A17, A21AlgebraFoundation & HigherGrades 2–6

Linear Equations

Q: Can the answer be negative or a fraction?

Yes. Equations often have answers like x = -3 or x = 52. Only reject such an answer if the context makes it impossible — a length cannot be negative.

Q: What is the difference between an equation and an identity?

An equation such as 2x + 1 = 7 is true for particular values (x = 3). An identity such as 2(x + 1) 2x + 2 is true for every value of x.

Solving an equation means finding the value of the letter that makes both sides equal. The whole topic rests on one rule — whatever you do to one side, do to the other — and it earns marks far beyond this lesson, because equations sit inside angle, perimeter, sequence and probability questions too.

45 min Video by Maths Genie AQA GCSE Maths

The big picture

Think of an equation as a set of scales in balance. Adding, subtracting, multiplying or dividing both sides by the same thing keeps it balanced while making it simpler, until the letter stands alone.

Undo the operations in the reverse order to the one they were done in. In $3 x + 5$ , the $x$ was multiplied by 3 and then had 5 added — so subtract 5 first, then divide by 3. On AQA papers, the harder marks are usually for forming the equation from words or a diagram, not for solving it.

Solving EquationsCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

What you'll learn

Solve one- and two-step equations using inverse operations
Solve equations with brackets and with the unknown on both sides
Clear fractions by multiplying every term by a common denominator
Form and solve equations from worded problems and diagrams

Keeping it balanced

Use inverse operations to get the letter on its own: the inverse of $+$ is $-$ , and the inverse of $\times$ is $\div$ . Apply each one to both sides, and write each step on a new line.

Deal with any adding or subtracting first, then the multiplying or dividing. Finally, check by substituting your answer back into the original equation.

Both sides stay equal at every step:

3 x + 5 = 20 \Rightarrow 3 x = 15 \Rightarrow x = 5

1Subtract 5 from both sides:

3 x = 15

2Divide both sides by 3:

x = 5

3Check:

3 (5) + 5 = 20

✓

Answer

x = 5

Your turnSame method, new numbers — try it before moving on.

Solve $4 x - 7 = 25$ .

Your answer

Your turnSame method, new numbers — try it before moving on.

Solve $\frac{x}{6} + 4 = 9$ .

Your answer

Tip — Answers do not have to be whole numbers. $2 x = 7$ gives $x = 3.5$ (or $\frac{7}{2}$ ) — do not round unless the question tells you to.

Brackets and unknowns on both sides

Expand any brackets first. Then, if the letter appears on both sides, subtract the smaller letter term from both sides so the letter is only on one side.

Collecting the letters on the side with the larger coefficient keeps the coefficient positive, which avoids sign errors.

1Expand:

5 x - 10 = 3 x + 8

2Subtract

3 x

from both sides:

2 x - 10 = 8

3Add 10:

2 x = 18

4Divide by 2:

x = 9

Answer

x = 9

Your turnSame method, new numbers — try it before moving on.

Solve $3 (2 x + 1) = 4 x + 15$ .

Your answer

Your turnSame method, new numbers — try it before moving on.

Solve $12 - x = 2 x - 9$ .

Your answer

Why "change the sign" works

Subtracting $3 x$ from both sides is exactly what people mean by "move $3 x$ over and change its sign". Saying it as an operation on both sides keeps the logic honest — and stops the classic slip of changing a sign that should not change.

Equations with fractions

Multiply every term on both sides by the lowest common multiple (LCM) of the denominators. All the fractions cancel in one step.

Put brackets round any numerator with more than one term before you multiply, so a minus sign in front applies to the whole numerator.

1The LCM of 2 and 3 is 6. Multiply both sides by 6:

3 (3 x - 1) = 2 (x + 5)

2Expand:

9 x - 3 = 2 x + 10

3Subtract

2 x

and add 3:

7 x = 13

Answer

x = \frac{13}{7}

Your turnSame method, new numbers — try it before moving on.

Solve $\frac{x}{2} + \frac{x}{3} = 10$ .

Your answer

Your turnSame method, new numbers — try it before moving on.

Solve $\frac{2 x + 1}{3} - \frac{x - 1}{2} = 2$ .

Your answer

Tip — $- 3 (x - 2)$ expands to $- 3 x + 6$ , not $- 3 x - 6$ . A negative times a negative is positive.

Forming equations

Choose a letter for the unknown and say what it stands for. Write every other quantity in terms of it, then use a fact from the question — a total, a perimeter, an angle sum, "the same as" — to write an equation.

