Mathematics Standard • Year 11 • Module 1 • Lesson 11
Intercepts and Linear Equations
Build fluency identifying the gradient (rate) and y-intercept (starting value) of y = mx + b, and writing equations from real contexts.
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 In the equation y = mx + b, what does m represent? What does b represent?
m = ____________________ b = ____________________
Q1.2 Identify the gradient and the y-intercept of C = 12 + 5h.
Gradient = ____________ y-intercept = ____________
Q1.3 The vertical intercept is the output when the input equals ____________.
2. Worked example — write a hire-cost equation
Follow each line of working. Every step has a reason on the right.
Problem. A bike hire costs $12 to begin, then $5 per hour. Write an equation for total cost C after h hours, then find the cost for 4 hours.
Step 1 — Identify the starting value (intercept b).
b = 12 (the fixed $12 fee, paid before riding begins)
Reason: the intercept is the output when the input (hours) is zero.
Step 2 — Identify the rate (gradient m).
m = 5 (dollars added for each extra hour)
Reason: the gradient is the change in cost per one-hour increase in time.
Step 3 — Write the equation in the form y = mx + b.
C = 12 + 5h
Reason: the constant 12 sits with b; the per-hour rate 5 multiplies h.
Step 4 — Substitute h = 4 to predict the 4-hour cost.
C = 12 + 5(4) = 12 + 20 = 32
Reason: brackets keep the multiplication explicit and stop slips.
Conclusion. The bike hire for 4 hours costs $32.
3. Faded example — fill in the missing steps
A savings plan is modelled by S = 75 + 20w, where S is savings in dollars after w weeks. Find S when w = 6 and interpret 75 and 20. Fill in each blank. 4 marks
Step 1 — Identify the intercept and gradient:
b = ____________ m = ____________
Step 2 — Substitute w = 6 with brackets:
S = 75 + 20(____) = ____________ + ____________
Step 3 — Calculate:
S = ____________
Step 4 — Interpret the numbers in plain English:
75 means: ____________________________________________
20 means: ____________________________________________
4. Graduated practice — intercepts and linear equations
Show your working in the space below each part. Always label units where given.
Foundation — read the equation (4 questions)
| Q | Problem | Answer |
|---|---|---|
| 4.1 1 | State the gradient and y-intercept of y = 4x + 7. | |
| 4.2 1 | State the gradient and y-intercept of C = 20 + 8t. | |
| 4.3 1 | Use C = 12 + 5h to find C when h = 3. | |
| 4.4 1 | Use S = 75 + 20w to find S when w = 4. |
Standard — write the equation from a context (6 questions)
Define the variables, then write y = mx + b. Substitute to find the requested value.
4.5 A taxi fare is $8 plus $2.50 per kilometre. Write an equation for cost C after k kilometres. 2 marks
4.6 A delivery cost is $15 plus $4 per zone. Write the equation for cost C after z zones, then find C for 6 zones. 2 marks
4.7 A gym membership costs $50 to join, then $25 per month. Write the equation for total cost M after n months. 2 marks
4.8 Pay is given by P = 40 + 18w. Explain in one sentence what 40 and 18 represent if P is pay after w weeks. 2 marks
4.9 A table shows outputs 10, 16, 22, 28 for inputs 0, 1, 2, 3. Write the linear equation. 2 marks
4.10 Distance from a table: d = 30 when t = 0, then d increases by 12 each minute. Write the equation and use it to find d when t = 5. 2 marks
Extension — predict beyond the table (2 questions)
4.11 A table has outputs 18, 25, 32, 39 for inputs 0, 1, 2, 3. (a) Write the equation. (b) Predict the output for input 7. (c) State, in one sentence, what the intercept represents if the input is "hours studied" and the output is "marks". 3 marks
4.12 A water tank starts with 240 L and drains at 3 L per minute. (a) Write a linear equation for volume V after t minutes. (b) Use the equation to predict V when t = 30. (c) State the meaning of the gradient as a rate. 3 marks
5. Self-check the easy 3
Tick the first three once you've checked your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — Meaning of m and b
m is the gradient (the rate of change — how much y changes per unit increase in x). b is the y-intercept (the starting value — the output when x = 0).
Q1.2 — C = 12 + 5h
Gradient = 5 (dollars per hour). y-intercept = 12 (the fixed starting cost in dollars).
Q1.3 — Vertical intercept
The vertical intercept is the output when the input equals zero.
Q3 — Faded savings example
Step 1: b = 75, m = 20.
Step 2: S = 75 + 20(6) = 75 + 120.
Step 3: S = 195.
Step 4: 75 means the person starts with $75 before any saving begins. 20 means savings increase by $20 each week.
Q4.1 — y = 4x + 7
Gradient = 4; y-intercept = 7.
Q4.2 — C = 20 + 8t
Gradient = 8; y-intercept = 20.
Q4.3 — C = 12 + 5(3)
C = 12 + 15 = $27.
Q4.4 — S = 75 + 20(4)
S = 75 + 80 = $155.
Q4.5 — Taxi fare
C = 8 + 2.50k (where C is cost in dollars and k is distance in kilometres).
Q4.6 — Delivery cost
C = 15 + 4z. For z = 6: C = 15 + 4(6) = 15 + 24 = $39.
Q4.7 — Gym membership
M = 50 + 25n (where M is total cost in dollars and n is the number of months).
Q4.8 — P = 40 + 18w
40 is the fixed starting pay (an allowance/bonus paid before any weeks are worked). 18 is the pay per week.
Q4.9 — Outputs 10, 16, 22, 28
Intercept (when input = 0) is 10. The output increases by 6 each step, so gradient = 6. y = 10 + 6x.
Q4.10 — Distance equation
d = 30 + 12t. For t = 5: d = 30 + 12(5) = 30 + 60 = 90 (units of distance).
Q4.11 — Marks vs hours
(a) Intercept = 18; outputs increase by 7 each step, so gradient = 7. y = 18 + 7x.
(b) y = 18 + 7(7) = 18 + 49 = 67 marks.
(c) The intercept 18 represents the mark a student would get with zero hours of study (e.g. a baseline from general knowledge).
Q4.12 — Draining tank
(a) V = 240 − 3t (the gradient is −3 because volume decreases by 3 L every minute).
(b) V = 240 − 3(30) = 240 − 90 = 150 L.
(c) The gradient −3 means the tank loses 3 litres for every 1 minute that passes.