KS3 · Maths

Solving linear equations involving brackets

'Double it, then add 4' and 'add 4, then double it' sound identical — but a bracket makes them different equations, and knowing why is the key to solving them.

Choose your first move

Expand it, or divide it away?

Every equation below has a bracket. Pick what you notice about it, then pick a first move and follow it to the answer. Then step back and try the other move.

What do you notice? → Your first move

Here's the big idea: there is more than one correct way to get rid of a bracket. Your job is to spot the neatest one.

8 worked routes.

On An equation with a bracket in it. 4 branches to choose from.

Every pair of routes ends at the same answer. That's no accident: any step you do to both sides keeps the equation true.

Predict, then check

Why does the bracket matter so much? Try writing each puzzle as an equation before you choose.

Puzzle A: 'I think of a number, multiply it by 2, then add 4. I get 10.' Puzzle B: 'I think of a number, add 4, then multiply by 2. I get 10.' Are the two numbers the same?

Expanding a bracket

Multiply every term inside

Expand −5(y + 1). Each box is the number outside the bracket times the term above it. Fill both boxes, then collect them into the expansion.

Grid: each cell is its row times its column
×	y	+1
−5

−5(y + 1) =

Type x² as x^2 if you cannot type ². Spaces do not matter.

Maths · Algebra

Keep the balance, clear the fraction

Tap through each line. Watch the operation on the right — it's always done to both sides.

GoalKeep 8 + 2 = 7 + 3 balanced

8 + 2 = 7 + 3

Both sides are 10. Think of an old-fashioned balance with equal weights in each pan.

Step 1 of 4

Both sides are 10. Think of an old-fashioned balance with equal weights in each pan.

More than one bracket

Your turn: fill the gaps

A rectangle has sides of (2x + 3) cm and (9 − x) cm. Its perimeter is 33 cm. Find x.

2(2x + 3) + 2(9 − x) = 33The perimeter is two of each side, so each side goes in a bracket with a 2 in front.
missing step

Which line is step 2?

4x + 6 + 18 − 2x = 334x + 3 + 18 − x = 334x + 6 + 18 + 2x = 33

What do you really think?

Do brackets always come off first?

Sam is solving 2(x + 5) + 3(x + 5) = 40.

Which is closest to what you think right now?

You always have to expand the brackets first. That's the rule.There's only one correct method. Solve it a different way and you'll get a different answer.(x + 5) can be treated as one thing: 2 lots plus 3 lots makes 5 lots of (x + 5).You add the brackets together, insides and all: 2(x + 5) + 3(x + 5) = 5(2x + 10).

How sure are you?

I'm sureI think soI'm guessing

You always have to expand the brackets first. That's the rule.

This is a common misconception

Expanding is always allowed, but it isn't a rule that it comes first. Look at Sam's equation: 2 lots of (x + 5) plus 3 lots of (x + 5) is just 5 lots of (x + 5) — like 2 apples + 3 apples = 5 apples. So 5(x + 5) = 40, then divide both sides by 5: x + 5 = 8, so x = 3.

There's only one correct method. Solve it a different way and you'll get a different answer.

This is a common misconception

Any step done to both sides keeps the equation true, so every valid route ends at the same answer. Expanding first: 2x + 10 + 3x + 15 = 40, so 5x + 25 = 40 and x = 3. Collecting first: 5(x + 5) = 40, so x = 3. Different paths, same x.

(x + 5) can be treated as one thing: 2 lots plus 3 lots makes 5 lots of (x + 5).

This matches the evidence

Exactly. Identical bracketed expressions are like terms, so they collect just like 2y + 3y = 5y. You get 5(x + 5) = 40, then x + 5 = 8 and x = 3. Collecting first also dodges sign slips when a bracket has a negative number in front of it.

You add the brackets together, insides and all: 2(x + 5) + 3(x + 5) = 5(2x + 10).

This is a common misconception

Only the multipliers add: 2 lots + 3 lots = 5 lots of the SAME (x + 5). The (x + 5) itself isn't doubled. 5(2x + 10) = 40 would give x = −1, and the check fails: 2 × (−1 + 5) + 3 × (−1 + 5) = 8 + 12 = 20, not 40.

Expanding is always allowed but never compulsory. Look first: identical brackets collect, common factors divide out — then take the route with the fewest chances to slip.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

A bracket means 'all of this, that many times' — and there's more than one way to undo it.

