1MA1 · A17AlgebraFoundation & HigherGrades 2–6

Linear Equations

An equation is a balance: whatever you do to one side, you must do to the other. Solving a linear equation — one where the unknown is not squared or multiplied by itself — means undoing the operations around the unknown, step by step, until it stands alone. The skill is simple to state and worth a lot of marks, because equations appear inside almost every other topic.

45 min Equations & Inequalities

The big picture

Every step in solving an equation keeps both sides equal while making the equation simpler. Inverse operations undo what has been done to the unknown, in reverse order: if $x$ was multiplied by 3 and then had 5 added, subtract 5 first and then divide by 3.

Edexcel frequently hides a linear equation inside a context — angles in a shape, ages, costs, perimeters — so forming the equation correctly from words or a diagram is often the harder part.

What you'll learn

Solve linear equations using inverse operations
Solve equations with brackets and with the unknown on both sides
Solve equations involving fractions
Form and solve linear equations from contexts

Inverse operations

Undo operations in the reverse order to the one in which they were applied. For $4x - 7 = 21$ : add 7 to get $4x = 28$ , then divide by 4 to get $x = 7$ .

Write each step on a new line so the working can be followed and checked.

Check by substituting back: $4(7) - 7 = 21$ ✓.

Example 1Solve

\dfrac{x}{5} + 3 = 11

1Subtract 3:

\dfrac{x}{5} = 8

2Multiply by 5.

Answer

x = 40

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

Solve $\dfrac{x}{4} - 2 = 5$ .

Your answer

Show worked solution

Solution

1Add 2: $\dfrac{x}{4} = 7$ .
2Multiply by 4: $x = 28$ .

Answer: $x = 28$

Brackets and unknowns on both sides

Expand any brackets first.

If the unknown appears on both sides, collect the unknown terms on the side where the coefficient stays positive, and the numbers on the other.

Answers can be negative or fractional — do not assume a whole number.

Example 1Solve

3(2x - 5) = 4(x + 1) - 7

1Expand:

6x - 15 = 4x + 4 - 7

, so

6x - 15 = 4x - 3

2Subtract

4x

2x - 15 = -3

3Add 15:

2x = 12

Answer

x = 6

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

Solve $2(3x + 1) = 5x + 9$ .

Your answer

Show worked solution

Solution

1Expand: $6x + 2 = 5x + 9$ .
2Subtract $5x$ and 2: $x = 7$ .

Answer: $x = 7$

Tip — Moving $4x$ from the right to the left means subtracting $4x$ from both sides. Say "subtract" rather than "move over and change sign" — it keeps the logic clear.

Equations with fractions

Multiply every term on both sides by the lowest common multiple of the denominators. This clears all the fractions at once.

Put brackets around numerators with more than one term before multiplying, to keep signs correct.

Example 1Solve

\dfrac{2x + 1}{3} - \dfrac{x - 2}{4} = 2

1Multiply every term by 12:

4(2x + 1) - 3(x - 2) = 24

2Expand:

8x + 4 - 3x + 6 = 24

3Simplify:

5x + 10 = 24

, so

5x = 14

Answer

x = \dfrac{14}{5} = 2.8

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

Solve $\dfrac{x + 3}{2} - \dfrac{x - 1}{5} = 3$ .

Your answer

A fraction like 13/3 is fine.

Show hint

Multiply every term by 10, and bracket the second numerator.

Show worked solution

Solution

1 $5(x + 3) - 2(x - 1) = 30$ .
2 $5x + 15 - 2x + 2 = 30$ , so $3x = 13$ .

Answer: $x = \dfrac{13}{3}$

Why this works

Multiplying $\dfrac{x - 2}{4}$ by 12 gives $3(x - 2)$ , and the minus sign in front then applies to both terms: $-3x + 6$ . Without the bracket, the $+6$ would wrongly become $-6$ .

