Solve: \(x + 5 = 12\)
Subtract 5 from both sides.
To undo \(+5\), subtract 5 from both sides: \(x = 12 - 5 = 7\).
Solving Linear Equations
Solving linear equations involves finding the value of a variable that makes an equation true. These problems often require skills from simplifying expressions and rearranging formulae, and are closely related to solving inequalities.
Overview
An equation says that two expressions are equal.
Solving an equation means finding the value of the variable that makes the equation true.
\( x + 5 = 12 \quad \Rightarrow \quad x = 7 \)
The key idea is balance.
Whatever you do to one side of the equation, you must also do to the other side.
What you should understand after this topic
- Understand what a linear equation is
- Solve one-step and two-step equations
- Keep both sides balanced
- Solve equations with brackets or variables on both sides
- Check answers by substitution
Key Definitions
Equation
A statement showing two expressions are equal.
Variable
A letter representing an unknown value.
Solve
Find the value of the variable that makes the equation true.
Linear Equation
An equation where the variable has power 1 only.
Balance
Both sides of the equation must stay equal at every step.
Check
Substitute your answer back into the original equation.
Key Rules
Do the same to both sides
If you subtract 5 on one side, subtract 5 on the other.
Undo operations in steps
Add or subtract first, then multiply or divide if needed.
Simplify each side if needed
Expand brackets or collect like terms before isolating the variable.
Always check the final value
Substitute it back into the original equation.
Quick Pattern Check
One-step
\(x + 4 = 9\)
Two-step
\(3x + 2 = 14\)
Brackets
\(2(x + 3) = 16\)
Variables on both sides
\(5x + 1 = 2x + 10\)
How to Solve
The balance idea
An equation works like balanced scales. Whatever you do to one side, you must do to the other.
\( x + 5 = 12 \)
Subtract 5 from both sides.
\( x = 7 \).
Key idea: Keep both sides equal at every step.
Step-by-step method
- Remove brackets if needed.
- Collect like terms.
- Move variables to one side.
- Move numbers to the other side.
- Divide to find the final answer.
- Check your solution.
One-step equations
\( \frac{x}{4} = 3 \)
Multiply both sides by 4.
Answer: \( x = 12 \).
Two-step equations
\( 3x + 2 = 14 \)
Subtract 2: \(3x = 12\).
Divide by 3: \(x = 4\).
Brackets and expanding
\( 4(x + 1) = 20 \)
Either divide first or expand first.
Exam tip: Choose the simpler method.
Variables on both sides
\( 5x + 2 = 2x + 11 \)
Move all \(x\) terms to one side.
Then solve as normal.
Exam thinking
Keep equations neat and step-by-step.
Watch negative signs carefully.
Exam tip: Always check your answer by substitution.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics.
Edexcel
Solve \( x + 7 = 15 \).
Edexcel
Solve \( 5x = 40 \).
Edexcel
Solve \( 3x - 4 = 11 \).
Edexcel
Solve \( 4(x + 2) = 20 \).
Edexcel
Solve \( \frac{x}{3} + 5 = 9 \).
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on reasoning and multi-step problem solving.
AQA
Solve \( 7x - 3 = 4x + 12 \).
AQA
Solve \( 5(2x - 1) = 3(x + 7) \).
AQA
Solve \( \frac{2x - 3}{4} = 5 \).
AQA
Three consecutive integers have a sum of 72. Form an equation and find the integers.
AQA
The perimeter of a rectangle is 54 cm. The length is \( 3x \) and the width is \( x + 3 \). Form an equation and find the value of \( x \).
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, structure, and contextual applications.
OCR
Solve \( 9 - 2x = 3x - 6 \).
OCR
Solve \( 4x + 7 = 3(2x - 5) \).
OCR
Solve \( 0.5x + 2.5 = 6 \).
OCR
A number is multiplied by 4 and then increased by 9. The result is 37. Find the number.
OCR
The angles of a triangle are \( x \), \( 2x \), and \( 3x \). Form an equation and find the value of \( x \).
Exam Checklist
Step 1
Simplify each side first if needed.
Step 2
Keep the equation balanced by doing the same to both sides.
Step 3
Isolate the variable step by step.
Step 4
Substitute your answer back in to check.
Most common exam mistakes
One-sided working
Changing one side only and breaking the balance.
Sign mistakes
Mixing up plus and minus when rearranging or simplifying.
Brackets skipped
Not expanding correctly before solving.
No check
Leaving an answer unverified when substitution would catch errors.
Common Mistakes
These are common mistakes students make when solving linear equations in GCSE Maths.
Not balancing both sides
Incorrect
A student changes only one side of the equation.
Correct
Whatever operation you perform, you must do it to both sides to keep the equation balanced.
