Equivalent equations and checking a solution
Solve linear equations by applying the same reversible operation to both sides. Check by substituting into the original equation, including negative and fractional solutions.
For learners aged 14+ · free · no account required
Language, curriculum and teaching reference are independent choices. For learners aged 14+: selected linear-equation skills or review. Read the selected programme notes for its level, version and extensions. This is a limited original series, not a full course or an official examination paper.
Brackets and linear equations connect to KS3 algebra; equations and identities connect to KS4. The three solution cases are explicit in Common Core, but are not a verbatim KS3 requirement.
Madrid 3.º ESO: review of first- and second-year linear equations (D.4). Check equivalent operations and substitution manually as well as the final answer. The mapping uses Decreto 65/2022; the mathematics section was also checked against Decreto 59/2024.
France 3e, 2026–27: selected first-degree equation skills and checking a proposed solution. The previous cycle-4 programme and annual expectations still apply; the new programme reaches 3e in 2028–29. Check the reasoning manually.
NRW Gymnasium G9: review of selected year 7/8 skills. Practise reversible operations and check the solution by substitution; review the steps manually. The real-number domain extends the first-stage rational-number scope; the consulted school-plan example introduces real numbers in year 9 (UV9.2).
Japan: review of selected first-year middle-school skills from the 2017 curriculum, fully introduced in 2021. Practise reversible operations and check the solution by substitution; review the steps manually. The real-number domain is an extension beyond the rational-number setting of the first year, for this 14+ series.
Korea: review of selected first-year middle-school skills from the 2022 curriculum, introduced for this year in March 2025. It reaches the third year only in March 2027. Practise reversible operations and check the solution by substitution; review the steps manually. The real-number domain is an extension beyond the rational-number setting of the first year, for this 14+ series.
Russia: mapping to the selected 2025 FRP edition, reviewing year 7 algebra. Practise reversible operations and check the solution by substitution; review the steps manually. The real-number domain extends the year 7 scope with a concept introduced in year 8.
CeDeC / INTEF is the August 2014 edition, used as a historical teaching reference. It is not a companion to the 2022 LOMLOE curriculum.
Sésamath 3e, Magnard, June 2021: selected sections of the student workbook (ISBN 9782210115811). This cross-reference does not cover the whole workbook.
QUA-LiS.NRW, 31 January 2020: a suggested school-plan example, especially UV7.6, page 34. It is a teaching reference, not a textbook or every school’s required sequence.
MEXT, July 2017: official mathematics curriculum commentary, selected first-year sections on pages 70–74. It is a teaching reference, not a publisher textbook companion.
Gyeonggi Office of Education, 10 March 2025: assessment guidance, mathematics chapter Ⅱ-1, pages 58–69. The criteria are a teaching reference; this checker does not assign its achievement levels.
FIPI, Moscow 2026: selected guidance on computation, checking answers, written models and units. It is a year 9 teaching reference, not a textbook or a complete OGE preparation course.
Tokyo Shoseki, 2025, 数学002-72: the publisher’s excerpts and teaching articles for pages 94, 96–97, 78, 101 and 104–108 were consulted. They connect equality-preserving operations, signs and coefficient transformations with modelling and checking the original situation. Classification into 0/1/infinitely many solutions and the real-number domain are MomentDesk extensions; this is a limited cross-reference, not the whole textbook.
Visang, Mathematics 1, first edition 1 March 2025, second printing 1 March 2026, ISBN 979-11-7316-034-9: pages 86–87, 89–91 and 92–93 were read in the publisher’s public e-book. They cover justified equation transformations, substitution, coefficients and a model checked against its situation and units. Classification into 0/1/infinitely many solutions and the real-number domain are MomentDesk extensions. Reasoning still needs manual assessment.
Cornelsen, Fundamente der Mathematik 7, NRW G9, first edition 2020, fifth printing 2026, ISBN 978-3-06-040177-2: the publisher’s open preview covers sections 6.2–6.6, pages 160–175. It explains justified transformations, special cases and modelling with units and interpretation. Its solution sets here use rational numbers; MomentDesk’s real-number domain is an extension. This limited reference keeps the original MomentDesk tasks and manual reasoning assessment.
Read the explanation
An equation asks when two expressions have the same value. Equivalent equations have the same solutions. The goal is to isolate the unknown while keeping that solution set.
Before you start
Signed-number arithmetic, fractions, decimals and the meaning of an unknown.
- Add or subtract the same expression on both sides. Combine like terms without changing their signs.
- Multiply or divide both sides by the same known nonzero number. Multiplying by zero loses information; dividing by zero is undefined.
- Record what you did to both sides. Moving a term is shorthand for adding or subtracting it, not a separate sign-changing rule.
- Substitute your result into the original equation and evaluate both sides. Equal values confirm that the candidate is a solution.
- In these first two topics, the unknown ranges over real numbers. No variable occurs in a denominator.
