🔮 Algebra Realm · Equations

Unknowns on both sides

Unknowns on both sides focuses on how to collect variable terms onto one side before isolating the unknown. In this lesson, focus on an equation is a balanced statement.

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Understand Unknowns on both sides

Unknowns on both sides focuses on how to collect variable terms onto one side before isolating the unknown. In this lesson, focus on an equation is a balanced statement.

An equation is a balanced statement. When the unknown appears on both sides, collect variable terms on one side before isolating it. For unknowns on both sides, the final written answer should make that exact relationship visible rather than hiding it inside an unexplained result.

Start here

Unknowns on both sides: An equation is a balanced statement. Substitute the proposed solution into the original equation and verify that both sides match. Keep the unknowns on both sides representation visible until the final line.

Picture the idea

Move algebra tiles across a balance scale and record each legal inverse operation as a solver step. Use the model to explain one change you notice while working on unknowns on both sides.

Check as you go

Substitute the proposed solution into the original equation and verify that both sides match. Write that check beside the final unknowns on both sides answer.

Key vocabulary

equationinverse operationbalanceunknownsolutionunknownsboth

Rules and key facts

5x + 2 = 3x + 12 gives 2x + 2 = 12, so x = 5.

  • Simplify either side if needed.
  • Choose an inverse operation that removes one obstacle around the unknown.
  • Apply the same operation to both sides.
  • Continue until the unknown is isolated, then substitute to check. Record the check explicitly for unknowns on both sides.

Step-by-step method

  1. Simplify either side if needed.
  2. Choose an inverse operation that removes one obstacle around the unknown.
  3. Apply the same operation to both sides.
  4. Continue until the unknown is isolated, then substitute to check. Record the check explicitly for unknowns on both sides.

What you need first

  • Recognise the vocabulary: equation, inverse operation, balance.
  • Be able to explain the purpose of unknowns on both sides before calculating.
  • Keep the relevant values, units and representation visible while you work.

Real-world use

  • Unknown prices
  • Formula calculations

Visual / interactive

See the idea, then move it around

Skip to Practice

Move algebra tiles across a balance scale and record each legal inverse operation as a solver step. Use the model to explain one change you notice while working on unknowns on both sides.

Interactive maths model Connected to this topic; move controls, check outputs, then earn XP only from verified actions.
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Worked examples

Examples, methods and exam thinking

Level 1 · Foundation

Understand the idea with small numbers, one representation and one clear step.

Level 2 · Secure

Use the standard Year 8 method with mixed examples and normal wording.

Level 3 · Challenge

Handle multi-step or less familiar questions and explain choices.

Level 4 · Exam-style

Solve a worded question, show reasoning, check accuracy and write a final sentence.

Foundation example

Build confidence

Given information: Unknowns on both sides — Solve 5x + 2 = 3x + 6. Method choice: use the unknowns on both sides method and show each step with the stated values. Calculation or reasoning: Subtract 3x: 2x + 2 = 6. Subtract 2, then divide by 2: x = 2. Final answer: 2. Check: substitute or compare with the original information to confirm the result fits the question.

  1. Simplify either side if needed.
  2. Choose an inverse operation that removes one obstacle around the unknown.
  3. Apply the same operation to both sides.
Secure example

Use the normal method

Given information: Unknowns on both sides — Solve 5x + 14 = 3x + 34. Method choice: use the unknowns on both sides method and show each step with the stated values. Calculation or reasoning: Subtract 3x: 2x + 14 = 34. Subtract 14, then divide by 2: x = 10. Final answer: 10. Check: substitute or compare with the original information to confirm the result fits the question.

Check: Divide by the remaining coefficient and check both original sides.

Challenge example

Stretch the idea

Given information: Unknowns on both sides — Solve 5x + 13 = 3x + 27. Method choice: use the unknowns on both sides method and show each step with the stated values. Calculation or reasoning: Subtract 3x: 2x + 13 = 27. Subtract 13, then divide by 2: x = 7. Final answer: 7. Check: substitute or compare with the original information to confirm the result fits the question.

Try explaining why each step works before checking the answer.

Exam-style example

Show your reasoning

Given information: Unknowns on both sides — Solve 5x + 12 = 3x + 20. Method choice: use the unknowns on both sides method and show each step with the stated values. Calculation or reasoning: Subtract 3x: 2x + 12 = 20. Subtract 12, then divide by 2: x = 4. Final answer: 4. Check: substitute or compare with the original information to confirm the result fits the question.

Exam tip: Choose the subtraction that leaves a positive x coefficient when possible.

Common mistakes

  • Changing one side only. This is a key trap when answering unknowns on both sides questions.
  • Skipping a line and losing a negative sign or denominator.

