What is substitution in algebra and how is it used in solving equations?

Substitution in algebra is a method used to solve equations by replacing a variable with a given value or another expression. This technique is particularly useful in solving systems of equations where one equation is solved for one variable, and that expression is substituted into another equation. This method simplifies the problem and helps find the values of the variables involved.

How do you perform substitution in a system of linear equations?

To perform substitution in a system of linear equations, first solve one of the equations for one of the variables. Then, substitute this expression into the other equation(s). This will result in an equation with one variable, which can be solved using standard algebraic methods. Once you find the value of one variable, substitute it back into the original equation to find the value of the other variable.

Can substitution be used to solve quadratic equations in the Cambridge IGCSE Mathematics curriculum?

Yes, substitution can be used to solve quadratic equations, especially when dealing with equations that can be simplified by substituting a variable with another expression. For example, if you have a quadratic equation in the form of ax^2 + bx + c = 0, and you can express x in terms of another variable, substitution can simplify the equation, making it easier to solve.

What are some common mistakes to avoid when using substitution in algebra?

Common mistakes when using substitution include not correctly solving for a variable before substituting, forgetting to substitute into all parts of the equation, and arithmetic errors during the substitution process. It's important to carefully follow each step, double-check calculations, and ensure that the substituted expression is correctly simplified.

Why is substitution an important skill in the Cambridge IGCSE Mathematics curriculum?

Substitution is an important skill in the Cambridge IGCSE Mathematics curriculum because it is a fundamental technique for solving various types of equations, including linear and quadratic equations. Mastery of substitution helps students develop problem-solving skills and understand the relationships between variables, which are essential for more advanced topics in mathematics.

Why is "Substituting a negative value without brackets" a common mistake in Substitution?

Why it happens: Students write -5^2 instead of (-5)^2 and end up with -25 instead of 25. How to avoid it: Always wrap the substituted value in brackets — especially when it's negative. Source: 0580 examiner reports — recurring across multiple sittings

Why is "Adding before squaring or multiplying" a common mistake in Substitution?

Why it happens: Reading left-to-right, students add adjacent terms before applying the indices/multiplications. How to avoid it: Underline the powers and multiplications in the substituted expression first; do those before any addition or subtraction.

Why is "Mishandling double negatives" a common mistake in Substitution?

Why it happens: - (-6) is written as -6 instead of +6. How to avoid it: Rewrite double signs explicitly: -(-6) = +6. Then continue.

Why is "Converting $\pi$ to a decimal when the question asks for "in terms of $\pi$"" a common mistake in Substitution?

Why it happens: Calculator habit — students hit the button to get a number. How to avoid it: Read the instruction. "In terms of " means leave the answer as k for some number k. Source: 0580/22 — geometry questions

Why is "Inverting the fraction in $\dfrac{a}{b/c}$" a common mistake in Substitution?

Why it happens: When the denominator is itself a fraction, students forget to flip-and-multiply. How to avoid it: Rewrite a/b/c as a · c/b before substituting numerical values.

AlgebraCambridge IGCSE Mathematics (0580)

Substitution

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next.

Previous Next

Learn this Topic

Start with the slides for the quick version, then go deeper with the full study notes.

Short Study Notes in the form of Slides

Read the notes first. If the method in a worked example clicks, you're ready for the questions.

Page 1 / 0

Detailed Study Notes

Full prose, callouts and a recap — built for A* mastery, not just a quick scan.

Take these study notes with you

Download a branded PDF — full prose, callouts, recap and memorise list for Substitution, ready to print or save offline.

Substitution into Algebraic Expressions — Cambridge IGCSE 0580 Maths Extended (2026)

Replace each variable in an expression with a numerical value, then evaluate. The arithmetic is the easy part — most lost marks come from sign mistakes and misreading BIDMAS.

What you’ll learn

Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).

C2.2 / E2.2 — Substitute numbers into expressions and formulae.

The substitution method

Replace each variable, wrap in brackets, then evaluate using BIDMAS.

Method.

Write down the expression.
For each variable, replace it with its value — wrapped in brackets.
Apply BIDMAS to evaluate.

Why brackets matter. Without them, a negative substitution can flip signs in unexpected ways:

$a = -3 \Rightarrow a^{2}$ . Without brackets: $-3^{2} = -(3^{2}) = -9$ . With brackets: $(-3)^{2} = 9$ . Different answers.

Worked. Find the value of $5x - 2y$ when $x = 3$ and $y = -4$ .

Substitute with brackets: $5(3) - 2(-4)$ .
Multiply: $15 - (-8) = 15 + 8$ .
$= 23$ .

Worked. Find the value of $2a^{2} - 3b$ when $a = -2$ and $b = 5$ .

Substitute: $2(-2)^{2} - 3(5)$ .
Indices first: $(-2)^{2} = 4$ , so $2(4) - 3(5) = 8 - 15$ .
$= -7$ .

Wrap every substitution in brackets, especially negatives.
Indices BEFORE multiplication.
$-3^{2} = -9$ , but $(-3)^{2} = 9$ .
Show every step — examiners credit method marks.

Substituting into formulae

Cambridge gives you a formula and a set of values; you compute. Common contexts: speed, area, volume, interest.

Many Paper 4 questions hand you a formula and a set of values. The skill is identifying which symbol gets which value, then evaluating.

Worked. The formula for the volume of a cone is $V = \dfrac{1}{3}\pi r^{2} h$ . Find $V$ when $r = 4$ cm and $h = 9$ cm. Take $\pi = 3.14$ .

