Find the value of \(3x + 2\) when \(x = 4\).
Replace x with 4, then calculate.
Substitute \(x = 4\): \(3 \times 4 + 2 = 12 + 2 = 14\).
Substitution
Substitution involves replacing variables with given values to evaluate expressions. This skill is widely used in algebra and problem solving across GCSE Maths.
Overview
Substitution means replacing a variable with a given number and then working out the value of the expression.
If \(x = 4\), then \(3x + 2 = 3(4) + 2 = 14\)
This topic is important because it appears throughout algebra, formulas, sequences, graphs and problem solving.
What you should understand after this topic
- Understand what substitution means
- Replace variables with numbers correctly
- Use brackets for negative values
- Work with powers and multiple variables
- Avoid common calculation mistakes
Key Definitions
Variable
A letter representing a value, such as \(x\) or \(y\).
Expression
A mathematical phrase made of numbers, letters and operations.
Substitute
Replace a letter with its given value.
Evaluate
Work out the final numerical answer.
Power
A small raised number showing repeated multiplication, such as in \(x^2\).
Bracket
Used to keep negative values clear when substituting.
Key Rules
Replace every letter carefully
If \(x = 3\), then \(2x + 1 = 2(3) + 1\).
Use brackets for negatives
If \(x = -2\), then \(x^2 = (-2)^2\).
Follow order of operations
Powers first, then multiplication, then addition or subtraction.
Different letters can have different values
If \(x = 2,\ y = 5\), then \(x + y = 7\).
Quick Pattern Check
One variable
\(4x + 3\)
Negative value
\(2x - 1\) when \(x = -3\)
Power involved
\(x^2 + 4\)
Two variables
\(3a + 2b\)
How to Solve
Step 1: Understand substitution
Substitution means replacing a variable with a number and then evaluating the expression.
If \(x = 5\), then \(x + 7 = 5 + 7 = 12\)
Exam tip: Always replace every occurrence of the variable.
Step 2: Replace the variable carefully
Find \(3x + 4\) when \(x = 2\)
Substitute: \(3(2) + 4\).
Multiply first: \(6 + 4\).
Answer: \(10\).
Step 3: Use brackets for negative numbers
Find \(2x + 5\) when \(x = -3\)
Write: \(2(-3) + 5\).
Multiply: \(-6 + 5\).
Answer: \(-1\).
Key idea: Always use brackets with negatives.
Step 4: Be careful with powers
Find \(x^2 + 1\) when \(x = -4\)
Write: \((-4)^2 + 1\).
Square first: \(16 + 1\).
Answer: \(17\).
Step 5: Substitute multiple variables
Find \(3a + 2b\) when \(a = 4,\ b = -1\)
Substitute: \(3(4) + 2(-1)\).
Calculate: \(12 - 2\).
Answer: \(10\).
Step 6: Fractions and expressions
Find \( \frac{x+4}{2} \) when \(x = 6\)
Substitute: \( \frac{6+4}{2} \).
Simplify: \( \frac{10}{2} = 5\).
Step 7: Order of operations still applies
Follow BIDMAS after substitution.
Multiply and divide before adding and subtracting.
Work inside brackets first.
Important
Substitution and order of operations go together. See order of operations.
Step 8: Exam method summary
- Write the expression clearly.
- Replace each variable with the given value.
- Use brackets for negative numbers.
- Follow the correct order of operations.
- Check your final answer.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics.
Edexcel
Find the value of \( 3x + 5 \) when \( x = 4 \).
Edexcel
Evaluate \( 2a^2 \) when \( a = 3 \).
Edexcel
Find the value of \( 4p - 3q \) when \( p = 5 \) and \( q = 2 \).
Edexcel
Evaluate \( x^2 + y^2 \) when \( x = 6 \) and \( y = 2 \).
Edexcel
Find the value of \( 5m - 2n + 3 \) when \( m = 4 \) and \( n = -1 \).
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on accurate substitution and order of operations.
AQA
Evaluate \( 3x^2 - 2x \) when \( x = 5 \).
AQA
Find the value of \( 2a^2b \) when \( a = 3 \) and \( b = 4 \).
AQA
Evaluate \( \frac{p + q}{r} \) when \( p = 8 \), \( q = 4 \), and \( r = 2 \).
AQA
Find the value of \( 4x^2 - y^2 \) when \( x = 3 \) and \( y = 5 \).
AQA
A student substitutes \( x = 2 \) into \( 5x^2 \) and writes the answer as 100. Explain the mistake and give the correct answer.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising substitution into formulae and real-life contexts.
OCR
The formula for the perimeter of a rectangle is \( P = 2(l + w) \). Find \( P \) when \( l = 7 \) cm and \( w = 3 \) cm.
OCR
The formula for the area of a triangle is \( A = \frac{1}{2}bh \). Find \( A \) when \( b = 10 \) cm and \( h = 6 \) cm.
OCR
The formula \( v = u + at \) gives the final velocity. Find \( v \) when \( u = 5 \), \( a = 3 \), and \( t = 4 \).
OCR
Given \( C = 2\pi r \), find \( C \) when \( r = 3 \). Give your answer in terms of \( \pi \).
