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Linear equations: transpose and verify

Solve equations with variable terms on both sides, explain each sign change and verify in the original equation. A 40-minute paper-first lesson.

40 minutes · Year 7, upper volume · text pp. 61–62 · foundation lesson 31

Textbook reference: Year 7, upper volume · Chapter 3 §3.2

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher referenceArrange in My Classroom

  1. 4′Check prerequisites
  2. 5′Observe and predict
  3. 8′Worked example and guided attempt
  4. 15′Independent practice
  5. 5′Discuss and correct
  6. 3′Exit questions

Observe · explain

A question before practising

Project 5x − 7 = 2x + 8. Ask the class to describe subtracting 2x on both sides, then adding 7 on both sides. Compare the full operations with the shorter transposition line.

Organise the key idea

Transposition abbreviates adding or subtracting the same term on both sides. A term changes sign when it is moved across the equality sign, not merely when it changes place on the same side.

Open the original chapterOpen the interactive inquiry

Use the lesson question here for paper discussion; the original activity opens separately.

Worked example1

Solve 5x − 7 = 2x + 8. Explain the operation behind each transposition.

5x−7=2x+85x-7=2x+8
Check your work
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Guided attempt2

Solve 4x + 9 = 7x − 6 by collecting the variable terms on the left. Complete each line.

4x+9=7x−64x+9=7x-6
  1. Subtract 7x on both sides: ____ x + 9 = −6.
  2. Subtract 9 on both sides: ____ x = ____.
  3. Divide both sides by ____: x = ____.
  4. Original left side = ____; original right side = ____.
Check your work
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Prerequisite3

For x + 7 = 2, which operation removes +7 while keeping both sides equal? Use it to find x.

x+7=2x+7=2
Check your work
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Prerequisite4

Solve −3x = 12 and check the sign of your solution.

−3x=12-3x=12
Check your work
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Independent practice5

Solve 6x − 5 = 3x + 10.

6x−5=3x+106x-5=3x+10
Check your work
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Independent practice6

Solve 2x + 11 = 5x − 7.

2x+11=5x−72x+11=5x-7
Check your work
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Independent practice7

Solve −4x + 3 = x − 12.

−4x+3=x−12-4x+3=x-12
Check your work
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Independent practice8

Solve 7 − 2x = 3x + 22.

7−2x=3x+227-2x=3x+22
Check your work
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Exit question9

Solve 8x + 1 = 5x − 14 and verify in the original equation.

8x+1=5x−148x+1=5x-14
Check your work
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Exit question10

A learner claims x = 7 solves 3x − 4 = x + 10. Verify by calculating both original sides. Is the claim correct?

3x−4=x+103x-4=x+10
Check your work
  • Calculate both original sides as 17.
  • Conclude that the equality holds at x = 7.
Further practice11

Solve 9x + 4 = 5x − 12.

9x+4=5x−129x+4=5x-12
Check your work
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Further practice12

Solve −2x − 9 = 4x + 3.

−2x−9=4x+3-2x-9=4x+3
Check your work
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Further practice13

Solve 12 − 5x = 2x − 9.

12−5x=2x−912-5x=2x-9
Check your work
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Further practice14

Solve 6x − 11 = −3x + 7.

6x−11=−3x+76x-11=-3x+7
Check your work
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Extension15

Solve 2x + 3x − 8 = 7x − 4x + 6. Collect like terms on each side before transposing.

2x+3x−8=7x−4x+62x+3x-8=7x-4x+6
Check your work
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Extension16

For 6x − 7 = 2x + 9, a learner writes 6x − 2x = 9 − 7 and then 4x = 2. Explain the first error and solve correctly.

6x−7=2x+96x-7=2x+9
Check your work
  • Identify the incorrect sign of the transposed constant and explain adding 7 on both sides.
  • Obtain x = 4 and check both original sides as 17.
Preparation and assessment notes

Prerequisites

Collecting like terms, signed-number operations and applying the same operation to both sides of an equality.

Look for in students’ work

Ask what was done to both sides before accepting a transposition line. Accept different valid orders. Verify using the original equation, not only the last simplified line. Do not introduce bracket removal, denominator clearing or systems.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Textbook reference and authorship

Original practice aligned to the referenced skills. Use your own textbook for the cited pages.

大湾区初中数学三维课堂（七年级上册）正文.pdf

Year 7, upper volume · text pp. 61–62 · foundation lesson 31

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Linear equations: transpose and verify

Year 7, upper volume · text pp. 61–62 · foundation lesson 31 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Solve 5x − 7 = 2x + 8. Explain the operation behind each transposition.

5x−7=2x+85x-7=2x+8
Guided attempt2

Solve 4x + 9 = 7x − 6 by collecting the variable terms on the left. Complete each line.

4x+9=7x−64x+9=7x-6
  1. Subtract 7x on both sides: ____ x + 9 = −6.
  2. Subtract 9 on both sides: ____ x = ____.
  3. Divide both sides by ____: x = ____.
  4. Original left side = ____; original right side = ____.
Prerequisite3

For x + 7 = 2, which operation removes +7 while keeping both sides equal? Use it to find x.

x+7=2x+7=2
Prerequisite4

Solve −3x = 12 and check the sign of your solution.

−3x=12-3x=12
g7-equations-transpose · 2026-10-14-guided-practice-v1 · Conditions and reasons matter; measurement is not proof.

Linear equations: transpose and verify

Year 7, upper volume · text pp. 61–62 · foundation lesson 31 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Solve 6x − 5 = 3x + 10.

6x−5=3x+106x-5=3x+10
Independent practice6

Solve 2x + 11 = 5x − 7.

2x+11=5x−72x+11=5x-7
Independent practice7

Solve −4x + 3 = x − 12.

−4x+3=x−12-4x+3=x-12
Independent practice8

Solve 7 − 2x = 3x + 22.

7−2x=3x+227-2x=3x+22
Exit question9

Solve 8x + 1 = 5x − 14 and verify in the original equation.

8x+1=5x−148x+1=5x-14
Exit question10

A learner claims x = 7 solves 3x − 4 = x + 10. Verify by calculating both original sides. Is the claim correct?

3x−4=x+103x-4=x+10
g7-equations-transpose · 2026-10-14-guided-practice-v1 · Conditions and reasons matter; measurement is not proof.

Linear equations: transpose and verify — Further practice

Year 7, upper volume · text pp. 61–62 · foundation lesson 31

Further practice11

Solve 9x + 4 = 5x − 12.

9x+4=5x−129x+4=5x-12
Further practice12

Solve −2x − 9 = 4x + 3.

−2x−9=4x+3-2x-9=4x+3
Further practice13

Solve 12 − 5x = 2x − 9.

12−5x=2x−912-5x=2x-9
Further practice14

Solve 6x − 11 = −3x + 7.

6x−11=−3x+76x-11=-3x+7
Extension15

Solve 2x + 3x − 8 = 7x − 4x + 6. Collect like terms on each side before transposing.

2x+3x−8=7x−4x+62x+3x-8=7x-4x+6
Extension16

For 6x − 7 = 2x + 9, a learner writes 6x − 2x = 9 − 7 and then 4x = 2. Explain the first error and solve correctly.

6x−7=2x+96x-7=2x+9

Linear equations: transpose and verify — Teacher answers

Year 7, upper volume · text pp. 61–62 · foundation lesson 31

Ask what was done to both sides before accepting a transposition line. Accept different valid orders. Verify using the original equation, not only the last simplified line. Do not introduce bracket removal, denominator clearing or systems.

Worked example1

Solve 5x − 7 = 2x + 8. Explain the operation behind each transposition.

5x−7=2x+85x-7=2x+8
Reference answer

x = 5.

  1. Subtract 2x from both sides: 3x − 7 = 8.
  2. Add 7 to both sides: 3x = 15. The short line is 5x − 2x = 8 + 7.
  3. Divide both sides by 3: x = 5.
  4. Check: left side 5 × 5 − 7 = 18; right side 2 × 5 + 8 = 18.
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Guided attempt2

Solve 4x + 9 = 7x − 6 by collecting the variable terms on the left. Complete each line.

4x+9=7x−64x+9=7x-6
  1. Subtract 7x on both sides: ____ x + 9 = −6.
  2. Subtract 9 on both sides: ____ x = ____.
  3. Divide both sides by ____: x = ____.
  4. Original left side = ____; original right side = ____.
Reference answer

x = 5.

  1. Subtract 7x: −3x + 9 = −6.
  2. Subtract 9: −3x = −15. Divide by −3: x = 5.
  3. Check: 4 × 5 + 9 = 29 and 7 × 5 − 6 = 29.
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Prerequisite3

For x + 7 = 2, which operation removes +7 while keeping both sides equal? Use it to find x.

x+7=2x+7=2
Reference answer

Subtract 7 from both sides, giving x = 2 − 7 = −5.

  1. Subtract 7 from both sides: x = 2 − 7 = −5.
  2. Check: −5 + 7 = 2.
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Prerequisite4

Solve −3x = 12 and check the sign of your solution.

−3x=12-3x=12
Reference answer

x = -4.

  1. Divide both sides by −3: x = −4.
  2. Check: −3 × (−4) = 12.
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Independent practice5

Solve 6x − 5 = 3x + 10.

6x−5=3x+106x-5=3x+10
Reference answer

x = 5.

  1. Subtract 3x and add 5 on both sides: 6x − 3x = 10 + 5.
  2. 3x = 15, so x = 5.
  3. Check: 6 × 5 − 5 = 25 and 3 × 5 + 10 = 25.
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Independent practice6

Solve 2x + 11 = 5x − 7.

