Practice Maths

Developing Linear Models from Context

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Key Terms

A linear model is a straight-line equation y = mx + c that represents a real-world situation.
m (gradient)
= rate of change = how much y changes for every 1-unit increase in x. It always has units (e.g. $/km, °C per hour, litres per minute).
c (y-intercept)
= initial value = the value of y when x = 0 (e.g. flag-fall charge, starting temperature, fixed fee).
To build a model: identify what changes (x), what depends on it (y), the rate (m), and the starting value (c).
Interpolation
: predicting within the known data range. Extrapolation: predicting outside the known range (less reliable).
Always check whether a linear model is appropriate: is the rate of change approximately constant?
Key Formula
• y = mx + c   (slope-intercept form)
• Gradient: m = (y2 − y1) ÷ (x2 − x1) = rise ÷ run
• Given two points (x1, y1) and (x2, y2): find m, then c = y1 − m×x1
• Units of m: [units of y] per [unit of x]
Component Meaning in context Example (taxi fare)
xIndependent variableDistance (km)
yDependent variableTotal fare ($)
mRate of change$2.50 per km
cInitial value (y-intercept)$3.00 flag fall
Modely = mx + cy = 2.5x + 3

Linear model: Taxi fare y = 2.5x + 3 (gradient triangle shows $2.50 per km)

x (km) y ($) 1 2 3 4 5 6 7 0 3 6 9 12 15 c = $3 (flag fall) run = 2 km rise = $5 m = 5÷2 = $2.50/km y = 2.5x + 3
Hot Tip Always write the units of the gradient. If y is dollars and x is kilometres, then m has units $/km. This helps you check your model makes sense. Also, check: does x = 0 give a sensible y-intercept? For a taxi, x = 0 means you haven’t gone anywhere but still pay the flag fall.

Worked Example 1 — Writing a model from a description

Question: A plumber charges a $60 call-out fee plus $85 per hour. Write a linear model for the total cost C in terms of hours worked h. Find the cost for 3 hours.

Identify: Dependent variable: C (total cost, $). Independent variable: h (hours).
Initial value (c): $60 (call-out fee, paid regardless of hours). Rate (m): $85/hr.

Model: C = 85h + 60

Predict: h = 3: C = 85(3) + 60 = 255 + 60 = $315

Worked Example 2 — Building a model from a table of values

Question: A car uses fuel at a constant rate. After 2 hours it has 52 L left; after 5 hours it has 28 L left. Write a linear model for fuel F (litres) in terms of time t (hours).

Find gradient: m = (28 − 52) ÷ (5 − 2) = −24 ÷ 3 = −8 L/hr (negative because fuel decreases)

Find c: Use point (2, 52): 52 = −8(2) + c → c = 52 + 16 = 68

Model: F = −8t + 68   (starts with 68 L, uses 8 L per hour)

Check: At t = 5: F = −8(5) + 68 = −40 + 68 = 28 L ✓

A linear model translates a real-world situation into the equation y = mx + c. Once you have the model, you can make predictions by substituting values.

Step 1: Identify the variables

Decide what x represents (the independent variable — the one you choose or control) and what y represents (the dependent variable — the one that results). Label both with units.

Step 2: Find the gradient m

The gradient is the rate of change. Look for phrases like “per km”, “each hour”, “for every unit”. If given two points, use m = (y&sub2; − y&sub1;) ÷ (x&sub2; − x&sub1;).

Step 3: Find the y-intercept c

The y-intercept is the value of y when x = 0. Look for “fixed fee”, “flag fall”, “initial amount”, “base charge”. If not directly given, substitute a known point into y = mx + c and solve for c.

Step 4: Write and use the model

Write y = mx + c with the actual variable names (e.g. C = 85h + 60). To predict: substitute the known value and evaluate. To find x given y: substitute y and solve for x.

Example: Electricity costs 28 cents per kWh plus a $35 daily supply charge. Write the model for daily cost C (in $) in terms of energy used E (kWh).
m = $0.28/kWh, c = $35 → C = 0.28E + 35
For E = 20 kWh: C = 0.28(20) + 35 = 5.60 + 35 = $40.60
Tip: If the rate is “per 100 km” or “per dozen”, convert it to a “per 1 unit” rate first. E.g. $12 per 100 km = $0.12 per km, so m = 0.12.

Evaluating whether a linear model is appropriate

A linear model is suitable when the rate of change is roughly constant. Check by computing gradients between successive points in a table — if they are all equal (or close), linear is appropriate. If they vary significantly, a non-linear model may be better.

Tip: When extrapolating (predicting beyond the data), state that the prediction assumes the linear pattern continues. This may not be realistic (e.g. a car can’t have negative fuel).

Mastery Practice

See Answers ➔
  1. Fluency

    Identify the gradient and y-intercept from each linear model, then state what each represents in context.

