Year 10 - Algebra

The Equation Apothecary

A focused exam-prep path for the same skills: checking a graphical solution, proving the exact point, and building simultaneous equations from a worded context.

2worked models
20new questions
10per chapter

The exam move

Simultaneous equations are not two separate answers. They ask for one point, or one pair of quantities, that makes both conditions true at the same time.

Proof beats a picture.
A graph can suggest an answer, but substitution or algebra proves it.
Variables carry meaning.
In worded problems, the final answer must return to the story.
Chapter I
Graph proof

Use a fresh exam-style graph question to learn how to test a claimed intersection and solve exactly.

introduces-procedure introduces-friction capstone
Chapter II
Worded modelling

Use a fresh exam-style modelling question to define variables, write two equations, solve, and state the answer.

confidence-builder consolidation capstone
Chapter I - Learn

Checking a Graphical Solution

In this exam-style task, a student has graphed two equations and used the intersection as the simultaneous solution. Your job is to decide whether that graph gives the correct answer, then prove it.

Worked model - graph proof

Solve the pair of equations shown in the diagram:

y = 2x - 5
x + y = 10

  1. Substitute y = 2x - 5 into x + y = 10.

  2. x + (2x - 5) = 10, so 3x - 5 = 10.

  3. 3x = 15, so x = 5.

  4. Substitute back: y = 2(5) - 5 = 5.

  5. The exact simultaneous solution is (5, 5).

The graph alone is not enough proof.
A drawn intersection must be checked against the equations. Here the true point is (5, 5). If a drawn intersection is not (5, 5), the graph reading is incorrect.

Proof sentence

The correct simultaneous solution is (5, 5). This is proven because y = 2(5) - 5 = 5, and 5 + 5 = 10. Therefore any graph showing a different intersection is incorrect.

What to do every time

  1. Write down the claimed point from the graph.

  2. Substitute that point into both given equations.

  3. If it fails either equation, reject the claim.

  4. Solve algebraically to find the exact point.

  5. Finish with a sentence that answers whether the student's graph was correct.

Chapter I: Graph proof
Chapter II - Learn

Building Equations from Words

This exam-style modelling task has two parts. First it uses one variable for a comparison statement. Then it uses a pair of simultaneous equations for two item types and a total value.

Part A - comparison equation

Maya and Noah collect 128 tokens between them. Maya collects 18 more tokens than Noah.

  1. Let n be the number Noah collects.

  2. Maya collects n + 18.

  3. Together: n + (n + 18) = 128.

  4. 2n + 18 = 128, so n = 55.

  5. Maya collects 73 tokens and Noah collects 55 tokens.

Part B - simultaneous equations

A school stall sells 130 tickets. Premium tickets cost $7 and standard tickets cost $4. The total collected is $730.

Let P = premium tickets and S = standard tickets
P + S = 130
7P + 4S = 730

  1. Use the counting equation: P + S = 130.

  2. Use the money equation: 7P + 4S = 730.

  3. Multiply the counting equation by 4: 4P + 4S = 520.

  4. Subtract from the money equation: 3P = 210.

  5. P = 70, then S = 60.

Final answer in context:
The stall sold 70 premium tickets and 60 standard tickets. Check: 70 + 60 = 130 and 7(70) + 4(60) = 730.

What to do every time

  1. Define the variables with units or object names.

  2. Write one equation for the total number of items.

  3. Write one equation for the total value, cost, or score.

  4. Eliminate or substitute to solve.

  5. State the answer using the story's words, not just x and y.

Chapter II: Worded modelling
Journal

Simultaneous Equations Summary

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