🔮 Algebra Realm · Equations

Solving by substitution

Solving by substitution focuses on how to test candidate values efficiently by replacing a variable everywhere it appears. In this lesson, focus on substitution replaces a letter with a known value.

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Understand Solving by substitution

Solving by substitution focuses on how to test candidate values efficiently by replacing a variable everywhere it appears. In this lesson, focus on substitution replaces a letter with a known value.

Substitution replaces a letter with a known value. Brackets protect negative or fractional substitutions, and evaluating both sides independently checks whether a proposed solution is valid. For solving by substitution, the final written answer should make that exact relationship visible rather than hiding it inside an unexplained result.

Start here

Solving by substitution: Substitution replaces a letter with a known value. Repeat the evaluation with explicit brackets and inspect each intermediate value. Keep the solving by substitution representation visible until the final line.

Picture the idea

Drop a value into each variable slot and watch a calculation tree evaluate one operation at a time. Use the model to explain one change you notice while working on solving by substitution.

Check as you go

Repeat the evaluation with explicit brackets and inspect each intermediate value. Write that check beside the final solving by substitution answer.

Key vocabulary

substituteevaluateexpressionsolutionchecksolvingby

Rules and key facts

For x = 3, 2x + 1 = 2(3) + 1 = 7.

  • Write the expression with the replacement in brackets.
  • Follow the operation order carefully.
  • Evaluate each side independently if checking an equation.
  • Compare the results and state the conclusion. Record the check explicitly for solving by substitution.

Step-by-step method

  1. Write the expression with the replacement in brackets.
  2. Follow the operation order carefully.
  3. Evaluate each side independently if checking an equation.
  4. Compare the results and state the conclusion. Record the check explicitly for solving by substitution.

What you need first

  • Recognise the vocabulary: substitute, evaluate, expression.
  • Be able to explain the purpose of solving by substitution before calculating.
  • Keep the relevant values, units and representation visible while you work.

Real-world use

  • Formula use
  • Checking models

Visual / interactive

See the idea, then move it around

Skip to Practice

Drop a value into each variable slot and watch a calculation tree evaluate one operation at a time. Use the model to explain one change you notice while working on solving by substitution.

Interactive maths model Connected to this topic; move controls, check outputs, then earn XP only from verified actions.
Responsive · validated · topic linked

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: Solving by substitution — Find 3x + 2 when x = 2. Method choice: use the solving by substitution method and show each step with the stated values. Calculation or reasoning: Substitute x = 2: 3 × 2 + 2 = 8. Final answer: 8. Check: substitute or compare with the original information to confirm the result fits the question.

  1. Write the expression with the replacement in brackets.
  2. Follow the operation order carefully.
  3. Evaluate each side independently if checking an equation.
Secure example

Use the normal method

Given information: Solving by substitution — Find 3x + 14 when x = 10. Method choice: use the solving by substitution method and show each step with the stated values. Calculation or reasoning: Substitute x = 10: 3 × 10 + 14 = 44. Final answer: 44. Check: substitute or compare with the original information to confirm the result fits the question.

Check: Compare the resulting values or check whether the equation is true.

Challenge example

Stretch the idea

Given information: Solving by substitution — Find 3x + 13 when x = 7. Method choice: use the solving by substitution method and show each step with the stated values. Calculation or reasoning: Substitute x = 7: 3 × 7 + 13 = 34. Final answer: 34. 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: Solving by substitution — Find 3x + 12 when x = 4. Method choice: use the solving by substitution method and show each step with the stated values. Calculation or reasoning: Substitute x = 4: 3 × 4 + 12 = 24. Final answer: 24. Check: substitute or compare with the original information to confirm the result fits the question.

Exam tip: Put negative substitutions in brackets every time.

Common mistakes

  • Dropping brackets around negative values. This is a key trap when answering solving by substitution questions.
  • Assuming a proposed answer is correct before evaluating both sides.

How to check your answer

Repeat the evaluation with explicit brackets and inspect each intermediate value. Write that check beside the final solving by substitution answer.

