Australian Curriculum v9 / ACiQ Year 7 Mathematics - Unit 0 - One-variable linear equations

One-variable linear equations

Solve one-variable linear equations with natural-number solutions and verify each solution by substitution.

Updated 2026-07-24 - 6 min read

Solve one-variable linear equations with natural-number solutions and verify each solution by substitution. This draft teaching resource is aligned to the cited Year 7 content descriptions and still requires named human curriculum approval.

Define each variable before operating. Every symbol, table entry and graph coordinate should retain one consistent meaning and unit.

1. Keep equations balanced

An equation states that two expressions are equal. Doing the same valid operation to both sides preserves that equality.

Decision check: Apply this lesson's rule to the worked example: Undo the operation attached to the variable on both sides. Identify the exact value, label or property that satisfies the rule.

2. Solve, model and verify two-step equations

Two-step equations are undone in reverse operation order. Verification is a separate substitution check, not a repeated guess.

Decision check: Apply this lesson's rule to the worked example: Solve with inverse operations, then substitute into the untouched original equation. Identify the exact value, label or property that satisfies the rule.

Worked examples with complete reasoning

Worked example 1

Question: Solve 2x + 5 = 19.

  1. Undo addition or subtraction first.
  2. Then undo multiplication or division.
  3. The solution is 7.

Answer: 7

Check: Reverse the order of operations while keeping both sides balanced.

Worked example 2

Question: Which value verifies 3x + 1 = 22?

  1. Substitute each candidate for the variable.
  2. 7 makes the left side equal the right side.

Answer: 7

Check: A solution must make the equation true when substituted.

Worked example 3

Question: Solve x + 7 = 19.

  1. Use the inverse operation on both sides of the equation.
  2. The solution is 12.

Answer: 12

Check: Undo the operation attached to the variable.

Method map for this topic

  1. Keep equations balanced: Undo the operation attached to the variable on both sides.
  2. Solve, model and verify two-step equations: Solve with inverse operations, then substitute into the untouched original equation.

Common mistakes and how to correct them

  • inverse operation: Not yet. Undo the operation attached to the variable. Recheck the inverse operation step, then try again.
  • two step equation: Not yet. Reverse the order of operations while keeping both sides balanced. Recheck the two step equation step, then try again.
  • verify solution: Not yet. A solution must make the equation true when substituted. Recheck the verify solution step, then try again.
  • word equation: Not yet. Define the unknown, then translate the relationships in order. Recheck the word equation step, then try again.
  • equation error: Not yet. Find the first inverse-operation error, then solve from that line. Recheck the equation error step, then try again.
  • equation model: Not yet. The fixed fee is added once; the per-session cost is multiplied by the unknown. Recheck the equation model step, then try again.

Practice

  1. Solve 2x + 5 = 19.
  2. Solve 3y - 4 = 20.
  3. Which value verifies 3x + 1 = 22?
  4. Which value verifies 4n - 5 = 27?
  5. Solve x + 7 = 19.
  6. Solve y - 9 = 6.

<details> <summary>Answers and reasoning</summary>

  1. 7 — Undo addition or subtraction first. Then undo multiplication or division. The solution is 7.
  2. 8 — Undo addition or subtraction first. Then undo multiplication or division. The solution is 8.
  3. 7 — Substitute each candidate for the variable. 7 makes the left side equal the right side.
  4. 8 — Substitute each candidate for the variable. 8 makes the left side equal the right side.
  5. 12 — Use the inverse operation on both sides of the equation. The solution is 12.
  6. 15 — Use the inverse operation on both sides of the equation. The solution is 15.

</details>

Checklist and vocabulary

| Skill | Observable evidence | | --- | --- | | solve two step equations integer | Solve two-step equations with integer coefficients and natural-number solutions, then check by substitution. | | find correct equation solving error | Find and correct an equation-solving error and verify the corrected solution. | | solve one step linear equations | Solve one-step linear equations using inverse operations on both sides. |

Retrieval prompts

  • State the defining relationship for each method above without looking.
  • Give one example where two visually similar representations mean different things.
  • Diagnose one misconception from the list and explain the first corrective step.
  • Name a check that is independent of simply repeating the same calculation.

Teach-back and self-check

Close the worked examples and explain one-variable linear equations as if you were helping another Year 7 student. Begin by naming the clue that tells you which relationship, representation or operation is needed. Then show one complete example with labels and units where they matter. At each line, explain what stayed equivalent and why the next step is allowed.

Use this four-part check before calling the method secure:

  1. Identify: What information is given, and what must be found or justified?
  2. Represent: Which number line, diagram, table, expression, graph or organised list makes the relationship visible?
  3. Reason: Which rule connects the representation to the required answer?
  4. Verify: Can you check by estimation, substitution, an inverse operation, a second representation or the original context?

If the explanation depends on “I just knew”, return to the method map and name the missing decision. A durable method is one you can explain, check and use in a changed context after a delay.

Transfer task

Model a changing school or household quantity using one-variable linear equations. Define every variable and unit, test at least two inputs, and explain what the outputs mean.

Sources