    1. (a) A tradesperson charges according to C = 75h + 50, where C is the total cost ($) and h is the number of hours worked.
    2. (b) A car's fuel tank follows F = −7d + 60, where F is fuel remaining (L) and d is distance driven (×10 km).
    3. (c) A mobile plan charges P = 0.15n + 20, where P is the monthly bill ($) and n is the number of text messages sent.
    4. (d) A bath is draining: V = −12t + 180, where V is the volume of water (litres) and t is time (minutes).
  2. Fluency

    Write a linear model y = mx + c for each situation. Clearly state what x and y represent.

    1. (a) A gym charges a $50 joining fee and $30 per month membership.
    2. (b) Water drains from a tank at 15 litres per minute. The tank initially holds 300 litres.
    3. (c) A runner burns 11 Calories per minute. They have already burned 44 Calories before you start timing.
    4. (d) A courier charges $8.00 per parcel plus a $5 booking fee per order.
  3. Fluency

    Find the gradient of the linear model passing through each pair of points and state the units.

    1. (a) (0 km, $3) and (4 km, $13) — y is fare in dollars, x is distance in km
    2. (b) (1 hr, 55 L) and (4 hr, 28 L) — y is fuel in litres, x is time in hours
    3. (c) (10 items, $48) and (25 items, $93) — y is total cost, x is number of items
    4. (d) (0 °C, 32 °F) and (100 °C, 212 °F) — Celsius to Fahrenheit conversion
  4. Fluency

    Use the linear model to make predictions.

    1. (a) C = 85h + 60. Find C when h = 4.5 hours.
    2. (b) F = −8t + 68. Find F when t = 6 hours. Is this answer realistic?
    3. (c) T = 1.8C + 32 (Celsius to Fahrenheit). Find T when C = 37 (body temperature).
    4. (d) Cost model C = 0.35n + 15. How many items (n) can be purchased for $50?
  5. Understanding

    Build a linear model from a table of values.

    Hint: Check that the differences in y are constant for equal steps in x (confirming linearity), then find m and c.
    Time (min), t05101520
    Distance (km), d06121824
    1. (a) Verify the data is linear by checking differences.
    2. (b) Write the linear model for d in terms of t.
    3. (c) Interpret the gradient in context.
    4. (d) Predict the distance travelled after 35 minutes. Is this interpolation or extrapolation?
  6. Understanding

    Compare two linear models.

    Electrician A charges $90/hr with a $40 call-out fee. Electrician B charges $70/hr with an $80 call-out fee.

    1. (a) Write linear models for the total cost of each electrician (CA and CB) in terms of hours h.
    2. (b) Calculate the cost of each for a 3-hour job.
    3. (c) Find the number of hours for which both electricians charge the same amount (set CA = CB and solve).
    4. (d) For jobs longer than this break-even time, which electrician is cheaper? Explain.
  7. Understanding

    Interpreting gradient and intercept in context.

    A scientist cools a liquid in a laboratory. The temperature T (°C) after t minutes follows the model T = −3.5t + 85.

    1. (a) What is the initial temperature of the liquid?
    2. (b) At what rate is the liquid cooling? Include units.
    3. (c) Find the temperature after 15 minutes.
    4. (d) After how many minutes does the liquid reach 0°C (freezing point)? Give your answer to 1 decimal place.
    5. (e) Is it realistic to use this model for very large values of t? Explain.
  8. Understanding

    Determine whether a linear model is appropriate.

    Method: Calculate the gradient between each consecutive pair of points. If all gradients are equal, the data is perfectly linear.
    x (items sold)0102030
    y (profit, $)−20050300550
    1. (a) Calculate the gradient between each pair of consecutive points.
    2. (b) Is a linear model appropriate? Justify your answer.
    3. (c) Write the linear model for profit P in terms of items sold n.
    4. (d) How many items must be sold for the business to break even (P = 0)?
  9. Problem Solving

    Developing and applying a real-world model.

    Challenge.

    A swimming pool is being filled using a hose. After 20 minutes, the pool contains 400 litres. After 50 minutes, it contains 1 000 litres.

    1. (a) Write a linear model for the volume V (litres) in terms of time t (minutes).
    2. (b) What does the y-intercept represent? Is it realistic in this context?
    3. (c) The pool holds 45 000 litres. How long will it take to fill? Give your answer in hours and minutes.
    4. (d) A second, larger hose fills at 25 L/min and starts when the first hose has been running for 30 minutes. Write a combined model for total volume after the second hose is added, and find the total time from the start until the pool is full.
  10. Problem Solving

    Two models, one decision.

    Challenge.

    Company A leases cars for a $500 deposit plus $0.18 per km driven. Company B charges no deposit but $0.28 per km. A business driver expects to drive approximately 4 500 km per month.

    1. (a) Write cost models CA and CB in terms of distance d (km).
    2. (b) At what distance are both companies equally expensive?
    3. (c) Which company should the business choose for 4 500 km per month, and by how much is it cheaper per month?
    4. (d) Over a 12-month lease, how much does the business save by choosing the better option?