Equations with Brackets
Two strategies for the same equation — expand then solve, or divide through first — and how to tell which one a given equation is asking for.
Printable Worksheets
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Before diving into the practice questions, take a moment to think about what you already know about this topic.
Know
- Brackets mean multiply the outside term by every term inside, that's the distributive law
- The distributive law: $a(b + c) = ab + ac$
Understand
- Two strategies exist: expand-then-solve and divide-first
- Strategy B (divide first) only works when the bracket stands alone, no extra terms outside
Can Do
- Solve equations like $3(x + 4) = 21$ using either strategy
- Choose the most efficient strategy for a given equation
Works for ALL brackets equations. Expand the brackets first using the distributive law, then solve the resulting two-step equation.
To expand $3(x + 4)$, multiply the 3 by every term inside the brackets: $3 \times x = 3x$ and $3 \times 4 = 12$. So $3(x + 4) = 3x + 12$. Then solve as usual.
Faster when it applies skip the expanding step entirely by dividing both sides by the coefficient first. Only works when the bracket stands alone (no extra terms outside).
OK for Strategy B: $3(x + 4) = 21$, bracket stands alone on the left. Divide both sides by 3.
Must use Strategy A: $3(x + 4) + 5 = 26$, the $+5$ outside means you cannot divide first. Expand instead.
Solve $4(2x - 3) = 20$ using both strategies side by side. Both give $x = 4$.
Watch Me Solve It · Worked example
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1Expand the brackets using the distributive law$2(3x + 1) = 20 \;\Rightarrow\; 6x + 2 = 20$Multiply the 2 by every term inside: $2 \times 3x = 6x$ and $2 \times 1 = 2$. Write the expanded equation.
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2Subtract 2 from both sides$6x + 2 - 2 = 20 - 2 \;\Rightarrow\; 6x = 18$Do the same thing to both sides to keep the equation balanced. Subtracting 2 isolates the $6x$ term.
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3Divide both sides by 6$6x \div 6 = 18 \div 6 \;\Rightarrow\; x = 3$Dividing both sides by the coefficient of $x$ gives the value of $x$.
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4Check by substituting $x = 3$ back into the original equation$2(3 \times 3 + 1) = 2(9 + 1) = 2(10) = 20$ ✓Always substitute back into the original equation (with brackets) to verify. Both sides equal 20, so $x = 3$ is correct.
Quick Check · 5 questions
Q1. Solve $5(x - 3) = 25$ using both methods (expand first AND divide first). Show all working.
Q2. Solve $\frac{x - 4}{3} = 6$. Show your working and check your answer.
Q3. A teacher has 6 identical packets of pencils. Each packet contains the same number of pencils plus 2 bonus pencils. In total there are 48 pencils. Write and solve an equation to find how many pencils are in each packet (not counting the bonus).
A perimeter equation with brackets
A perimeter equation with brackets. A rectangle has width $x$ centimetres and length $x+3$ centimetres. Its perimeter is 30 centimetres.
Form an equation using all four sides, expand and solve it, and give the rectangle’s dimensions. Check that your dimensions give the stated perimeter.
Show your working first, then open the worked solution to check your reasoning.
Check the worked solution
Two lengths and two widths give $2(x+3)+2x=30$. Expand: $2x+6+2x=30$, so $4x+6=30$. Subtract 6 and divide by 4 to get $x=6$. The width is 6 cm and the length is 9 cm. The perimeter check is $2(9)+2(6)=18+12=30$ cm. Both dimensions are positive.
Key Concept
Review the main ideas from this lesson.
Formulas
Key formulas and rules.
Watch Out
Common mistakes to avoid.
Check
Always verify your answers.
Practice
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Next
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Interactive: Algebra Machine
Substitute numbers into algebraic expressions and see them evaluate step by step.
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Brain Trainer
Set a timer for 4 minutes. Solve as many as you can. Check your answers after.