Australian Curriculum v9 / ACiQ Year 8 Mathematics - Unit 2 - Linear modelling in applied contexts

Linear modelling in applied contexts

Form and solve linear equations from practical situations and interpret the solution in its context.

Updated 2026-07-26 - 13 min read

Linear modelling in applied contexts is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Translate contexts into linear equations

Define the unknown with units, represent fixed and variable parts, and make the equation match the stated total or condition. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by translate contexts into linear equations and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: The language describes relationships: rates multiply quantities, fixed amounts are added once, and totals form the equality.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

A taxi costs $4 plus $2.50 per kilometre. Which equation gives a $24 fare for k kilometres?

Step 1 - identify the governing idea: Define the unknown with units, represent fixed and variable parts, and make the equation match the stated total or condition.

Step 2 - apply it to this evidence: The fixed fee is added once and the rate multiplies kilometres.

Result: 4 + 2.5k = 24

The relationship is visible in the working: The fixed fee is added once and the rate multiplies kilometres. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 6.5k = 24 — It does not agree with the required relationship: The fixed fee is added once and the rate multiplies kilometres.
  • 4k + 2.5 = 24 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2.5(k + 4) = 24 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Three identical notebooks and a $5 pen cost $23. Which equation models notebook price n?

Step 1 - identify the governing idea: Define the unknown with units, represent fixed and variable parts, and make the equation match the stated total or condition.

Step 2 - apply it to this evidence: Three notebooks cost 3n and the pen adds $5.

Result: 3n + 5 = 23

The relationship is visible in the working: Three notebooks cost 3n and the pen adds $5. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • n + 8 = 23 — It does not agree with the required relationship: Three notebooks cost 3n and the pen adds $5.
  • 5n + 3 = 23 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 3(n + 5) = 23 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

A rectangle has perimeter 46 cm and length 15 cm. Which equation finds width w?

Step 1 - identify the governing idea: Define the unknown with units, represent fixed and variable parts, and make the equation match the stated total or condition.

Step 2 - apply it to this evidence: Rectangle perimeter is twice the sum of length and width.

Result: 2(15 + w) = 46

The relationship is visible in the working: Rectangle perimeter is twice the sum of length and width. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 15w = 46 — It does not agree with the required relationship: Rectangle perimeter is twice the sum of length and width.
  • 15 + w = 46 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2(15w) = 46 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Solve and interpret linear models

Solve the model, then state what the solution means with its unit and check it satisfies contextual constraints. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by solve and interpret linear models and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: A contextual solution needs meaning, units and a check that it is possible in the situation.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

Using 4 + 2.5k = 24, find k.

Step 1 - identify the governing idea: Solve the model, then state what the solution means with its unit and check it satisfies contextual constraints.

Step 2 - apply it to this evidence: Subtracting 4 and dividing by 2.5 gives k = 8 kilometres.

Result: 8 km

The relationship is visible in the working: Subtracting 4 and dividing by 2.5 gives k = 8 kilometres. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 10 km — It does not agree with the required relationship: Subtracting 4 and dividing by 2.5 gives k = 8 kilometres.
  • 7.6 km — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 50 km — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

A gym charges $30 joining plus $12 per week. After how many weeks is the total $126?

Step 1 - identify the governing idea: Solve the model, then state what the solution means with its unit and check it satisfies contextual constraints.

Step 2 - apply it to this evidence: 30 + 12w = 126 gives 12w = 96 and w = 8.

Result: 8 weeks

The relationship is visible in the working: 30 + 12w = 126 gives 12w = 96 and w = 8. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 13 weeks — It does not agree with the required relationship: 30 + 12w = 126 gives 12w = 96 and w = 8.
  • 10.5 weeks — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 7 weeks — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Tickets cost $7 each with a $10 booking fee. A budget is $50. What is the greatest whole number of tickets?

Step 1 - identify the governing idea: Solve the model, then state what the solution means with its unit and check it satisfies contextual constraints.

Step 2 - apply it to this evidence: 10 + 7t ≤ 50 gives t ≤ 40/7 ≈ 5.71, so at most 5 whole tickets can be bought.

Result: 5

The relationship is visible in the working: 10 + 7t ≤ 50 gives t ≤ 40/7 ≈ 5.71, so at most 5 whole tickets can be bought. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 6 — It does not agree with the required relationship: 10 + 7t ≤ 50 gives t ≤ 40/7 ≈ 5.71, so at most 5 whole tickets can be bought.
  • 7 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 5.71 tickets — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Compare linear models

Write each option as fixed cost plus rate times quantity, then compare at the same quantity or solve where the models are equal. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by compare linear models and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: A smaller rate can be offset by a larger fixed cost; cost depends on the quantity used.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

Plan A costs 10 + 3x and Plan B costs 22 + 2x. At what x are costs equal?

Step 1 - identify the governing idea: Write each option as fixed cost plus rate times quantity, then compare at the same quantity or solve where the models are equal.

Step 2 - apply it to this evidence: Set 10 + 3x = 22 + 2x and solve to get x = 12.

Result: 12

The relationship is visible in the working: Set 10 + 3x = 22 + 2x and solve to get x = 12. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 6 — It does not agree with the required relationship: Set 10 + 3x = 22 + 2x and solve to get x = 12.
  • 10 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 22 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

At x = 5, which is cheaper: A = 8 + 4x or B = 15 + 3x?

Step 1 - identify the governing idea: Write each option as fixed cost plus rate times quantity, then compare at the same quantity or solve where the models are equal.

Step 2 - apply it to this evidence: A costs 28 and B costs 30.

Result: Plan A

The relationship is visible in the working: A costs 28 and B costs 30. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Plan B — It does not agree with the required relationship: A costs 28 and B costs 30.
  • They are equal — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • It cannot be decided — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

A hire option is $40 + $6 per hour. Another is $10 per hour. Which is cheaper for 12 hours?

Step 1 - identify the governing idea: Write each option as fixed cost plus rate times quantity, then compare at the same quantity or solve where the models are equal.

Step 2 - apply it to this evidence: The first costs $112 and the second $120.

Result: The first option by $8

The relationship is visible in the working: The first costs $112 and the second $120. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • The second option by $8 — It does not agree with the required relationship: The first costs $112 and the second $120.
  • The first option by $40 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • They are equal — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. Three identical notebooks and a $5 pen cost $23. Which equation models notebook price n?
  2. A gym charges $30 joining plus $12 per week. After how many weeks is the total $126?
  3. At x = 5, which is cheaper: A = 8 + 4x or B = 15 + 3x?

Answers

  1. 3n + 5 = 23 — Three notebooks cost 3n and the pen adds $5.
  2. 8 weeks — 30 + 12w = 126 gives 12w = 96 and w = 8.
  3. Plan A — A costs 28 and B costs 30.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources