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Equations: meaning, solutions and equal operations

Distinguish an expression from an equation in an unknown, test a solution and explain operations on both sides.

40 minutes · Year 7, upper volume · text pp. 55–58 · foundation lessons 28–29

Textbook reference: Year 7, upper volume · Chapter 3 · Linear equations

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the upper-volume plan

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

Test x = 4 in 3x + 2 = 14. Then change only one side and ask whether the same value still solves the new equation.

Organise the key idea

A solution makes the original sides equal. Adding or subtracting the same value, or multiplying or dividing by a nonzero value, preserves the solution set.

Open the original chapterOpen the interactive inquiry

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

Worked example1

Is x = 4 a solution of the equation? Calculate both original sides to justify your answer.

3x+2=143x+2=14
Check your work
  • Use the original equality and apply each operation to both sides.
  • Obtain the solution and check it in the original equation or situation.
Guided attempt2

Three identical notebooks and a single 2-dollar packing fee cost 17 dollars. Let x be the price of one notebook. Write an equation and find x.

  1. Notebook cost = ____; total equation: ____ = 17.
  2. After subtracting 2: ____ x = ____.
  3. x = ____ dollars; check: ____.
Check your work
  • Use the original equality and apply each operation to both sides.
  • Obtain the solution and check it in the original equation or situation.
Prerequisite3

Is 2x + 3, by itself, an equation in an unknown? Identify the missing feature.

Check your work
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Prerequisite4

Is 7 + 5 = 12 an equation containing an unknown, as required by this lesson’s definition? Explain.

Check your work
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Independent practice5

Solve, then verify by substitution.

2x−3=72x-3=7
Check your work
  • Use the original equality and apply each operation to both sides.
  • Obtain the solution and check it in the original equation or situation.
Independent practice6

From x + 6 = 10, a learner subtracts 6 from both sides and writes x = 4. Does this preserve the solutions? Explain.

Check your work
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Independent practice7

From x/3 = 4, a learner multiplies both sides by 3 and obtains x = 12. Is this equivalent?

Check your work
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Independent practice8

From x + 3 = 8, a learner subtracts 3 only on the left and writes x = 8. Explain why this does not preserve the equation.

Check your work
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Exit question9

A number plus 7 equals 19. Let the number be x, write an equation and find it.

Check your work
  • Use the original equality and apply each operation to both sides.
  • Obtain the solution and check it in the original equation or situation.
Exit question10

Verify whether x = −2 solves the equation. Show the original left and right sides.

4x+1=−74x+1=-7
Check your work
  • Use the original equality and apply each operation to both sides.
  • Obtain the solution and check it in the original equation or situation.
Further practice11

Does 5x + 2 = 17 fit the definition of an equation in an unknown? Identify both required features.

Check your work
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Further practice12

From x − 4 = 9, add 4 to both sides. Is the result x = 13 equivalent?

Check your work
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Further practice13

From −2x = 10, divide both sides by −2. Explain why x = −5 is equivalent.

Check your work
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Further practice14

Multiplying x = 3 by 0 gives 0 = 0. Do these two equations have the same solutions? Give a concrete test value.

Check your work
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Extension15

Does every real value of x solve x + x = 2x? Explain without choosing just one test value.

x+x=2xx+x=2x
Check your work
  • Explain equality of the two expressions for every real value, not just a numerical trial.
Extension16

Can 2x + 1 = 2x + 4 have a solution? Apply the same subtraction on both sides and explain.

2x+1=2x+42x+1=2x+4
Check your work
  • Derive the contradiction and conclude no solution.
Preparation and assessment notes

Prerequisites

Signed arithmetic, substitution and the meaning of an equality sign.

Look for in students’ work

Use the prerequisite checks to choose support, project the worked and guided items, then let learners work on paper. Select supplements or extensions as needed. No learner answers are saved by this presentation.

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

A 40-minute selection of original core practice after studying the cited textbook lessons. It is not a replacement for every lesson in the chapter.

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. 55–58 · foundation lessons 28–29

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

Equations: meaning, solutions and equal operations

Year 7, upper volume · text pp. 55–58 · foundation lessons 28–29 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Is x = 4 a solution of the equation? Calculate both original sides to justify your answer.

3x+2=143x+2=14
Guided attempt2

Three identical notebooks and a single 2-dollar packing fee cost 17 dollars. Let x be the price of one notebook. Write an equation and find x.

