra zone, starting from $15 at zero zones.
(iii) 15 + 4z = 43 ⇒ 4z = 28 ⇒ z = 7 zones.
Problem 5 — Draining rainwater tank
Set up. Intercept 240, gradient −3 (water is leaving).
(i) V = 240 − 3t. The gradient is negative because the volume decreases as time passes — the tank is losing water, not gaining it.
(ii) V = 240 − 3(25) = 240 − 75 = 165 L.
(iii) 240 − 3t = 0 ⇒ 3t = 240 ⇒ t = 80 minutes (= 1 h 20 min) before the tank is empty.