Practice Maths

Creating and Interpreting Linear Models

Key Ideas

Key Terms

Linear model
An equation of the form y = mx + c used to represent a real-world relationship where one quantity changes at a constant rate.
Gradient
The rate of change (m) — how much the dependent variable changes for each one-unit increase in the independent variable.
y-intercept
The value of y when x = 0 (c) — often represents a fixed starting value such as a flat fee or initial quantity.
Domain
The set of meaningful input values for a model within its real-world context (e.g. 0 ≤ h ≤ 8 for working hours).
Range
The set of output values produced by the model for all values in the domain.
Multiple representations
A linear model can be expressed as an equation, a table of values, or a graph — all carrying equivalent information.
Modelling cycle
The process: (1) identify variables, (2) write an equation, (3) make predictions, (4) interpret results, (5) validate in context.

The Modelling Cycle

1. Identify variables.   2. Write the equation.   3. Make predictions.   4. Interpret results.   5. Validate in context.

Hot Tip A model is only valid within its context. You can’t have negative time, a negative number of people, or more water than a tank holds. Always check whether your answer makes sense in the real world.

Worked Example

Question: A car hire company charges $40 per day plus a $25 insurance fee. Write a model for the total cost C for d days. Find the cost for 5 days and how many days can be hired for $185.

C = 40d + 25.   Gradient = 40 (cost per day), y-intercept = 25 (fixed fee).
For 5 days: C = 40(5) + 25 = $225.
For $185: 40d + 25 = 185 → 40d = 160 → d = 4 days.

What Is a Linear Model?

A linear model is an equation of the form y = mx + b that represents a real-world relationship between two quantities. It is called "linear" because, when graphed, it produces a straight line. Linear models are used whenever one quantity changes at a constant rate relative to another — like the cost of hiring a tradesperson (fixed call-out fee + hourly rate), or the temperature dropping by a consistent number of degrees each hour.

The two quantities are usually the independent variable (x — what you can control or measure, like hours or distance) and the dependent variable (y — what changes as a result, like cost or distance travelled).

Creating a Linear Equation from a Context

To write a linear model from a word problem, identify: (1) the fixed starting value (this is b, the y-intercept), and (2) the constant rate of change (this is m, the gradient).

Example: A plumber charges a $60 call-out fee plus $80 per hour. Write an equation for the total cost C after h hours.
Fixed fee = $60 (this is b). Hourly rate = $80 (this is m). Equation: C = 80h + 60.

To find the cost for 3 hours: C = 80(3) + 60 = 240 + 60 = $300.

You can also work backwards: if a customer's bill is $220, how many hours was the job? 220 = 80h + 60 → 80h = 160 → h = 2 hours.

Interpreting the Gradient and y-intercept

In a real-world linear model, both m and b have specific meanings that you should be able to explain in words — not just as numbers.

The gradient m is the rate of change. It tells you how much y increases (or decreases) for each one-unit increase in x. In the plumber example, m = 80 means the cost increases by $80 for every additional hour of work.

The y-intercept b is the value of y when x = 0. It often represents an initial or fixed quantity — a starting amount before any "rate" is applied. In the plumber example, b = 60 means the customer pays $60 even before any work begins (just for the plumber to show up).

Using a Linear Model to Make Predictions

Once you have the equation, you can use it to make predictions for any x value. Substitute the x value and calculate y. This works for both interpolation (predicting within the known data range) and extrapolation (predicting beyond the known data range).

Example: A mobile phone plan costs $30 per month plus $0.10 per text message. Model: Cost = 0.10n + 30, where n = number of texts. Predicted cost for 500 texts: 0.10(500) + 30 = 50 + 30 = $80.

Extrapolation is riskier than interpolation — the model might not hold true far beyond the data range (e.g. the plumber might charge a different rate for jobs over 8 hours).

