Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 1 - Linear inequalities and simultaneous equations

Linear inequalities and simultaneous equations

Solve linear inequalities and pairs of simultaneous linear equations algebraically and graphically.

Updated 2026-07-26 - 12 min read

Linear inequalities and simultaneous equations 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.

Solve and graph linear inequalities

Maintain equivalence as with equations and reverse the inequality only when multiplying or dividing by a negative. 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 graph linear inequalities 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: Negative scaling reverses order, so the sign must reverse at that step.

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

Solve 4x − 7 ≤ 13.

Step 1 - identify the governing idea: Maintain equivalence as with equations and reverse the inequality only when multiplying or dividing by a negative.

Step 2 - apply it to this evidence: Add 7 and divide by positive 4.

Result: x ≤ 5

The relationship is visible in the working: Add 7 and divide by positive 4. 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:

  • x ≥ 5 — It does not agree with the required relationship: Add 7 and divide by positive 4.
  • x ≤ 20 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x = 5 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Solve −3x > 12.

Step 1 - identify the governing idea: Maintain equivalence as with equations and reverse the inequality only when multiplying or dividing by a negative.

Step 2 - apply it to this evidence: Dividing by −3 reverses > to <.

Result: x < −4

The relationship is visible in the working: Dividing by −3 reverses > to <. 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:

  • x > −4 — It does not agree with the required relationship: Dividing by −3 reverses > to <.
  • x < 4 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x = −4 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Solve 2 − 5x ≥ 17.

Step 1 - identify the governing idea: Maintain equivalence as with equations and reverse the inequality only when multiplying or dividing by a negative.

Step 2 - apply it to this evidence: Subtract 2, then divide 15 by −5 and reverse the sign.

Result: x ≤ −3

The relationship is visible in the working: Subtract 2, then divide 15 by −5 and reverse the sign. 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:

  • x ≥ −3 — It does not agree with the required relationship: Subtract 2, then divide 15 by −5 and reverse the sign.
  • x ≤ 3 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x ≥ 19/5 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Solve simultaneous linear equations

Use substitution or elimination to find values satisfying both equations at the same time. 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 simultaneous 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: A simultaneous solution must make both equations true; one equation alone has infinitely many pairs.

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

Solve x + y = 11 and x − y = 3.

Step 1 - identify the governing idea: Use substitution or elimination to find values satisfying both equations at the same time.

Step 2 - apply it to this evidence: Adding equations gives 2x = 14, so x = 7 and y = 4.

Result: (7, 4)

The relationship is visible in the working: Adding equations gives 2x = 14, so x = 7 and y = 4. 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:

  • (4, 7) — It does not agree with the required relationship: Adding equations gives 2x = 14, so x = 7 and y = 4.
  • (8, 3) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (11, 3) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Solve y = 2x + 1 and y = 10 − x.

Step 1 - identify the governing idea: Use substitution or elimination to find values satisfying both equations at the same time.

Step 2 - apply it to this evidence: Equating expressions gives 2x + 1 = 10 − x, so x = 3 and y = 7.

Result: (3, 7)

The relationship is visible in the working: Equating expressions gives 2x + 1 = 10 − x, so x = 3 and y = 7. 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:

  • (7, 3) — It does not agree with the required relationship: Equating expressions gives 2x + 1 = 10 − x, so x = 3 and y = 7.
  • (11/3, 25/3) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (2, 5) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Solve 2x + 3y = 13 and x + y = 5.

Step 1 - identify the governing idea: Use substitution or elimination to find values satisfying both equations at the same time.

Step 2 - apply it to this evidence: Doubling the second equation and subtracting gives y = 3, then x = 2.

Result: (2, 3)

The relationship is visible in the working: Doubling the second equation and subtracting gives y = 3, then x = 2. 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:

  • (3, 2) — It does not agree with the required relationship: Doubling the second equation and subtracting gives y = 3, then x = 2.
  • (1, 4) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (5, 1) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Interpret intersections and constraints

A graph intersection satisfies both relations; an inequality solution is a region or interval whose boundaries and context must be checked. 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 interpret intersections and constraints 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 exact intersection must satisfy both equations, and contextual constraints may restrict otherwise valid algebraic values.

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

Lines y = x + 2 and y = 8 − x intersect where?

Step 1 - identify the governing idea: A graph intersection satisfies both relations; an inequality solution is a region or interval whose boundaries and context must be checked.

Step 2 - apply it to this evidence: Set x + 2 = 8 − x to get x = 3 and y = 5.

Result: (3, 5)

The relationship is visible in the working: Set x + 2 = 8 − x to get x = 3 and y = 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:

  • (5, 3) — It does not agree with the required relationship: Set x + 2 = 8 − x to get x = 3 and y = 5.
  • (2, 6) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (4, 4) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

A ticket number t satisfies 6t + 15 ≤ 60 and t is a whole number. What is the maximum t?

Step 1 - identify the governing idea: A graph intersection satisfies both relations; an inequality solution is a region or interval whose boundaries and context must be checked.

Step 2 - apply it to this evidence: t ≤ 7.5, so the greatest whole number is 7.

Result: 7

The relationship is visible in the working: t ≤ 7.5, so the greatest whole number is 7. 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:

  • 7.5 — It does not agree with the required relationship: t ≤ 7.5, so the greatest whole number is 7.
  • 8 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 10 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Which point satisfies y ≥ 2x and y < 5?

Step 1 - identify the governing idea: A graph intersection satisfies both relations; an inequality solution is a region or interval whose boundaries and context must be checked.

Step 2 - apply it to this evidence: For (2,4), 4 ≥ 4 and 4 < 5.

Result: (2, 4)

The relationship is visible in the working: For (2,4), 4 ≥ 4 and 4 < 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:

  • (3, 5) — It does not agree with the required relationship: For (2,4), 4 ≥ 4 and 4 < 5.
  • (4, 6) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (1, 1) — 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. Solve −3x > 12.
  2. Solve y = 2x + 1 and y = 10 − x.
  3. A ticket number t satisfies 6t + 15 ≤ 60 and t is a whole number. What is the maximum t?

Answers

  1. x < −4 — Dividing by −3 reverses > to <.
  2. (3, 7) — Equating expressions gives 2x + 1 = 10 − x, so x = 3 and y = 7.
  3. 7 — t ≤ 7.5, so the greatest whole number is 7.

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