How to check your answer

Substitute the proposed solution into the original equation and verify that both sides match. Write that check beside the final unknowns on both sides answer.

Extension challenge

Create a unknowns on both sides problem with a tempting incorrect answer. Solve it, apply the check, and explain exactly where the incorrect method breaks down.

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Unknowns on both sides challenge

Use balance battle controls to solve three checked unknowns on both sides rounds. Solve at least two of three marked rounds and use feedback to correct any error.

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Challenge Unknowns on both sides Arcane Boss

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Unknowns on both sides Arcane Boss

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Study cards

Core idea

Unknowns on both sides: An equation is a balanced statement. Substitute the proposed solution into the original equation and verify that both sides match. Keep the unknowns on both sides representation visible until the final line.

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Key vocabulary

equation · inverse operation · balance · unknown · solution · unknowns · both

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Rules

Simplify either side if needed. Choose an inverse operation that removes one obstacle around the unknown. Apply the same operation to both sides. Continue until the unknown is isolated, then substitute to check. Record the check explicitly for unknowns on both sides.

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Formula / fact

5x + 2 = 3x + 12 gives 2x + 2 = 12, so x = 5.

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Foundation example

Given information: Unknowns on both sides — Solve 5x + 2 = 3x + 6. Method choice: use the unknowns on both sides method and show each step with the stated values. Calculation or reasoning: Subtract 3x: 2x + 2 = 6. Subtract 2, then divide by 2: x = 2. Final answer: 2. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Secure example

Given information: Unknowns on both sides — Solve 5x + 14 = 3x + 34. Method choice: use the unknowns on both sides method and show each step with the stated values. Calculation or reasoning: Subtract 3x: 2x + 14 = 34. Subtract 14, then divide by 2: x = 10. Final answer: 10. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Challenge example

Given information: Unknowns on both sides — Solve 5x + 13 = 3x + 27. Method choice: use the unknowns on both sides method and show each step with the stated values. Calculation or reasoning: Subtract 3x: 2x + 13 = 27. Subtract 13, then divide by 2: x = 7. Final answer: 7. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Exam-style example

Given information: Unknowns on both sides — Solve 5x + 12 = 3x + 20. Method choice: use the unknowns on both sides method and show each step with the stated values. Calculation or reasoning: Subtract 3x: 2x + 12 = 20. Subtract 12, then divide by 2: x = 4. Final answer: 4. Check: substitute or compare with the original information to confirm the result fits the question.

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Common mistake

Changing one side only. This is a key trap when answering unknowns on both sides questions.

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Exam tip

For unknowns on both sides, show the key representation before the final calculation. Use this final check: Substitute the proposed solution into the original equation and verify that both sides match.

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Real-world use

Unknown prices, Formula calculations

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Checklist

I can explain unknowns on both sides, use the method, check for mistakes, and answer an exam-style question.

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Flashcards

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Help for Unknowns on both sides

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Simple explanation

Unknowns on both sides: An equation is a balanced statement. Substitute the proposed solution into the original equation and verify that both sides match. Keep the unknowns on both sides representation visible until the final line.

Think of unknowns on both sides as a careful model: make the important values visible, change one thing at a time, and use the check to prove the answer fits.

Step-by-step breakdown

  1. Simplify either side if needed.
  2. Choose an inverse operation that removes one obstacle around the unknown.
  3. Apply the same operation to both sides.
  4. Continue until the unknown is isolated, then substitute to check. Record the check explicitly for unknowns on both sides.

Hint 1

Start by naming the given information and the exact result required for unknowns on both sides.

Hint 2

Simplify either side if needed.

Full worked solution

Given information: Unknowns on both sides — Solve 5x + 2 = 3x + 6. Method choice: use the unknowns on both sides method and show each step with the stated values. Calculation or reasoning: Subtract 3x: 2x + 2 = 6. Subtract 2, then divide by 2: x = 2. Final answer: 2. Check: substitute or compare with the original information to confirm the result fits the question.

Method: Simplify either side if needed. → Choose an inverse operation that removes one obstacle around the unknown. → Apply the same operation to both sides. → Continue until the unknown is isolated, then substitute to check. Record the check explicitly for unknowns on both sides.

Common mistake warning

Changing one side only. This is a key trap when answering unknowns on both sides questions.

Choose a support button above when you need a nudge.

Mastery milestones

Badges reward learning, not locked clicking

  • I can explain unknowns on both sides in my own words.
  • I can use these words accurately: equation, inverse operation, balance.
  • I can follow the 4-step method without guessing.
  • I can avoid this mistake: Changing one side only.
  • I can apply this check: Substitute the proposed solution into the original equation and verify that both sides match.
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