Worked examples
Subtract 4 from both sides, then divide by 3. Check the original equation.
3x + 4 = 28
3x = 24
x = 8
3(8) + 4 = 28
Divide both sides by −5. The negative sign belongs to the whole coefficient.
−5x = 15
x = −3
−5(−3) = 15
Common mistakes
- Changing one side only can produce an equation with different solutions.
- A negative coefficient does not force a positive answer. Substitute the signed value before accepting it.
Explain your reasoning
Name the operation on both sides, keep signs, and show the original-equation check.
The checker assesses a number or a selected option. It does not grade your reasoning, written equation or units. Use the rubric with your tutor or for self-checking. Ten tasks are a small practice set, not proof of mastery.
Practise
Enter a number without x = or a unit. Exact equal values are accepted: integers, decimals, fractions and mixed numbers. A decimal point or comma is accepted; ambiguous input such as 1,234 needs a choice of meaning. Keep units and reasoning in your written work.
Your answers stay in this page's memory until you reload or leave. No saved draft, account result or automatic homework submission is created.
Your written work and answers
Tasks and explanations
Solve: 4x + 7 = 31. Enter x.
Answer and explanation
6
Subtract 7 from both sides, then divide by 4.
4x = 24
x = 6
4(6) + 7 = 31
Find x: 18 − 3x = 33.
Answer and explanation
−5
Subtract 18, then divide by −3. Check the negative value.
−3x = 15
x = −5
18 − 3(−5) = 33
Solve: 5x − 8 = 2x + 13.
Answer and explanation
7
Subtract 2x from both sides and add 8 to both sides.
3x = 21
x = 7
35 − 8 = 14 + 13
Find x: + 2 = 0.
Answer and explanation
−8
Subtract 2, then multiply both sides by 4.
= −2
x = −8
+ 2 = 0
Solve: 0.6x + 1.4 = 4.1.
Answer and explanation
4.5
Subtract the constant, then divide by the nonzero coefficient. Use the exact decimal value.
0.6x = 2.7
x = 4.5
0.6(4.5) + 1.4 = 4.1
Find x: 3x + 2 = x − 7.
Answer and explanation
−4.5
Subtract x and 2 from both sides, then divide by 2.
2x = −9
x = −4.5
−11.5 = −11.5
Solve: − = 5. An exact fraction or an equal decimal is accepted.
Answer and explanation
−
Multiply by 3, then divide by −2. Keep the sign.
−2x = 15
x = −
−(−) = 5
Check the proposed value in 2x + 5 = −1; x = −2. Which equality is obtained?
- −4 = −1
- −1 = −1
- 1 = −1
- 9 = −1
Answer and explanation
1 = −1
Substitute −2 into the original left side. It equals 1, while the right side is −1, so the proposed value is not a solution.
2(−2) + 5 = 1
1 ≠ −1
For 7x − 9 = 12, which line follows from adding 9 to both sides?
- 7x = 3
- 7x = 21
- 7x − 18 = 3
- 7x − 18 = 21
Answer and explanation
7x = 21
Add 9 to each side; the left constant cancels.
7x − 9 + 9 = 12 + 9
7x = 21
For 2x + 3 = 11, which line is NOT equivalent?
- 3x + 1 = 13
- 3x − 1 = 11
- 4x + 1 = 17
- 4x − 1 = 13
Answer and explanation
4x − 1 = 13
The original equation has solution 4. Substitute it into each option; 4x − 1 = 13 is false at this value and has a different solution.
2x = 8
x = 4
4(4) − 1 = 15
15 ≠ 13
Explain your reasoning
A learner rewrites 6x + 5 = 17 as 6x = 22. Explain the error, solve the original equation and verify the solution.
Answer and explanation
Subtract 5, not add it: 6x = 12, so x = 2. Check: 6(2) + 5 = 17.
The written response should identify the unequal operation, show subtraction on both sides, divide by a nonzero coefficient and substitute into the original equation.
Worksheet
Curriculum source
Korea · first middle-school year review · 2022
[9수02-03] / [9수02-04] · 2026-10-12
Curriculum sourceTextbook / teaching reference: OpenStax · Elementary Algebra 2e
Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis · Edition: 2020-04-22
The linked sections were consulted as a teaching reference. These explanations and tasks are original MomentDesk work. This is a topic cross-reference, not a complete textbook companion.
Textbook / teaching reference: CeDeC / INTEF · Matemáticas 3° E.S.O.
CeDeC / INTEF · Edition: 2014-08
The linked sections were consulted as a teaching reference. These explanations and tasks are original MomentDesk work. This is a topic cross-reference, not a complete textbook companion.
Textbook / teaching reference: Sésamath 3e · Cahier élève
Association Sésamath · Magnard · Edition: 2021-06
The linked sections were consulted as a teaching reference. These explanations and tasks are original MomentDesk work. This is a topic cross-reference, not a complete textbook companion.