Limitations of Linear Models

Not every real-world relationship is perfectly linear. Linear models are approximations — they work well over a limited range but may break down outside that range. For example, the cost of electricity might increase more than linearly once you use very large amounts, or a taxi fare might have different rates at night compared to daytime.

When using a model to make predictions, always consider: Is the relationship actually linear? Is my prediction within a reasonable range? Are there other factors the model doesn't account for?

Key tip: When writing a linear model from a word problem, the most common mistake is mixing up m and b. Ask yourself: "What is present even when x = 0?" That is b. "How much does y change for each extra unit of x?" That is m. If a question says "increases by $15 for each additional km," then m = 15. If it says "plus a $20 booking fee," then b = 20.

Mastery Practice

  1. Write a linear equation to model each situation. Identify the gradient and y-intercept and explain what each means in context. Fluency

    1. A plumber charges $60 call-out fee plus $50 per hour.
    2. A gym membership costs $30 sign-up fee plus $15 per week.
    3. A car is 200 km from home and travels at 80 km/h towards home.
    4. A candle is 20 cm tall and burns at 2 cm per hour.
    5. A phone plan charges $0.25 per minute with no fixed fee.
    6. Water in a tank is 500 L and drains at 30 L per minute.
    7. A baker makes $4 profit per cake and has overhead costs of $80 per day.
    8. A student earns $12 per hour and already has $50 in savings.

  2. For each table of values, write the equation, then use it to find the specified value. Fluency

    1. Hours (h)0123
      Cost ($C)20355065
      Find C when h = 7.
    2. Days (d)0123
      Distance (km)30022014060
      Find the distance after 4 days.
    3. Items (n)1234
      Price ($P)7121722
      Find the price of 10 items.
    4. Minutes (t)051015
      Volume (L)04080120
      How long to fill 400 L?
    5. Km (k)0246
      Fare ($F)3.507.1010.7014.30
      Find the fare for a 10 km trip.
    6. Hours (h)0123
      Height (cm)25232119
      When will the candle height reach 7 cm?
    7. Weeks (w)0123
      Savings ($)80105130155
      After how many weeks will savings reach $330?
    8. People (p)10203040
      Cost ($C)4507009501200
      Find the cost for 55 people.

  3. For each model, state the gradient and y-intercept and explain what each means in context. Fluency

    1. C = 50h + 80  (C = cost in dollars, h = hours)
    2. D = 300 − 60t  (D = distance from destination in km, t = hours)
    3. P = 4n − 80  (P = profit in dollars, n = items sold)
    4. V = 800 − 25t  (V = volume of water in litres, t = minutes)
    5. T = −10 + 3h  (T = temperature in °C, h = hours after midnight)
    6. S = 12w + 50  (S = total savings in dollars, w = weeks)
    7. C = 1.80k + 3.50  (C = taxi cost in dollars, k = kilometres)
    8. H = 30 − 2.5t  (H = candle height in cm, t = hours)

  4. For each pair of linear models, find when the two models give the same value. Interpret your answer in context. Understanding

    1. Plan A: C = 30 + 10h; Plan B: C = 15h.  (C = cost, h = hours)
    2. Company X: C = 50d + 100; Company Y: C = 40d + 200.  (C = cost, d = days)
    3. Runner A: D = 8t; Runner B: D = 5t + 9.  (D = distance in km, t = hours)
    4. Pool A: V = 200t; Pool B: V = 150t + 500.  (V = volume in L, t = minutes)
    5. Electrician P: C = 60h + 50; Electrician Q: C = 45h + 95.  (C = cost, h = hours)
    6. Savings A: S = 20w + 100; Savings B: S = 30w.  (S = savings, w = weeks)

