Equations with Brackets and Pronumerals on Both Sides
Every question with a fully worked solution.
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Theory
To solve an equation with brackets, expand every bracket first, collect the like terms, then solve as usual. This Year 9 maths page shows how to expand correctly, including a negative or a bare minus in front of a bracket, two brackets, pronumerals on both sides, and a bracket over a denominator.
To expand a bracket, multiply every term inside by the term outside. \(3(x + 4)\) becomes \(3x + 12\).
A negative outside changes the sign of every term inside. \(-2(x - 5)\) becomes \(-2x + 10\).
A bare minus in front of a bracket means \(-1\). \(-(x + 3)\) becomes \(-x - 3\).
Like terms have the same pronumeral part. Collect them on each side, such as \(4n + 3n = 7n\).
Then solve the equation exactly as before: gather the pronumerals on one side, the numbers on the other, and divide.
Expanding a bracket:
| Bracket | Expands to | Note |
|---|---|---|
| \(5(x + 2)\) | \(5x + 10\) | both terms multiplied |
| \(5(x - 2)\) | \(5x - 10\) | the sign travels with the \(2\) |
| \(-5(x - 2)\) | \(-5x + 10\) | both signs change |
| \(-(x - 2)\) | \(-x + 2\) | a bare minus is \(-1\) |
How to solve an equation with brackets
- Clear any denominator under a bracket by multiplying both sides by it.
- Expand every bracket. Watch the signs when the number outside is negative.
- Collect like terms on each side.
- Solve: take the smaller pronumeral term off both sides, move the numbers to the other side, then divide. Check by substituting.
(i) \(3(x + 4) = 27\)
(ii) \(-(x - 6) = 3x - 2\).
(i) Expand the bracket, then solve as usual.
| \(3x + 12\) | \(=\) | \(27\) |
| \(3x\) | \(=\) | \(15\) |
| \(x\) | \(=\) | \(\boldsymbol{5}\) |
(ii) A bare minus means \(-1\). It changes both signs inside.
| \(-x + 6\) | \(=\) | \(3x - 2\) |
| \(6\) | \(=\) | \(4x - 2\) |
| \(8\) | \(=\) | \(4x\) |
| \(x\) | \(=\) | \(\boldsymbol{2}\) |
Check (ii): LHS \(= -(2 - 6) = 4\). RHS \(= 6 - 2 = 4\). \(\checkmark\)
Expand, then take the smaller pronumeral term, \(3y\), off both sides.
| \(8y - 12\) | \(=\) | \(3y + 13\) |
| \(5y - 12\) | \(=\) | \(13\) |
| \(5y\) | \(=\) | \(25\) |
| \(y\) | \(=\) | \(\boldsymbol{5}\) |
Check: LHS \(= 4(10 - 3) = 28\). RHS \(= 15 + 13 = 28\). \(\checkmark\)
Expand both brackets. Collect the like terms before moving anything.
| \(4n - 4 + 3n + 6\) | \(=\) | \(30\) |
| \(7n + 2\) | \(=\) | \(30\) |
| \(7n\) | \(=\) | \(28\) |
| \(n\) | \(=\) | \(\boldsymbol{4}\) |
Check: \(4 \times 3 + 3 \times 6 = 12 + 18 = 30\) \(\checkmark\)
The second bracket is multiplied by \(-5\). So \(-5 \times w = -5w\) and \(-5 \times (-4) = +20\).
| \(6w + 2 - 5w + 20\) | \(=\) | \(30\) |
| \(w + 22\) | \(=\) | \(30\) |
| \(w\) | \(=\) | \(\boldsymbol{8}\) |
Check: \(2 \times 25 - 5 \times 4 = 50 - 20 = 30\) \(\checkmark\)
Writing \(-20\) instead of \(+20\) is the most common mistake here.
Common pitfalls
Frequently asked questions
How do you solve equations with brackets?
Expand every bracket, collect like terms on each side, then solve: pronumerals to one side, numbers to the other, then divide.
What happens when there is a negative in front of a bracket?
The negative multiplies every term inside, so every sign inside changes. \(-3(x - 4) = -3x + 12\).
What does a minus sign in front of a bracket mean?
A bare minus means \(-1\) times the bracket. It changes the sign of every term inside: \(-(x + 3) = -x - 3\).
Do you have to expand the brackets first?
Usually, yes. If the bracket sits over a denominator, multiply by the denominator first. If the number outside divides the other side exactly, you can divide first instead: \(4(x - 2) = 20\) gives \(x - 2 = 5\).
How do you check an equation with brackets?
Substitute your answer into the original equation, brackets and all. Work out each side. They should be equal.