Solve it, then answer the question that was actually asked. If it asks for the longest side, finding $x$ is only part of the way there.

The perimeter is 32 cm:

(x + 3) + 2 x + (2 x - 1) = 32

1Add the sides:

x + 3 + 2 x + 2 x - 1 = 32

2Simplify:

5 x + 2 = 32

, so

5 x = 30

and

x = 6

3Sides:

6 + 3 = 9

2 (6) = 12

2 (6) - 1 = 11

AnswerThe longest side is 12 cm.

Your turnSame method, new numbers — try it before moving on.

I think of a number, multiply it by 4 and subtract 9. The result is the same as doubling the number and adding 13. What is the number?

Your answer

Tip — Check your answer in the story, not just the equation: $9 + 12 + 11 = 32$ ✓.

Think like an examiner

Show each operation as a new line. A correct first step earns a method mark even if a later slip loses the answer.
Leave answers as exact fractions like $\frac{13}{7}$ unless the question asks for a decimal.
In worded or diagram questions, a final answer of " $x = 6$ " often scores nothing if the question asked for a length or an angle.

The method

same operation on both sides

The balance rule — every step, every term.

a x + b = c \Rightarrow x = \frac{c - b}{a}

Undo

+ b

first, then

\times a

\times LCM of denominators

Clears every fraction in one step.

Watch out for these

Dividing only one term:

4 x + 8 = 20 \Rightarrow x + 8 = 5

Subtract 8 first (

4 x = 12

), or divide every term by 4 (

x + 2 = 5

Expanding

- 3 (x - 2)

- 3 x - 6

Negative × negative is positive:

- 3 x + 6

Multiplying only the fractions by the LCM and forgetting the whole number on the other side.

Every term on both sides gets multiplied:

\frac{x}{2} + \frac{x}{3} = 10

becomes

3 x + 2 x = 60

From

5 x = 30

, writing

x = 150

x = 25

5 x

means

5 \times x

, so divide by 5:

x = 6

Stretch yourself

A rectangle has length $(3 x - 2)$ cm and width $(x + 1)$ cm. A square has side $(x + 3)$ cm. The rectangle and the square have the same perimeter. Which has the larger area, and by how much?

Hint — Form an equation from the perimeters, solve for $x$ , then work out both areas.

Questions students ask

Which side should I collect the

x

terms on?

Either side gives the right answer. Collecting on the side with the larger

x

coefficient keeps it positive, so there is less chance of a sign error.

Can the answer be negative or a fraction?

Yes. Equations often have answers like

x = - 3

x = \frac{5}{2}

. Only reject such an answer if the context makes it impossible — a length cannot be negative.

What is the difference between an equation and an identity?

An equation such as

2 x + 1 = 7

is true for particular values (

x = 3

). An identity such as

2 (x + 1) \equiv 2 x + 2

is true for every value of

x

Key takeaways

Keep the balance: do the same operation to both sides.
Undo operations in reverse order — addition and subtraction before multiplication and division.
Expand brackets, then collect the letter terms on one side.
Clear fractions by multiplying every term by the LCM of the denominators.
When forming equations, define the letter, use a fact from the question, and answer what was asked.

Exam-style questions

Practise it the way it is examined

4 original questions · 15 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 33 marks

Solve $4 (x - 3) = 2 x + 7$ .[3]

Question 2Grade 45 marks

A rectangle has length $(3 x + 2)$ cm and width $(x + 4)$ cm. Its perimeter is 44 cm.

(a)Work out the value of $x$ .[3]
(b)Work out the area of the rectangle.[2]

Question 3Grade 54 marks

The angles of a quadrilateral, in degrees, are $x$ , $2 x + 10$ , $3 x - 30$ and $80$ .

Work out the size of the largest angle.[4]

Question 4Grade 63 marks

Solve $\frac{2 x + 3}{4} - \frac{x - 1}{3} = 2$ .[3]

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Ready to practise Linear Equations? Pick a mode and earn XP & Dobloons.

Next step

Practise this topic

Real past-paper questions on Linear Equations, marked mark-by-mark. How you do feeds straight into your weak-topic list, so your revision keeps targeting what actually needs work.

Practise this topic

Linear Equations

Keeping it balanced

Brackets and unknowns on both sides

Equations with fractions

Forming equations

Practise it the way it is examined

Skill drill

Practise this topic

Related topics

Linear Equations

Keeping it balanced

Brackets and unknowns on both sides

Equations with fractions

Forming equations

Practise it the way it is examined

Skill drill

Practise this topic

Related topics