  1. Notebook cost = ____; total equation: ____ = 17.
  2. After subtracting 2: ____ x = ____.
  3. x = ____ dollars; check: ____.
Prerequisite3

Is 2x + 3, by itself, an equation in an unknown? Identify the missing feature.

Prerequisite4

Is 7 + 5 = 12 an equation containing an unknown, as required by this lesson’s definition? Explain.

g7-equations-meaning · 2026-10-14-upper-core-v1 · Conditions and reasons matter; measurement is not proof.

Equations: meaning, solutions and equal operations

Year 7, upper volume · text pp. 55–58 · foundation lessons 28–29 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Solve, then verify by substitution.

2x−3=72x-3=7
Independent practice6

From x + 6 = 10, a learner subtracts 6 from both sides and writes x = 4. Does this preserve the solutions? Explain.

Independent practice7

From x/3 = 4, a learner multiplies both sides by 3 and obtains x = 12. Is this equivalent?

Independent practice8

From x + 3 = 8, a learner subtracts 3 only on the left and writes x = 8. Explain why this does not preserve the equation.

Exit question9

A number plus 7 equals 19. Let the number be x, write an equation and find it.

Exit question10

Verify whether x = −2 solves the equation. Show the original left and right sides.

4x+1=−74x+1=-7
g7-equations-meaning · 2026-10-14-upper-core-v1 · Conditions and reasons matter; measurement is not proof.

Equations: meaning, solutions and equal operations — Further practice

Year 7, upper volume · text pp. 55–58 · foundation lessons 28–29

Further practice11

Does 5x + 2 = 17 fit the definition of an equation in an unknown? Identify both required features.

Further practice12

From x − 4 = 9, add 4 to both sides. Is the result x = 13 equivalent?

Further practice13

From −2x = 10, divide both sides by −2. Explain why x = −5 is equivalent.

Further practice14

Multiplying x = 3 by 0 gives 0 = 0. Do these two equations have the same solutions? Give a concrete test value.

Extension15

Does every real value of x solve x + x = 2x? Explain without choosing just one test value.

x+x=2xx+x=2x
Extension16

Can 2x + 1 = 2x + 4 have a solution? Apply the same subtraction on both sides and explain.

2x+1=2x+42x+1=2x+4

Equations: meaning, solutions and equal operations — Teacher answers

Year 7, upper volume · text pp. 55–58 · foundation lessons 28–29

Use the prerequisite checks to choose support, project the worked and guided items, then let learners work on paper. Select supplements or extensions as needed. No learner answers are saved by this presentation.

Worked example1

Is x = 4 a solution of the equation? Calculate both original sides to justify your answer.

3x+2=143x+2=14
Reference answer

Yes. The left side is 3 × 4 + 2 = 14 and the right side is 14.

x=4x=4
  1. Substitute the proposed value into the entire left side.
  2. Both sides equal 14, so the equality holds.
  • Use the original equality and apply each operation to both sides.
  • Obtain the solution and check it in the original equation or situation.
Guided attempt2

Three identical notebooks and a single 2-dollar packing fee cost 17 dollars. Let x be the price of one notebook. Write an equation and find x.

  1. Notebook cost = ____; total equation: ____ = 17.
  2. After subtracting 2: ____ x = ____.
  3. x = ____ dollars; check: ____.
Reference answer

3x + 2 = 17; x = 5 dollars.

x=5x=5
  1. The repeated cost is 3x; add the fixed fee once: 3x + 2 = 17.
  2. Subtract 2 from both sides: 3x = 15. Divide by 3: x = 5.
  3. Check: 3 × 5 + 2 = 17.
  • Use the original equality and apply each operation to both sides.
  • Obtain the solution and check it in the original equation or situation.
Prerequisite3

Is 2x + 3, by itself, an equation in an unknown? Identify the missing feature.

Reference answer

No. It is an expression; there is no equality sign joining two sides.

  1. The letter x is present.
  2. An equation also requires an equality, which is absent here.
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Prerequisite4

Is 7 + 5 = 12 an equation containing an unknown, as required by this lesson’s definition? Explain.

Reference answer

It is a true equality, but it contains no unknown, so it is not an equation in an unknown in this lesson.

  1. The equality sign is present and the two sides agree.
  2. No unknown quantity appears.
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Independent practice5

Solve, then verify by substitution.

