Linear Relations, Equations and Inequalities — AC9M8A02

• Recognise linear relations from constant first differences and straight-line graphs.
• Graph linear relations and simple horizontal or vertical boundaries.
• Solve linear equations and one-variable inequalities using inverse operations and graphical reasoning.
• Verify equation solutions by substitution and interpret inequality solutions as sets of values.

Recall Cartesian coordinates, substitution, integer arithmetic, equivalent expressions and the idea that applying the same inverse operation to both sides preserves equality.

A relation is linear when equal changes in the independent variable produce constant changes in the dependent variable. In a table this appears as constant first differences; on a graph the points lie on a straight line.

An equation asks for values that make two expressions equal. An inequality such as 2x+1<9 describes a set of values, not one isolated answer. The Year 8 Australian descriptor explicitly connects graphical and algebraic solution methods and substitution checks.

For two expressions, the equation solution is the x-coordinate where their graphs intersect. Algebra, graph and substitution should agree.

• A linear graph must pass through the origin. Only zero-intercept relations do.
• An inequality has one answer. It usually represents an interval or set.
• Solving means changing one side only. Equality is preserved by equivalent operations on both sides.
• A graph is decorative. Straightness, intersections, boundaries and regions are mathematical evidence.

1. Complete a table for y=3x+2 and identify the first difference.
2. Plot three points for y=−x+4 and explain why the graph is linear.
3. Solve 4x−5=19 and verify by substitution.
4. Solve 3x+2<14 and represent the solution on a number line.

1. Decide whether x:0,1,2,3 and y:5,8,11,14 is linear.
2. Graph y=2x+3 for −2≤x≤3.
3. Solve 5x+4=29 and check.
4. Solve 4x−7<13.
5. Explain how the intersection of y=2x+5 and y=17 solves 2x+5=17.
6. Compare x=3 and y=3 on the Cartesian plane.
7. Solve 2x+9=5x−6 and state how a graph verifies it.

Plan A costs C=15+4d and Plan B costs C=27+2d. Determine when the plans cost the same, interpret the graph intersection and identify which plan is cheaper on each side.

1. How can a table indicate linearity? Constant first differences for equally spaced x-values show constant rate of change.
2. What does solving an equation mean? Finding value(s) that make the two sides equal.
3. Why is an inequality answer often a range? Many values can satisfy a comparison.
4. How can a graph verify a solution? The algebraic solution matches an intersection or boundary; substitution confirms it numerically.

1. A table has x=1,2,3,4 and y=7,11,15,19. Show it is linear, form a rule and predict y at x=10.
   Use first differences and a rule consistent with every row.
2. Solve 3x+5=2x+17 and verify by substitution and graph.
   All representations should give the same x-value.
3. Solve 5x−2<18 and explain why the answer is a set.
   Show the inverse-operation chain and interval.

• I recognise constant rate of change.
• I graph linear relations accurately.
• I solve equations and inequalities using justified steps.
• I verify solutions graphically and by substitution.

Exit ticket: What is the difference between the solution to an equation and the solution to an inequality?

Australian Curriculum v9.0 — AC9M8A02: graph linear relations on the Cartesian plane using digital tools where appropriate; solve linear equations and one-variable inequalities using graphical and algebraic techniques; verify solutions by substitution.

Victorian Curriculum F–10 Version 2.0 — Level 8, VC2M8A02: Exact relationship.

NSW Mathematics K–10 Syllabus (2022) — Stage 4, MA4-EQU-C-01; MA4-LIN-C-01; MAO-WM-01: Partial. NSW Stage 4 covers equations and graphical linear relationships, but AC9M8A02 is broader because it explicitly includes one-variable inequalities and the full graph–algebra verification sequence.

• AC9M8A01 — linear expressions
• AC9M8A03 — linear modelling
• AC9M8A04 — parameter experiments

• Australian Curriculum v9.0
• Victorian Curriculum F–10 Version 2.0 Mathematics
• NSW Mathematics K–10 Syllabus (2022)

Curriculum equivalents for Graph linear relations on the Cartesian plane using digital tools...

Mapped skill: graph linear relations on the Cartesian plane using digital tools where appropriate; solve linear equations and one-variable inequalities using graphical and algebraic techniques; verify solutions by substitution

These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.

Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.