OCR
The formula \( s = ut + \frac{1}{2}at^2 \) gives the displacement. Find \( s \) when \( u = 2 \), \( a = 4 \), and \( t = 3 \).
Exam Checklist
Step 1
Read the value of each variable carefully.
Step 2
Replace every variable in the expression.
Step 3
Use brackets when a substituted value is negative.
Step 4
Work out the answer using the correct order of operations.
Most common exam mistakes
Missing a bracket
Especially dangerous when squaring or multiplying negatives.
Wrong order
Doing addition before multiplication or powers.
Wrong value
Using the value for \(a\) in place of \(b\), or vice versa.
Incomplete substitution
Replacing one variable but forgetting another one in the same expression.
Common Mistakes
These are common mistakes students make when substituting values into expressions in GCSE Maths.
Not replacing every variable
Incorrect
A student substitutes for some variables but leaves others unchanged.
Correct
Replace every occurrence of the variable with the given value throughout the expression.
Missing brackets with negative numbers
Incorrect
A student substitutes a negative number without brackets.
Correct
Always use brackets when substituting negative values. For example, substitute \(x = -3\) as \((-3)\), especially when squaring.
Incorrect order of operations
Incorrect
A student performs calculations in the wrong order after substitution.
Correct
Follow BIDMAS: brackets, indices, division and multiplication, then addition and subtraction.
Confusing powers and multiplication
Incorrect
A student treats \(x^2\) as \(2x\).
Correct
\(x^2\) means \(x \times x\), not \(2x\). Apply the correct operation after substitution.
Using the wrong value for a variable
Incorrect
A student substitutes values into the wrong variables.
Correct
Match each value carefully to its corresponding variable before substituting.
Try It Yourself
Practise substituting values into algebraic expressions.
Foundation
Foundation Practice
Substitute values into simple expressions.
Find the value of \(5a\) when \(a = 6\).
Multiply 5 by 6.
\(5 \times 6 = 30\).
Find the value of \(2y + 3\) when \(y = 5\).
Substitute y = 5.
\(2 \times 5 + 3 = 10 + 3 = 13\).
Find the value of \(4x - 1\) when \(x = 3\).
Multiply first, then subtract.
\(4 \times 3 - 1 = 12 - 1 = 11\).
Find the value of \(x^2\) when \(x = 6\).
Square means multiply by itself.
\(6^2 = 6 \times 6 = 36\).
Find the value of \(2x + 3\) when \(x = 7\).
Multiply then add.
\(2 \times 7 + 3 = 14 + 3 = 17\).
A student says \(2x + 3 = 2 + 3 = 5\) when \(x = 2\). What mistake did they make?
2x means 2 × x.
They should calculate \(2 \times 2 + 3 = 7\), not \(2 + 3\).
Find the value of \(10 - x\) when \(x = 4\).
Substitute 4 into x.
\(10 - 4 = 6\).
Find the value of \(3a + 2\) when \(a = 8\).
Multiply 3 by 8 first.
\(3 \times 8 + 2 = 24 + 2 = 26\).
Find the value of \(x^2 + 1\) when \(x = 5\).
Square first, then add 1.
\(5^2 + 1 = 25 + 1 = 26\).
Higher
Higher Practice
Substitute into expressions with powers, brackets and negatives.
Find the value of \(x^2 + 3x\) when \(x = -2\).
Be careful with negatives and powers.
\((-2)^2 = 4\), and \(3(-2) = -6\). So \(4 - 6 = -2\).
Find the value of \(2x^2 - 3x\) when \(x = -3\).
Square first, then multiply.
\((-3)^2 = 9\), so \(2 × 9 = 18\). Then \(-3(-3) = 9\). So \(18 + 9 = 27\).
Find the value of \((x + 2)^2\) when \(x = 3\).
Work inside the bracket first.
\((3 + 2)^2 = 5^2 = 25\).
Find the value of \((x - 4)^2\) when \(x = 6\).
Evaluate bracket first.
\((6 - 4)^2 = 2^2 = 4\).
Find the value of \(3x^2 + 2y\) when \(x = 2\) and \(y = 5\).
Substitute both values.
\(3(2^2) + 2(5) = 3×4 + 10 = 12 + 10 = 22\).
Find the value of \(x^2 - y^2\) when \(x = 5\), \(y = 3\).
Square both values separately.
\(5^2 - 3^2 = 25 - 9 = 16\).
A student calculates \((-3)^2 = -9\). What is the mistake?
Negative × negative = positive.
\((-3)^2 = 9\), not -9.
Find the value of \(2(x + 3)\) when \(x = 4\).
Work inside the bracket first.
\(2(4 + 3) = 2 × 7 = 14\).
Find the value of \(x^3\) when \(x = -2\).
Multiply -2 three times.
\((-2)^3 = -8\).
Find the value of \(3x^2 + 4x + 1\) when \(x = 2\).
Work in order: square → multiply → add.
\(3(4) + 8 + 1 = 12 + 8 + 1 = 21\).
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What does substitution mean?
Replacing variables with numbers.
Why are brackets important?
They ensure negative numbers are handled correctly.
How do I check my answer?
Recalculate carefully and follow order of operations.