Sign errors
Incorrect
A student makes mistakes with positive and negative numbers.
Correct
Take care with signs when adding, subtracting, multiplying or dividing. Double-check each step.
Forgetting to expand brackets
Incorrect
A student tries to solve without removing brackets.
Correct
Expand brackets first before collecting like terms and solving the equation.
Moving terms without understanding
Incorrect
A student moves terms across the equals sign incorrectly.
Correct
Instead of “moving”, think in terms of inverse operations. Apply the same operation to both sides.
Not checking the solution
Incorrect
A student does not verify the final answer.
Correct
Substitute your answer back into the original equation to check it is correct.
Try It Yourself
Practise solving linear equations step by step.
Foundation
Foundation Practice
Solve one-step and two-step equations.
Solve: \(x - 4 = 9\)
Add 4 to both sides.
\(x - 4 = 9\). Add 4 to both sides: \(x = 13\).
Solve: \(3x = 18\)
Divide both sides by 3.
\(3x = 18\). Divide both sides by 3: \(x = 6\).
Solve: \(\frac{x}{5} = 4\)
Multiply both sides by 5.
\(\frac{x}{5} = 4\). Multiply both sides by 5: \(x = 20\).
Solve: \(2x + 3 = 11\)
Undo the +3 first, then divide by 2.
Subtract 3 from both sides: \(2x = 8\). Then divide by 2: \(x = 4\).
Solve: \(5x - 2 = 18\)
Add 2 first.
\(5x - 2 = 18\). Add 2 to both sides: \(5x = 20\). Divide by 5: \(x = 4\).
Solve: \(4x + 7 = 23\)
Subtract 7 first.
Subtract 7: \(4x = 16\). Divide by 4: \(x = 4\).
Solve: \(6x + 1 = 31\)
Undo the +1 first.
\(6x + 1 = 31\). Subtract 1: \(6x = 30\). Divide by 6: \(x = 5\).
A student solves \(2x + 5 = 17\) and gets \(x = 11\). What mistake did they make?
After subtracting 5, the equation is \(2x = 12\).
The student likely did \(17 - 5 = 12\), but then forgot to divide by 2. The correct answer is \(x = 6\).
Solve: \(9x - 6 = 30\)
Add 6, then divide by 9.
Add 6 to both sides: \(9x = 36\). Divide by 9: \(x = 4\).
Higher
Higher Practice
Solve equations with brackets, negatives, fractions and unknowns on both sides.
Solve: \(3x + 5 = x + 17\)
Collect x terms on one side and numbers on the other.
Subtract \(x\) from both sides: \(2x + 5 = 17\). Subtract 5: \(2x = 12\). Divide by 2: \(x = 6\).
Solve: \(5x - 4 = 2x + 11\)
Move the smaller x term first.
Subtract \(2x\): \(3x - 4 = 11\). Add 4: \(3x = 15\). Divide by 3: \(x = 5\).
Solve: \(2(x + 3) = 18\)
You can divide by 2 first or expand first.
Divide both sides by 2: \(x + 3 = 9\). Subtract 3: \(x = 6\).
Solve: \(4(x - 2) = 20\)
Divide by 4 first.
Divide both sides by 4: \(x - 2 = 5\). Add 2: \(x = 7\).
Solve: \(3(2x - 1) = 15\)
Divide by 3 first, then solve.
Divide by 3: \(2x - 1 = 5\). Add 1: \(2x = 6\). Divide by 2: \(x = 3\).
Solve: \(7 - 2x = 15\)
Subtract 7 first. Be careful with \(-2x\).
Subtract 7: \(-2x = 8\). Divide by \(-2\): \(x = -4\).
Solve: \(\frac{x}{3} + 4 = 10\)
Subtract 4 first, then multiply by 3.
Subtract 4: \(\frac{x}{3} = 6\). Multiply by 3: \(x = 18\).
Solve: \(\frac{x - 1}{4} = 3\)
Multiply both sides by 4 first.
Multiply both sides by 4: \(x - 1 = 12\). Add 1: \(x = 13\).
A student solves \(4(x - 2) = 20\) and writes \(4x - 2 = 20\). What mistake did they make?
When expanding, 4 must multiply both \(x\) and \(-2\).
\(4(x - 2)\) expands to \(4x - 8\), not \(4x - 2\). The correct solution is \(x = 7\).
Solve: \(2x + 9 = 5x - 6\)
Move x terms to one side and numbers to the other.
Subtract \(2x\): \(9 = 3x - 6\). Add 6: \(15 = 3x\). Divide by 3: \(x = 5\).
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
How do I solve equations?
Use inverse operations to isolate the variable.
What is the goal?
To get the variable on its own.
How do I check?
Substitute the answer back into the equation.