What you need to know

2(x + 4) = 10 means (x + 4) twice. It is not the same equation as 2x + 4 = 10.
Expanding multiplies every term inside the bracket: 5(x − 10) = 5x − 50 and −5(y + 1) = −5y − 5.
Doing the same thing to both sides — adding, subtracting, multiplying or dividing — keeps an equation balanced.
If the number outside the bracket divides the other side exactly, or both sides share a common factor, dividing first is usually quickest.
If it doesn't divide neatly, expanding first keeps whole numbers: 4(x + 11) = 15 becomes 4x + 44 = 15, so x = −29/4.
If the number outside is a fraction, multiply both sides by its reciprocal.

The big picture

A bracket in an equation means 'everything inside, that many times'. To solve, keep both sides equal while you remove it: expand the bracket, divide both sides by the number outside, or multiply by the reciprocal of a fraction. With several brackets, expand and collect into ax + b = c; identical brackets can be collected first. Every valid route gives the same answer, so choose the neatest.

Key points

1Order matters: 'multiply by 2, then add 4' is 2x + 4, but 'add 4, then multiply by 2' is 2(x + 4).

2There is more than one correct way to remove a bracket, and every valid route reaches the same solution.

3With several brackets: expand each one, collect like terms into ax + b = c, then solve.

4Identical bracketed expressions are like terms: 2(x + 5) + 3(x + 5) = 5(x + 5).

5Check a solution by putting it back into both sides of the original equation.

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Worked example

Problem

Solve 3(2x + 5) = 5(x + 4).

⚠ Watch out

Multiplying only the first term in the bracket: writing 4(y + 2) as 4y + 2, or turning 2(x + 4) = 10 into x + 8 = 10. The number outside multiplies every term inside, signs included: −5(y + 1) = −5y − 5.

🧠

Memory hook

Bracket = 'all of it, that many times'. To undo it, look before you leap: divide if it's tidy, expand if it isn't.

✓

Check yourself

Solve 5(x − 10) = 45 in two ways: first divide both sides by 5, then start again and expand first. Which felt quicker? (Both routes should give x = 19.)

Flashcards

(15)

What does 2(x + 4) mean?

Two lots of (x + 4): everything inside the bracket is doubled, so it's 2x + 8.

Write 'add 3 to x, then multiply by 5' as an expression.

5(x + 3). The bracket shows the adding happens first. (5x + 3 means 'times 5, then add 3'.)

Expand 4(y + 2).

4y + 8. The 4 multiplies both terms inside, so it is not 4y + 2.

In 2(7x + 4) + 3x, what does the 2 multiply?

Only the 7x and the 4. The 3x is outside the bracket, so the whole thing is 14x + 8 + 3x = 17x + 8.

Why does multiplying both sides by the same number keep an equation true?

Equal amounts times the same number stay equal: 2(8 + 2) = 2(7 + 3) gives 20 = 20. Use different multipliers and you get 20 = 30 — no longer equal.

When is dividing by the number outside the bracket a good first step?

When it divides the other side exactly: 2(x + 7) = 30 gives x + 7 = 15, so x = 8.

When is expanding the better first step?

When the number outside doesn't divide the other side neatly. 3(x + 4) = 20 becomes 3x + 12 = 20: whole numbers until the very last step, x = 8/3.

Can two different correct methods give two different answers?

No. Any step done to both sides keeps the equation true, so every valid route reaches the same solution.

How do you remove a fraction in front of a bracket, like ⅗(7x − 4) = 21?

Multiply both sides by the reciprocal, 5/3: 7x − 4 = 35, so 7x = 39 and x = 39/7.

How do you clear the fraction in 2e + 3 = ¼e?

Multiply every term on both sides by 4: 8e + 12 = e.

What can you do first with 3(2x − 1) = 9(x − 2)?

Divide both sides by the common factor 3: 2x − 1 = 3(x − 2). Smaller numbers, same solution (x = 5).

Solving with several brackets: what's the order?

Expand each bracket, collect like terms, rearrange into ax + b = c, then solve.

Why can't you collect like terms in 3(x + 4) + 5(x + 2) straight away?

The terms are locked inside different brackets. Expand first: 3x + 12 + 5x + 10 = 8x + 22.

What is 2(x + 5) + 3(x + 5), simplified?

5(x + 5). Identical brackets are like terms: 2 lots + 3 lots = 5 lots.

How do you check a solution to an equation with brackets?

Substitute it into both sides of the original equation. For 3(6x + 2) = 60, x = 3 gives 3 × 20 = 60 ✓

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Expand it, or divide it away?

Multiply every term inside

Keep the balance, clear the fraction

Your turn: fill the gaps

Do brackets always come off first?

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