Year 10 Mathematics · AC9M10A02

Linear Inequalities and Simultaneous Equations

Solve situations with more than one condition. Learn when elimination, substitution or a graph is most efficient, how an intersection represents a common solution, and…

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Key conceptTeach from the board

1. An inequality describes a set

The statement x < 4 means every real number less than 4 is a solution. A strict sign (< or >) excludes the boundary; ≤ or ≥ includes it. When multiplying or dividing both sides by a negative number, reverse the inequality because the order of numbers on the number line reverses under multiplication by a negative.

2. Inequalities in 2 variables describe regions

For 2x + 3y < 12, first draw the boundary 2x + 3y = 12. Because the inequality is strict, the boundary is not included. Test a point not on the line, often (0,0). Since 0 < 12 is true, the solution region is the side containing the origin.

3. A simultaneous solution satisfies both equations

If two straight-line equations are both true, their common ordered pair is where the lines intersect. Algebraically, elimination combines equations to remove one variable; substitution replaces one variable with an equivalent expression. Graphing makes the meaning visible but may give only an approximate solution unless the intersection is clear.

4. Method choice matters

Use substitution when a variable is already isolated or easy to isolate. Use elimination when coefficients already match or can be made to match simply. Use graphs when interpretation, feasibility or an approximate intersection is central to the question.

Worked examplesWe do

Worked examples

AC9M10A02 - Linear Inequalities and Simultaneous Equations
Example 1

Solve −4x≤20 and explain the sign change.

Example 2

Solve x+y=12 and x−y=2 by elimination.

Example 3

Solve y=3x−4 and x+y=12 by substitution.

Example 4

For x+2y<8, state the boundary type and which side is shaded after testing (0,0).

Curriculum examplesCopied content

Australian Curriculum:AC9M10A02, Year 10 Algebra — solving linear inequalities and simultaneous linear equations in 2 variables, interpreting graphical solutions and communicating results in context.

Victoria:VC2M10A08, Level 10 Algebra covers linear inequalities, including number-line and Cartesian-plane representations; VC2M10A09 covers simultaneous linear equations using algebraic and graphical methods. Together they provide a strong alignment.

NSW: Stage 5 distributes this content mainly through Paths. MA5-EQU-P-01 includes linear inequalities, MA5-EQU-P-02 includes linear simultaneous equations, and MA5-FNC-P-01 supports graphing inequalities in one and 2 variables. MAO-WM-01 supports communicating and justifying solutions. This is not a single-outcome equivalence.

Alignment explanation: The lesson's one- and two-variable inequality work maps directly to the Australian descriptor and Victorian A08, while elimination/substitution/graph interpretation maps to Australian A02 and Victorian A09. NSW places much of the equivalent complexity in Stage 5 Paths, so the page identifies those Path outcomes explicitly rather than calling them Core.

Lesson componentAustralian CurriculumVictoriaNSW
Linear inequalitiesAC9M10A02VC2M10A08MA5-EQU-P-01
2-variable inequality regionsAC9M10A02VC2M10A08MA5-FNC-P-01
Simultaneous equationsAC9M10A02VC2M10A09MA5-EQU-P-02
Graphical/context interpretationAC9M10A02 + proficiencyVC2M10A09MA5-FNC-P-01 + MAO-WM-01
Questions and answersWith answers
What does an intersection mean?
It is an ordered pair satisfying both equations at the same time.
When should I use elimination?
When one variable has equal or easily matched coefficients, making it efficient to remove.
Why does an inequality sign reverse after dividing by a negative?
Multiplying by a negative reverses number order: for example 2<5 but −2>−5.
How do I choose which inequality region to shade?
Test a point not on the boundary and shade the side where the inequality is true.
Can simultaneous equations have no solution?
Yes. Distinct parallel lines never intersect. Coincident lines represent infinitely many common solutions.
Practice and reviewReady for practice
  • Forgetting to reverse the sign: only reverse it when multiplying or dividing both sides by a negative.
  • Using a solid line for x+y<5: strict inequalities exclude the boundary, so use a dashed boundary.
  • Shading by eye: use a test point unless the correct side is immediately justified.
  • Finding x but not y: a simultaneous solution needs both coordinates.
  • Eliminating the wrong way: if coefficients have the same sign, subtract; if opposite, add—or scale first.
  • Ignoring context: negative or fractional solutions can be algebraically valid but impossible for counts or physical constraints.
Curriculum alignmentStart here
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