2x−3=72x-3=7
Reference answer

x = 5.

x=5x=5
  1. Add 3 on both sides: 2x = 10.
  2. Divide by 2: x = 5; check 2 × 5 − 3 = 7.
  • Use the original equality and apply each operation to both sides.
  • Obtain the solution and check it in the original equation or situation.
Independent practice6

From x + 6 = 10, a learner subtracts 6 from both sides and writes x = 4. Does this preserve the solutions? Explain.

Reference answer

Yes. Both sides have the same subtraction, and both equations have solution x = 4.

  1. (x + 6) − 6 = 10 − 6.
  2. Simplifying gives x = 4.
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Independent practice7

From x/3 = 4, a learner multiplies both sides by 3 and obtains x = 12. Is this equivalent?

Reference answer

Yes. Multiplication by the same nonzero number 3 preserves the solution x = 12.

  1. Multiply the complete left and right sides by 3.
  2. The left becomes x and the right becomes 12.
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Independent practice8

From x + 3 = 8, a learner subtracts 3 only on the left and writes x = 8. Explain why this does not preserve the equation.

Reference answer

The right side must also lose 3. The correct result is x = 5. At x = 8, the original left side is 11, not 8.

  1. The valid operation is (x + 3) − 3 = 8 − 3.
  2. It gives x = 5. The proposed x = 8 fails the original equation.
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Exit question9

A number plus 7 equals 19. Let the number be x, write an equation and find it.

Reference answer

x + 7 = 19, so x = 12.

x=12x=12
  1. Translate the sentence directly: x + 7 = 19.
  2. Subtract 7 on both sides: x = 12. Check 12 + 7 = 19.
  • Use the original equality and apply each operation to both sides.
  • Obtain the solution and check it in the original equation or situation.
Exit question10

Verify whether x = −2 solves the equation. Show the original left and right sides.

4x+1=−74x+1=-7
Reference answer

Yes: 4 × (−2) + 1 = −7, equal to the right side.

x=−2x=-2
  1. Substitute −2 with its sign: 4 × (−2) + 1.
  2. The left side is −8 + 1 = −7.
  • Use the original equality and apply each operation to both sides.
  • Obtain the solution and check it in the original equation or situation.
Further practice11

Does 5x + 2 = 17 fit the definition of an equation in an unknown? Identify both required features.

Reference answer

Yes. It has an equality sign and an unknown x.

  1. The equality sign joins two expressions.
  2. The left side contains unknown x.
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Further practice12

From x − 4 = 9, add 4 to both sides. Is the result x = 13 equivalent?

Reference answer

Yes. The same addition removes −4 from the left and gives 13 on the right.

  1. (x − 4) + 4 = 9 + 4.
  2. Both equations have solution x = 13.
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Further practice13

From −2x = 10, divide both sides by −2. Explain why x = −5 is equivalent.

Reference answer

−2 is nonzero. Dividing both complete sides by −2 gives x = −5 with the same solution.

  1. The left coefficient becomes 1.
  2. 10 ÷ (−2) = −5; substitution gives 10.
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Further practice14

Multiplying x = 3 by 0 gives 0 = 0. Do these two equations have the same solutions? Give a concrete test value.

Reference answer

No. For example, x = 4 satisfies 0 = 0 but not x = 3. Multiplying by zero loses the original restriction.

  1. The original equation allows only x = 3.
  2. The new identity is true for every x, including 4.
  • State the applicable definition or condition accurately.
  • Support the conclusion with the given numbers, a check or a concrete counterexample.
Extension15

Does every real value of x solve x + x = 2x? Explain without choosing just one test value.

x+x=2xx+x=2x
Reference answer

Yes. Combining the identical terms gives 2x = 2x for every real x.

  1. The left side is 2x by the definition of adding like terms.
  2. Both sides are the same expression for every real value.
  • Explain equality of the two expressions for every real value, not just a numerical trial.
Extension16

Can 2x + 1 = 2x + 4 have a solution? Apply the same subtraction on both sides and explain.

2x+1=2x+42x+1=2x+4
Reference answer

No solution: subtracting 2x on both sides would require 1 = 4.

  1. Subtract 2x on both sides.
  2. The false statement 1 = 4 cannot be repaired by choosing x.
  • Derive the contradiction and conclude no solution.