Year 7 Mathematics · AC9M7A03

Solving One-variable Linear Equations

We are learning to solve one-variable linear equations with natural number solutions and verify the solution by substitution

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Learning goalsSay it simply

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 conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Solve 3x+5=26

  1. Subtract 5 from both sides.
  2. Simplify to 3x=21.
  3. Divide both sides by 3.
  4. Substitute the solution into the original equation.

Final answer: x=7

Check: 3(7)+5=26.

Example 2

Solve 4(x−2)=24

  1. Divide both sides by 4.
  2. Simplify to x−2=6.
  3. Add 2 to both sides.
  4. 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.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: Subtract 7: 5x=35. Divide by 5: x=7. Check: 5×7+7=42, so the solution is verified.
  3. 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.

  1. Plan: Choose operations, then calculate the matching right side.
  2. Work: One valid equation is 5x−8=37: adding 8 and dividing by 5 gives x=9, and substitution checks 5(9)−8=37.
  3. 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.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7A03 - Solving One-variable Linear Equations
Example 1

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.

Example 2

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

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

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.

Curriculum examplesCopied content

Content description: solve one-variable linear equations with natural number solutions; verify the solution by substitution.

Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.

Practice and reviewReady for practice

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.

Curriculum alignmentStart here

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.
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