Forming equations

Choose a letter for the unknown and say what it represents. Write the other quantities in terms of it, then use a fact from the question — a total, a perimeter, an angle sum — to form an equation.

After solving, answer the question that was actually asked.

Example 1The angles of a triangle are

x

2x + 15

and

3x - 21

degrees. Find the size of the largest angle.

1Angles sum to

180

x + 2x + 15 + 3x - 21 = 180

6x - 6 = 180

, so

x = 31

3Angles:

31^{\circ}

77^{\circ}

72^{\circ}

AnswerThe largest angle is

77^{\circ}

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

A rectangle has length $(3x - 1)$ cm and width $(x + 2)$ cm. Its perimeter is 38 cm. Find $x$ .

Your answer

Show worked solution

Solution

1 $2(3x - 1) + 2(x + 2) = 38$ , so $8x + 2 = 38$ .
2 $8x = 36$ and $x = 4.5$ .

Answer: $x = 4.5$

Tip — Check your answer in the original context: here the angles $31 + 77 + 72$ do sum to $180$ .

Think like an examiner

Write each operation as a new line of working.
Leave answers as exact fractions unless a decimal is requested.
In context questions, answer the specific quantity asked for.

Solving equations

ax + b = c \Rightarrow x = \tfrac{c - b}{a}

One-step and two-step equations.

\text{Multiply by LCM of denominators}

Clearing fractions.

Watch out for these

Dividing only one term:

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

Subtract 8 first, or divide every term by 4.

Expanding

-3(x - 2)

-3x - 6

Negative times negative:

-3x + 6

Multiplying only the fractions by the LCM.

Multiply every term, including whole-number terms.

Giving

x = 31

when the question asks for the largest angle.

Substitute back and answer the question.

Stretch yourself

Amira is three times as old as her son. In 12 years’ time she will be twice as old as he is. How old is each of them now?

Hint — Let the son be $s$ years old now. Write both ages in 12 years, then use "twice as old".

Reveal solution

SolutionNow: son

s

, Amira

3s

. In 12 years: son

s + 12

, Amira

3s + 12

. So

3s + 12 = 2(s + 12) = 2s + 24

, giving

s = 12

. The son is 12 and Amira is 36. Check: in 12 years they are 24 and 48, and

48 = 2 \times 24

✓.

\blacksquare

Questions students ask

Which side should I collect the unknowns on?

Either works, but choosing the side with the larger coefficient keeps the coefficient positive and avoids sign errors.

Can the answer be negative?

Yes, unless the context makes that impossible (such as a length). Always check against the context.

What is the difference between an equation and an identity?

An equation is true only for certain values; an identity is true for all values.

Key takeaways

Keep both sides balanced; use inverse operations in reverse order.
Expand brackets, then collect unknowns on one side.
Clear fractions by multiplying every term by the LCM of the denominators.
Form equations by defining the unknown and using a fact from the question.
Check by substituting back.

Exam-style questions

Practise it the way it is examined

3 original questions · 10 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 $7x - 4 = 3(x + 8)$ .[3]
Show mark scheme
- M1Expands: $3x + 24$
- M1Collects terms: $4x = 28$
- A1 $x = 7$
Answer: $x = 7$

Question 2Grade 54 marks

An isosceles triangle has two equal sides of length $(2x + 3)$ cm and $(5x - 9)$ cm. Its third side is $(x + 6)$ cm.

Work out the perimeter of the triangle.[4]
Show mark scheme
- M1 $2x + 3 = 5x - 9$
- A1 $x = 4$
- M1Sides $11, 11, 10$
- A1Perimeter $32$ cm
Answer: $32$ cm

Question 3Grade 63 marks

Solve $\dfrac{3x - 2}{4} = \dfrac{x + 5}{3}$ .[3]
Show mark scheme
- M1 $3(3x - 2) = 4(x + 5)$
- M1 $9x - 6 = 4x + 20$
- A1 $x = 5.2$
Answer: $x = 5.2$

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

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

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