2x+11=5x−72x+11=5x-7
Reference answer

x = 6.

  1. Subtract 5x and 11 on both sides: 2x − 5x = −7 − 11.
  2. −3x = −18. Divide by −3: x = 6.
  3. Check: 2 × 6 + 11 = 23 and 5 × 6 − 7 = 23.
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Independent practice7

Solve −4x + 3 = x − 12.

−4x+3=x−12-4x+3=x-12
Reference answer

x = 3.

  1. Subtract x and 3 on both sides: −4x − x = −12 − 3.
  2. −5x = −15. Divide by −5: x = 3.
  3. Check: −4 × 3 + 3 = −9 and 3 − 12 = −9.
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Independent practice8

Solve 7 − 2x = 3x + 22.

7−2x=3x+227-2x=3x+22
Reference answer

x = -3.

  1. Subtract 3x and 7 on both sides: −2x − 3x = 22 − 7.
  2. −5x = 15. Divide by −5: x = −3.
  3. Check: 7 − 2 × (−3) = 13 and 3 × (−3) + 22 = 13.
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Exit question9

Solve 8x + 1 = 5x − 14 and verify in the original equation.

8x+1=5x−148x+1=5x-14
Reference answer

x = -5.

  1. Subtract 5x and 1 on both sides: 3x = −15.
  2. Divide by 3: x = −5.
  3. Check: 8 × (−5) + 1 = −39 and 5 × (−5) − 14 = −39.
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Exit question10

A learner claims x = 7 solves 3x − 4 = x + 10. Verify by calculating both original sides. Is the claim correct?

3x−4=x+103x-4=x+10
Reference answer

Yes. At x = 7, the left side is 3 × 7 − 4 = 17 and the right side is 7 + 10 = 17.

  1. Use the original equation, including both constants.
  2. Left side: 3 × 7 − 4 = 17. Right side: 7 + 10 = 17.
  3. The two sides are equal, so x = 7 is a solution.
  • Calculate both original sides as 17.
  • Conclude that the equality holds at x = 7.
Further practice11

Solve 9x + 4 = 5x − 12.

9x+4=5x−129x+4=5x-12
Reference answer

x = -4.

  1. Subtract 5x and 4 on both sides: 4x = −16.
  2. Divide by 4: x = −4.
  3. Check: 9 × (−4) + 4 = −32 and 5 × (−4) − 12 = −32.
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Further practice12

Solve −2x − 9 = 4x + 3.

−2x−9=4x+3-2x-9=4x+3
Reference answer

x = -2.

  1. Subtract 4x and add 9 on both sides: −6x = 12.
  2. Divide by −6: x = −2.
  3. Check: −2 × (−2) − 9 = −5 and 4 × (−2) + 3 = −5.
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Further practice13

Solve 12 − 5x = 2x − 9.

12−5x=2x−912-5x=2x-9
Reference answer

x = 3.

  1. Subtract 2x and 12 on both sides: −7x = −21.
  2. Divide by −7: x = 3.
  3. Check: 12 − 5 × 3 = −3 and 2 × 3 − 9 = −3.
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Further practice14

Solve 6x − 11 = −3x + 7.

6x−11=−3x+76x-11=-3x+7
Reference answer

x = 2.

  1. Add 3x and 11 on both sides: 9x = 18.
  2. Divide by 9: x = 2.
  3. Check: 6 × 2 − 11 = 1 and −3 × 2 + 7 = 1.
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Extension15

Solve 2x + 3x − 8 = 7x − 4x + 6. Collect like terms on each side before transposing.

2x+3x−8=7x−4x+62x+3x-8=7x-4x+6
Reference answer

x = 7.

  1. Collect on each side: 5x − 8 = 3x + 6.
  2. Subtract 3x and add 8 on both sides: 2x = 14.
  3. Divide by 2: x = 7.
  4. Check: 2 × 7 + 3 × 7 − 8 = 27 and 7 × 7 − 4 × 7 + 6 = 27.
  • Combine only like terms and keep both sides equal.
  • Find the stated value and check it in the original equation.
Extension16

For 6x − 7 = 2x + 9, a learner writes 6x − 2x = 9 − 7 and then 4x = 2. Explain the first error and solve correctly.

6x−7=2x+96x-7=2x+9
Reference answer

Moving −7 to the right requires +7, corresponding to adding 7 on both sides. Thus 4x = 16 and x = 4; both original sides equal 17.

  1. The wrong line keeps −7 after it crosses the equality sign.
  2. Subtract 2x and add 7 on both sides: 6x − 2x = 9 + 7.
  3. 4x = 16, so x = 4.
  4. Check: 6 × 4 − 7 = 17 and 2 × 4 + 9 = 17.
  • Identify the incorrect sign of the transposed constant and explain adding 7 on both sides.
  • Obtain x = 4 and check both original sides as 17.