Textbook / teaching reference: QUA-LiS.NRW · Beispiel für einen schulinternen Lehrplan
QUA-LiS.NRW · Edition: 2020-01-31
The linked sections were consulted as a teaching reference. These explanations and tasks are original MomentDesk work. This is a topic cross-reference, not a complete textbook companion.
Textbook / teaching reference: 文部科学省 · 中学校学習指導要領解説 数学編
文部科学省 · Edition: 2017-07
The linked sections were consulted as a teaching reference. These explanations and tasks are original MomentDesk work. This is a topic cross-reference, not a complete textbook companion.
Textbook / teaching reference: 경기도교육청 · 성취평가, 이렇게 실천해요! [중학교편]
경기도교육청 · Edition: 2025-03-10
The linked sections were consulted as a teaching reference. These explanations and tasks are original MomentDesk work. This is a topic cross-reference, not a complete textbook companion.
Textbook / teaching reference: ФИПИ · Методические рекомендации · Математика
И. В. Ященко · И. Р. Высоцкий · П. И. Самсонов · Edition: 2026
The linked sections were consulted as a teaching reference. These explanations and tasks are original MomentDesk work. This is a topic cross-reference, not a complete textbook companion.
Textbook / teaching reference: 東京書籍 · 新編 新しい数学 1
清水美憲・真島秀行ほか108名 · Edition: 2025 · 令和7年度
The linked sections were consulted as a teaching reference. These explanations and tasks are original MomentDesk work. This is a topic cross-reference, not a complete textbook companion.
Textbook / teaching reference: Cornelsen · Fundamente der Mathematik 7 · NRW G9
Andreas Pallack (Hrsg.) · Autorenteam · Edition: 2020
The linked sections were consulted as a teaching reference. These explanations and tasks are original MomentDesk work. This is a topic cross-reference, not a complete textbook companion.
Textbook / teaching reference: 비상교육 · 중학교 수학 1
이진호・김선희・이소민・황윤희・이윤경 · Edition: 2025-03-01
The linked sections were consulted as a teaching reference. These explanations and tasks are original MomentDesk work. This is a topic cross-reference, not a complete textbook companion.
Next topic: Brackets and the number of solutions
Worksheet: Equivalent equations and checking a solution
Name the operation on both sides, keep signs, and show the original-equation check.
Solve: 4x + 7 = 31. Enter x.
Equation, working, units and checkAnswer:
Find x: 18 − 3x = 33.
Equation, working, units and checkAnswer:
Solve: 5x − 8 = 2x + 13.
Equation, working, units and checkAnswer:
Find x: + 2 = 0.
Equation, working, units and checkAnswer:
Solve: 0.6x + 1.4 = 4.1.
Equation, working, units and checkAnswer:
Find x: 3x + 2 = x − 7.
Equation, working, units and checkAnswer:
Solve: − = 5. An exact fraction or an equal decimal is accepted.
Equation, working, units and checkAnswer:
Check the proposed value in 2x + 5 = −1; x = −2. Which equality is obtained?
- −4 = −1
- −1 = −1
- 1 = −1
- 9 = −1
Equation, working, units and checkAnswer:
For 7x − 9 = 12, which line follows from adding 9 to both sides?
- 7x = 3
- 7x = 21
- 7x − 18 = 3
- 7x − 18 = 21
Equation, working, units and checkAnswer:
For 2x + 3 = 11, which line is NOT equivalent?
- 3x + 1 = 13
- 3x − 1 = 11
- 4x + 1 = 17
- 4x − 1 = 13
Equation, working, units and checkAnswer:
Answer key: Equivalent equations and checking a solution
6
Subtract 7 from both sides, then divide by 4.
4x = 24
x = 6
4(6) + 7 = 31
−5
Subtract 18, then divide by −3. Check the negative value.
−3x = 15
x = −5
18 − 3(−5) = 33
7
Subtract 2x from both sides and add 8 to both sides.
3x = 21
x = 7
35 − 8 = 14 + 13
−8
Subtract 2, then multiply both sides by 4.
= −2
x = −8
+ 2 = 0
4.5
Subtract the constant, then divide by the nonzero coefficient. Use the exact decimal value.
0.6x = 2.7
x = 4.5
0.6(4.5) + 1.4 = 4.1
−4.5
Subtract x and 2 from both sides, then divide by 2.
2x = −9
x = −4.5
−11.5 = −11.5
−
Multiply by 3, then divide by −2. Keep the sign.
−2x = 15
x = −
−(−) = 5
1 = −1
Substitute −2 into the original left side. It equals 1, while the right side is −1, so the proposed value is not a solution.
2(−2) + 5 = 1
1 ≠ −1
7x = 21
Add 9 to each side; the left constant cancels.
7x − 9 + 9 = 12 + 9
7x = 21
4x − 1 = 13
The original equation has solution 4. Substitute it into each option; 4x − 1 = 13 is false at this value and has a different solution.
2x = 8
x = 4
4(4) − 1 = 15
15 ≠ 13