Example 1
Solve 3x+5=26
- Subtract 5 from both sides.
- Simplify to 3x=21.
- Divide both sides by 3.
- Substitute the solution into the original equation.
Final answer: x=7
Check: 3(7)+5=26.
What students learn in AC9M7A03
We are learning to solve one-variable linear equations with natural number solutions and verify the solution by substitution.
In this topic, the one-variable linear equations are deliberately chosen to have natural-number solutions. A solution is the value that makes both sides equal, and every solving line must preserve that equality.
Use inverse operations in reverse order and perform the same operation on both sides. Brackets are handled as a grouped operation before isolating the variable.
Substitution is an independent check: replacing the variable with the proposed solution must produce the same numerical value on both sides.
Success criteria
- I can solve one-variable linear equations using balanced inverse operations.
- I can explain why each equation line is equivalent to the previous line.
- I can verify a natural-number solution by substitution.
Key vocabulary
- linear equation
- equation where the variable has power one.
- equivalent equations
- equations with the same solution set.
- solution
- value making the original equation true.
Visual models and representations
Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.
4 worked numerical & application examples
Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.
Example 2
Solve 4(x−2)=24
- Divide both sides by 4.
- Simplify to x−2=6.
- Add 2 to both sides.
- Verify in the original equation.
Final answer: x=8
Check: 4(8−2)=24.
Example 3
Application problem 1
Problem: Solve 5x+7=42 using equal operations on both sides, then verify your solution by substitution.
- Plan: Represent the information first, then calculate, interpret and independently check the result.
- Work: Subtract 7: 5x=35. Divide by 5: x=7. Check: 5×7+7=42, so the solution is verified.
- Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Final answer: Subtract 7: 5x=35. Divide by 5: x=7. Check: 5×7+7=42, so the solution is verified.
Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Example 4
Application problem 2
Problem: Give an equation with natural-number solution 9 and verify it.
- Plan: Choose operations, then calculate the matching right side.
- Work: One valid equation is 5x−8=37: adding 8 and dividing by 5 gives x=9, and substitution checks 5(9)−8=37.
- Interpret: The complete model for “Give an equation with natural-number solution 9 and verify it.” shows One valid equation is 5x−8=37: adding 8 and dividing by 5 gives x=9, and substitution checks 5(9)−8=37.
Final answer: One valid equation is 5x−8=37: adding 8 and dividing by 5 gives x=9, and substitution checks 5(9)−8=37.
Check: The complete model for “Give an equation with natural-number solution 9 and verify it.” shows One valid equation is 5x−8=37: adding 8 and dividing by 5 gives x=9, and substitution checks 5(9)−8=37.
Common misconceptions
Common mistake: Term moved by changing sign without explanation.
Correction: Apply the same inverse operation to both sides.
Common mistake: Distributive brackets ignored.
Correction: 2(x+3)=2x+6.
Common mistake: Division applied to one term only.
Correction: Divide every term or simplify the whole side correctly.
10 important problems to solve
Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.
1. Solve x+9=15.
Check answer
Answer: x=6
Hint: Undo the addition of 9.
Why: Using the method “Undo the addition of 9.” gives x=6.
2. Solve 5x=35.
Check answer
Answer: x=7
Hint: Divide both sides by 5.
Why: Using the method “Divide both sides by 5.” gives x=7.
3. Does x=4 solve 2x+3=11?
Check answer
Answer: Yes; 2(4)+3=11.
Hint: Substitute 4 for x.
Why: Using the method “Substitute 4 for x.” gives Yes; 2(4)+3=11.
4. Solve 2x+7=25.
Check answer
Answer: x=9
Hint: Subtract 7, then divide by 2.
Why: Using the method “Subtract 7, then divide by 2.” gives x=9.
5. Solve 6x−5=31.
Check answer
Answer: x=6
Hint: Add 5, then divide by 6.
Why: Using the method “Add 5, then divide by 6.” gives x=6.
6. Solve 3(x+2)=27.
Check answer
Answer: x=7
Hint: Divide by 3 before subtracting 2.
Why: Using the method “Divide by 3 before subtracting 2.” gives x=7.
7. Correct this solution: 4x+3=19, so x=19−3.
Check answer
Answer: The subtraction step gives 4x=16, but multiplication by 4 must also be undone; dividing both sides by 4 gives x=4, checked by 4(4)+3=19.
Hint: Do not forget to undo multiplication by 4.
Why: Equivalent equations preserve equality, so both inverse steps are needed to obtain and verify the solution x=4.
8. A $6 entry fee plus $4 per ride totals $30. How many rides?
Check answer
Answer: Let r be the rides: 6+4r=30, so 4r=24 and r=6 rides; checking gives $6+4($6)=$30.
Hint: Translate fixed and per-ride costs into an equation.
Why: The complete model for “A $6 entry fee plus $4 per ride totals $30. How many rides?” shows Let r be the rides: 6+4r=30, so 4r=24 and r=6 rides; checking gives $6+4($6)=$30.
9. Give an equation with natural-number solution 9 and verify it.
Check answer
Answer: One valid equation is 5x−8=37: adding 8 and dividing by 5 gives x=9, and substitution checks 5(9)−8=37.
Hint: Choose operations, then calculate the matching right side.
Why: The complete model for “Give an equation with natural-number solution 9 and verify it.” shows One valid equation is 5x−8=37: adding 8 and dividing by 5 gives x=9, and substitution checks 5(9)−8=37.
10. Solve 5x+7=42 using equal operations on both sides, then verify your solution by substitution.
Check answer
Answer: Subtract 7: 5x=35. Divide by 5: x=7. Check: 5×7+7=42, so the solution is verified.
Hint: Represent the information first, then calculate, interpret and independently check the result.
Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Curriculum coverage and elaborations
Content description: solve one-variable linear equations with natural number solutions; verify the solution by substitution.
International curriculum mapping
The Australian Curriculum code above is exact. Victorian Year 7, NSW Stage 4, US Grade 7, England Key Stage 3, New Zealand Level 4 and comparable international curricula contain broadly related learning, but code-to-code equivalence varies by jurisdiction.
🎥 Optional Video Lesson
The SkillrHub lesson remains the primary learning resource. This optional video reinforces the explanation; you can complete the lesson and practice without watching.
Before you watch:
- Pause after each worked example.
- Try the examples yourself.
- Return to the SkillrHub lesson before continuing.
Recommended: Solving Two-Step Equations
Math Antics — Undo operations to solve a linear equation and verify the result.
As you watch: Why must the same operation be applied to both sides of an equation?
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Curriculum equivalents: Victoria, NSW and international
Curriculum equivalents for Solve one-variable linear equations with natural number solutions; verify the...
Mapped skill: solve one-variable linear equations with natural number solutions; verify the solution 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.
| Region | Curriculum framework | Closest level or code |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M7A03 · Year 7 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M7A03 · Level 7 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA4-EQU-C-01 + MA4-LIN-C-01 · Stage 4 |
| United States (USA) | Common Core State Standards for Mathematics | Grade 7 |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 7 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 8, Key Stage 3 |
| India | NCERT / CBSE — Mathematics | Class 7 |
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.
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Topic reference: AC9M7A03 — Solving One-variable Linear Equations
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