Linear equations

Edexcel International GCSE Mathematics A· 2.4 (2016 spec)· 12 min read

1. Solving Basic Linear Equations (Foundation Tier)★★☆☆☆⏱ 3 min

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📘 Definition

Linear equation in one unknown

An equation where the highest power of the single variable is 1, with no products or powers of the variable.

Example:

$5 x + 8 = 12$ is linear; $x^{2} + 3 = 7$ is not

The golden rule for solving all linear equations is: perform the identical operation to both sides of the equals sign to isolate the unknown. Follow these standard steps: 1. Expand any brackets 2. Collect terms with the unknown on one side, constants on the other 3. Divide by the coefficient of the unknown to find the solution.

📐 Worked Example

Solve $7 (x + 3) = 5 x - 8$

1
Expand the left-hand bracket
$7x + 21 = 5x - 8$
2
Subtract 5x from both sides to collect x terms on the left
$2x + 21 = -8$
3
Subtract 21 from both sides to collect constants on the right
$2x = -29$
4
Divide both sides by 2 to isolate x
$x = -\frac{29}{2}$

Exam tip:

Double check bracket expansion: multiply every term inside the bracket by the coefficient outside, not just the first term.

2. Linear Equations with Fractional Coefficients★★★☆☆Foundation / Higher only⏱ 3 min

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For equations containing fractions, eliminate denominators first by multiplying every term in the equation by the lowest common multiple (LCM) of all denominators. This simplifies the equation to an integer form that you can solve using the basic steps above.

📐 Worked Example

(Foundation) Solve $\frac{4 x + 5}{2} = 3$

1
Multiply all terms by the only denominator, 2, to eliminate the fraction
$4x + 5 = 6$
2
Subtract 5 from both sides
$4x = 1$
3
Divide by 4 to isolate x
$x = \frac{1}{4}$

📐 Worked Example

(Higher) Solve $\frac{2 x - 3}{6} + \frac{x + 2}{3} = \frac{5}{2}$

1
Find LCM of denominators 6, 3, 2 = 6. Multiply every term by 6
$2x - 3 + 2(x + 2) = 15$
2
Expand the remaining bracket
$2x - 3 + 2x + 4 = 15$
3
Collect like terms
$4x + 1 = 15$
4
Isolate x and simplify the solution
$4x = 14 \implies x = \frac{7}{2}$

Exam tip:

For Higher tier questions with multiple denominators, always use the LCM of all denominators to eliminate fractions in one step, rather than multiplying by each denominator individually.

3. Setting Up Linear Equations from Context★★★☆☆⏱ 3 min

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Many exam questions present a geometric or real-world scenario and ask you to find an unknown value. Your first task is to translate the written information into a linear equation, then solve it using the steps you have learned.

📐 Worked Example

The three interior angles of a triangle are $a^{\circ}$ , $(a + 10)^{\circ}$ , and $(a + 20)^{\circ}$ . Find the value of $a$ .

1
Recall the sum of interior angles of a triangle is 180°, so write the equation
$a + (a + 10) + (a + 20) = 180$
2
Collect like terms
$3a + 30 = 180$
3
Subtract 30 from both sides
$3a = 150$
4
Divide by 3 to isolate a
$a = 50$

Exam tip:

If the question asks you to 'set up and solve' an equation, you will get separate marks for writing the correct equation and solving it, so never skip writing the equation step.

4. Practice and Verification★★★★☆⏱ 3 min

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✓ Quick check

Solve $3 (2 x - 5) = 9$
- $x = 4$
- $x = 2$
- $x = - 4$
- $x = - 2$
Reveal answer
$x=4$ —
Expand to get $6 x - 15 = 9$ , so $6 x = 24$ and $x = 4$ . Correct!
What is the first step to solve $\frac{x}{3} + \frac{x - 1}{2} = 4$ ?
- Multiply all terms by 6
- Subtract $\frac{x - 1}{2}$ from both sides
- Add the fractions on the left
- Divide both sides by 4
Reveal answer
Multiply all terms by 6 —
The LCM of 3 and 2 is 6, so multiplying all terms by 6 eliminates all fractions in one step, per Edexcel recommended method.

5. Common Pitfalls

Wrong move:

Forgetting to multiply constant terms by the LCM when eliminating fractions, e.g. for $\frac{x}{2} + 3 = 5$ , only multiplying $\frac{x}{2}$ by 2 to get $x + 3 = 5$

Why:

You must apply the same operation to every term on both sides to maintain equation balance

Correct move:

Multiply every term by 2 to get $x + 6 = 10$ , so $x = 4$

Wrong move:

Incorrect bracket expansion, e.g. $7 (x + 3) = 7 x + 3$

Why:

The coefficient outside the bracket must be distributed to every term inside the bracket

Correct move:

Multiply 7 by x and 7 by 3 to get $7 x + 21$

Wrong move:

Sign errors when moving terms across the equals sign, e.g. $2 x + 21 = 5 x - 8$ becomes $2 x + 5 x = - 8 + 21$

Why:

To move a term across the equals sign you subtract it from both sides, so its sign flips

Correct move:

Rearrange to $2 x - 5 x = - 8 - 21$ , so $- 3 x = - 29$ and $x = \frac{29}{3}$

Wrong move:

Giving rounded decimal answers instead of exact fractions when not requested

Why:

Edexcel awards full marks only for exact answers unless rounding is explicitly specified

Correct move:

Simplify all fractions fully and present them as your final answer unless instructed otherwise

Wrong move:

Skipping writing the equation for context questions, jumping straight to the final answer

Why:

Method marks are awarded for deriving the correct equation, even if your final solution is wrong

Correct move:

Always write the full linear equation you derived from the context before solving it

6. Quick Reference Cheatsheet

Step	Foundation Tier	Higher Tier
1	Expand all brackets first	Find LCM of all denominators, multiply every term by LCM to eliminate fractions
2	Collect variable terms on one side, constants on the other	Expand any remaining brackets
3	Divide by coefficient of variable to isolate unknown	Collect like terms, isolate variable as per foundation steps
4	Check via substitution, present exact simplified fraction	Check via substitution, present exact simplified fraction
Context questions	Translate scenario to equation first, then solve	Same as foundation, expect multi-step scenarios

7. Frequently Asked

Do I need to show full working for linear equation questions?

Yes. Edexcel awards method marks for correct steps even if your final answer is incorrect. Always write every step (bracket expansion, fraction elimination, term collection) clearly.

When can I give a decimal answer instead of a fraction?

Only if the question explicitly asks for a decimal or rounded value. For all other cases, present your final answer as a fully simplified exact fraction, e.g. $\frac{7}{2}$ instead of 3.5.

Going deeper

official_resourceEdexcel 4MA1 2016 SpecificationFull specification for Equations, formulae and identities unit

What's Next

Mastering linear equations is the foundation for all higher algebra topics in Edexcel IGCSE Maths A, including simultaneous equations, quadratic equations, and algebraic graphs. You will encounter linear equations across both calculator and non-calculator papers, often combined with geometry, statistics, or real-world problem-solving contexts. Once you are confident solving linear equations and setting them up from context, you can move on to simultaneous linear equations, which require solving two equations with two unknowns, and rearranging formulae, which uses the same 'same operation to both sides' rule you learned here. Make sure to practice past paper questions regularly to build speed and accuracy, and always check your working for common mistakes like incorrect bracket expansion or sign errors.