Extension challenge

Create a solving by substitution problem with a tempting incorrect answer. Solve it, apply the check, and explain exactly where the incorrect method breaks down.

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Solving by substitution challenge

Use substitution lock controls to solve three checked solving by substitution rounds. Solve at least two of three marked rounds and use feedback to correct any error.

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Study cards

Core idea

Solving by substitution: Substitution replaces a letter with a known value. Repeat the evaluation with explicit brackets and inspect each intermediate value. Keep the solving by substitution representation visible until the final line.

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Key vocabulary

substitute · evaluate · expression · solution · check · solving · by

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Rules

Write the expression with the replacement in brackets. Follow the operation order carefully. Evaluate each side independently if checking an equation. Compare the results and state the conclusion. Record the check explicitly for solving by substitution.

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Formula / fact

For x = 3, 2x + 1 = 2(3) + 1 = 7.

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Foundation example

Given information: Solving by substitution — Find 3x + 2 when x = 2. Method choice: use the solving by substitution method and show each step with the stated values. Calculation or reasoning: Substitute x = 2: 3 × 2 + 2 = 8. Final answer: 8. Check: substitute or compare with the original information to confirm the result fits the question.

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Secure example

Given information: Solving by substitution — Find 3x + 14 when x = 10. Method choice: use the solving by substitution method and show each step with the stated values. Calculation or reasoning: Substitute x = 10: 3 × 10 + 14 = 44. Final answer: 44. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Challenge example

Given information: Solving by substitution — Find 3x + 13 when x = 7. Method choice: use the solving by substitution method and show each step with the stated values. Calculation or reasoning: Substitute x = 7: 3 × 7 + 13 = 34. Final answer: 34. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Exam-style example

Given information: Solving by substitution — Find 3x + 12 when x = 4. Method choice: use the solving by substitution method and show each step with the stated values. Calculation or reasoning: Substitute x = 4: 3 × 4 + 12 = 24. Final answer: 24. Check: substitute or compare with the original information to confirm the result fits the question.

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Common mistake

Dropping brackets around negative values. This is a key trap when answering solving by substitution questions.

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Exam tip

For solving by substitution, show the key representation before the final calculation. Use this final check: Repeat the evaluation with explicit brackets and inspect each intermediate value.

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Real-world use

Formula use, Checking models

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Checklist

I can explain solving by substitution, use the method, check for mistakes, and answer an exam-style question.

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Flashcards

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Help for Solving by substitution

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Simple explanation

Solving by substitution: Substitution replaces a letter with a known value. Repeat the evaluation with explicit brackets and inspect each intermediate value. Keep the solving by substitution representation visible until the final line.

Think of solving by substitution 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. Write the expression with the replacement in brackets.
  2. Follow the operation order carefully.
  3. Evaluate each side independently if checking an equation.
  4. Compare the results and state the conclusion. Record the check explicitly for solving by substitution.

Hint 1

Start by naming the given information and the exact result required for solving by substitution.

Hint 2

Write the expression with the replacement in brackets.

Full worked solution

Given information: Solving by substitution — Find 3x + 2 when x = 2. Method choice: use the solving by substitution method and show each step with the stated values. Calculation or reasoning: Substitute x = 2: 3 × 2 + 2 = 8. Final answer: 8. Check: substitute or compare with the original information to confirm the result fits the question.

Method: Write the expression with the replacement in brackets. → Follow the operation order carefully. → Evaluate each side independently if checking an equation. → Compare the results and state the conclusion. Record the check explicitly for solving by substitution.

Common mistake warning

Dropping brackets around negative values. This is a key trap when answering solving by substitution questions.

Choose a support button above when you need a nudge.

Mastery milestones

Badges reward learning, not locked clicking

  • I can explain solving by substitution in my own words.
  • I can use these words accurately: substitute, evaluate, expression.
  • I can follow the 4-step method without guessing.
  • I can avoid this mistake: Dropping brackets around negative values.
  • I can apply this check: Repeat the evaluation with explicit brackets and inspect each intermediate value.
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