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5.3 Solving Equations with Brackets

GCSE Maths Foundation · Edexcel · Chapter 5: Equations, Inequalities and Sequences

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: expanding brackets to solve equations, negative signs outside brackets, and word problems. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Solve equations with brackets by expanding first
  • Solve equations with brackets on both sides
  • Expand a bracket with a negative number outside it correctly
  • Form and solve equations with brackets from words, shapes and ages

Key words

  • Expand: Multiply out a bracket: 3(x + 4) = 3x + 12.
  • Bracket: Shows that everything inside is treated as one quantity.
  • Simplify: Collect like terms to make an expression shorter.
  • Negative multiplier: A negative number in front of a bracket, e.g. −3(x − 2). It changes the sign of every term inside.

Expand, then solve

Multiply every term inside the bracket by the number outside. Then solve the equation in the usual way.

Worked example: one bracket

Solve 3(x + 4) = 27.
Expand: 3x + 12 = 27
Subtract 12: 3x = 15. Divide by 3: x = 5

Answer: x = 5

Key fact: Another way: when the number outside divides the other side exactly, divide first. 3(x + 4) = 27 gives x + 4 = 9, so x = 5.

Worked example: expand, then collect like terms

Solve 4(2x + 1) + 3 = 31.
Expand: 8x + 4 + 3 = 31, so 8x + 7 = 31
Subtract 7: 8x = 24. Divide by 8: x = 3

Answer: x = 3

Brackets on both sides

Worked example: expand both brackets

Solve 5(x − 2) = 2(x + 7).
Expand: 5x − 10 = 2x + 14
Subtract 2x: 3x − 10 = 14. Add 10: 3x = 24. Divide by 3: x = 8
Check: 5 × 6 = 30 and 2 × 15 = 30 ✓

Answer: x = 8

A negative sign outside the bracket

A negative number outside a bracket multiplies every term inside, so it changes every sign.

  • −3(x − 2); Expanded −3x + 6; Why −3 × −2 = +6
  • −2(x + 5); Expanded −2x − 10; Why −2 × +5 = −10
  • −(x − 4); Expanded −x + 4; Why the − means −1 ×

Worked example: subtracting a bracket

Solve 20 − 3(x − 2) = 11.
Expand: 20 − 3x + 6 = 11, so 26 − 3x = 11
Add 3x: 26 = 11 + 3x. Subtract 11: 15 = 3x. x = 5

Answer: x = 5

Worked example: two brackets, one subtracted

Solve 2(3x + 1) − (x − 4) = 21.
Expand: 6x + 2 − x + 4 = 21, so 5x + 6 = 21
5x = 15, x = 3

Answer: x = 3

Problems that need brackets

Use a bracket when a whole amount is multiplied: “add 6, then multiply by 4” is 4(n + 6), not 4n + 6.

Worked example: area of a rectangle

6(x + 3) = 54
Divide by 6: x + 3 = 9, so x = 6. The length is 9 cm.

Answer: x = 6

Worked example: ages

Priya is 3 times as old as her brother. In 8 years’ time she will be twice as old as him. How old is Priya now?
Brother = b, Priya = 3b. In 8 years: 3b + 8 = 2(b + 8)
3b + 8 = 2b + 16, so b = 8. Priya is 3 × 8 = 24.

Answer: Priya is 24

Common mistakes

  • Only multiplying the first term: 2(x + 5) is 2x + 10, not 2x + 5.
  • Sign errors with a negative outside: −3(x − 2) = −3x + 6.
  • Forgetting that −(x − 4) means −1 × (x − 4) = −x + 4.
  • Writing 4n + 6 for “add 6 then multiply by 4”. You need the bracket: 4(n + 6).

Revision questions

Try these, then check your answers below each one.

Question 1. Solve 3(x + 5) = 24.

Answer: 3

Working: x + 5 = 8, so x = 3.

Question 2. Solve 4(2y - 3) = 28.

Answer: 5

Working: 2y - 3 = 7, 2y = 10, y = 5.

Question 3. Solve 5(x - 1) = 3(x + 3). (Choose from: x = 7, x = 5, x = 4, x = 2.)

Answer: x = 7

Working: 5x - 5 = 3x + 9, 2x = 14, x = 7.

Related lessons

Continue in the full course

This lesson is section 5.3 (EGCMF0503) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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