  5. For each model, identify and explain a sensible domain restriction. Then state the corresponding range. Understanding

    1. A candle is 25 cm tall and burns at 2.5 cm/hour: H = 25 − 2.5t
    2. A pool holds 10 000 L and fills at 400 L/min: V = 400t
    3. A car park charges $3 entry plus $2 per hour. The car park closes after 12 hours: C = 2h + 3
    4. A student scores marks at a rate of 3 marks per question on a 20-question test: M = 3q
    5. Temperature rises from −5°C at 2°C/hour until it reaches 15°C: T = −5 + 2h
    6. A recipe can be scaled for between 1 and 8 people. One serve costs $4.50: C = 4.5p

  6. Complete each full modelling task. Problem Solving

    1. Phone plans: Plan A charges $25 per month plus $0.15 per minute of calls. Plan B charges $40 per month for unlimited calls.
      1. Write a model for each plan’s monthly cost C in terms of minutes used m.
      2. Find the number of minutes where both plans cost the same.
      3. Erica uses about 90 minutes per month. Which plan is better for her?
      4. State the domain for each model and explain why it makes sense.
    2. Car hire: Cars R Us charges $60 per day with no deposit. Hire4Less charges $40 per day plus a $100 refundable deposit.
      1. Write a total cost model for each company (ignoring the refundable deposit for Hire4Less).
      2. How many days until Hire4Less becomes cheaper?
      3. For a 7-day hire, how much do you save by choosing the cheaper company?
    3. Swimming pool: A pool of capacity 24 000 L is being filled by two hoses. Hose A delivers 300 L/min and Hose B delivers 200 L/min. Only Hose A is used for the first 20 minutes, then both are used together.
      1. How much water is in the pool after the first 20 minutes?
      2. Write a model for the volume V in terms of total minutes t (for t > 20).
      3. When is the pool full?
    4. Gardening costs: A gardener charges $45 per hour. She estimates her jobs take between 2 and 6 hours. Materials cost an extra $30 flat fee on every job.
      1. Write a model for the total job cost C in terms of hours h.
      2. State the domain and range for this model.
      3. A customer has a budget of $200. Can the gardener complete a 4-hour job within budget? Show working.
  7. Break-even Analysis. A business makes revenue and incurs costs, both modelled linearly.

    A food truck has fixed daily costs of $120 and earns $8 revenue per meal sold. Problem Solving

    1. Write a model for the daily cost C (it is constant, just the fixed costs).
    2. Write a model for the daily revenue R in terms of meals sold m.
    3. Find the break-even point — how many meals must be sold to cover costs?
    4. The owner wants to make at least $200 profit per day. Write and solve an inequality to find how many meals are needed. State the domain restriction.
  8. Critique a Model. Read the scenario and evaluate the model given.

    A student models the height of a sunflower as H = 3.5w − 10, where H is height in cm and w is weeks after planting. Problem Solving

    1. What does the gradient 3.5 represent in this context?
    2. What does the y-intercept −10 represent? Does it make physical sense? Explain.
    3. Find the week when the model predicts the plant first appears above the ground (H > 0).
    4. The sunflower reaches a maximum height of about 200 cm and then stops growing. What is the domain restriction on this model? Explain why the model breaks down beyond this point.
  9. Model from Data. Build a linear model from a real data table, then use it to predict.

    The table shows the distance a cyclist is from home at various times. Problem Solving

    Time (hours)0123
    Distance from home (km)60453015
    1. Find the gradient and write the linear model D = mt + c.
    2. Interpret the gradient and y-intercept in context.
    3. At what time does the cyclist arrive home? Show your working.
    4. Is this model valid for t > 4? Explain why or why not.
  10. Energy Bills. Two electricity providers charge differently.

    Provider A charges a $25 monthly supply fee plus $0.28 per kWh used. Provider B charges no supply fee but $0.35 per kWh. Problem Solving

    1. Write a cost model for each provider (C = cost, k = kWh used).
    2. Find the number of kWh at which both providers charge the same amount.
    3. A household uses 380 kWh per month. Which provider is cheaper, and by how much?
    4. Sketch a rough graph showing both models. Label the intersection point and explain what it means in context.
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