$V = \dfrac{1}{3} \times 3.14 \times (4)^{2} \times 9$ .
Indices: $(4)^{2} = 16$ .
$V = \dfrac{1}{3} \times 3.14 \times 16 \times 9 = \dfrac{452.16}{3} = 150.72$ cm $^{3}$ .

Worked. Find $A$ when $P = 250$ , $r = 5$ , $n = 4$ in $A = P\left(1 + \dfrac{r}{100}\right)^{n}$ .

$A = 250 \times \left(1 + \dfrac{5}{100}\right)^{4} = 250 \times (1.05)^{4}$ .
$(1.05)^{4} \approx 1.21551$ .
$A \approx 250 \times 1.21551 \approx 303.88$ .

Read the formula carefully — match each letter to its value.
If the question says 'take $\pi = 3.14$ ', use that, not the calculator's $\pi$ .
State units in the final answer.
Round only at the end of a multi-step calculation.

Substituting fractions and surds

Same method, brackets even more important. Keep exact form unless asked otherwise.

When the value to substitute is a fraction or a surd, brackets are non-negotiable.

Worked. Find $3x^{2} + 2x$ when $x = \dfrac{1}{2}$ .

$3\left(\dfrac{1}{2}\right)^{2} + 2\left(\dfrac{1}{2}\right)$ .
$3 \times \dfrac{1}{4} + 1 = \dfrac{3}{4} + 1 = \dfrac{7}{4}$ .

Worked. Find $a^{2} - b$ when $a = \sqrt{5}$ and $b = 2$ .

$(\sqrt{5})^{2} - 2$ .
$5 - 2 = 3$ .

The square root squared cancels neatly. Cambridge loves this.

Fractions and surds: always wrap in brackets.
$(\sqrt{n})^{2} = n$ — useful simplification.
Keep exact form for final answers when the question allows it.

How it’s examined

Substitution appears on every paper, mostly as 1-2 mark steps inside a larger problem. Paper 2 gives a formula plus values and asks for a numerical answer. Paper 4 hides substitution inside formulae for area, volume, interest, motion, etc. Examiner reports flag negative-substitution sign errors and missed brackets as the recurring failures.

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E2.2); 0580/22 May/Jun 2024 — Q15 (substitution into formula); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

Master this Topic

Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

Take this whole topic with you

Download a branded revision sheet — worked examples, formulae, definitions and common mistakes for Substitution, ready to print or save as PDF.

Step-by-step worked examples — Substitution

Step-by-step solutions to past-paper-style questions on substitution, written exactly the way a tutor would explain them at the board.

1Substitute integers into a linear expression
Core• substitution, linear
Question
Find the value of $5x - 3y$ when $x = 4$ and $y = -2$ .
Step-by-step solution
1. Step 1
  Replace each variable with its value, keeping the operation signs.
  $5x - 3y = 5(4) - 3(-2)$
2. Step 2
  Multiply each term, careful with the negative.
  $= 20 - (-6) = 20 + 6 = 26$
Answer
$26$
Examiner tip
Always wrap the substituted value in brackets, especially for negatives. Writing $5 \times 4 - 3 \times -2$ without brackets is the most frequent slip in the examiner reports.
2Substitute into expressions with squares and cubes
Core• Adapted from 0580/22 May/Jun 2023 Q4• substitution, powers
Question
Evaluate $a^{2} - 2ab + b^{2}$ when $a = 3$ and $b = -5$ .
Step-by-step solution
1. Step 1
  Substitute, using brackets for the negative.
  $(3)^{2} - 2(3)(-5) + (-5)^{2}$
2. Step 2
  Apply BIDMAS: powers first, then multiplications.
  $= 9 - (-30) + 25$
3. Step 3
  Simplify the double negative.
  $= 9 + 30 + 25 = 64$
Answer
$64$
Examiner tip
$(-5)^{2}$ is $25$ , not $-25$ . Bracketing the substitution is what protects the sign on the square.
3Substitute into a physics-style formula
Extended• Adapted from 0580/42 Oct/Nov 2024 Q3• formula, substitution
Question
The formula $s = ut + \dfrac{1}{2}at^{2}$ gives the distance travelled. Find $s$ when $u = 5$ , $a = 2$ and $t = 4$ .
Step-by-step solution
1. Step 1
  Substitute carefully.
  $s = (5)(4) + \dfrac{1}{2}(2)(4)^{2}$
2. Step 2
  Powers first, then multiplications.
  $= 20 + \dfrac{1}{2}(2)(16)$
3. Step 3
  Simplify.
  $= 20 + 16 = 36$
Answer
$s = 36$
Examiner tip
Mark schemes credit the BIDMAS layout. Doing $5 \times 4 + \frac{1}{2} \times 2 \times 4 \times 4$ in one line is more error-prone — write the powers step out.
4Substitute fractions into a rational expression
Extended• substitution, fractions
Question
Find the value of $\dfrac{2x + y}{x - y}$ when $x = \dfrac{1}{2}$ and $y = -\dfrac{1}{4}$ .
Step-by-step solution
1. Step 1
  Substitute, again with brackets.
  $\dfrac{2(\tfrac{1}{2}) + (-\tfrac{1}{4})}{(\tfrac{1}{2}) - (-\tfrac{1}{4})}$
2. Step 2
  Numerator: $2 \times \tfrac{1}{2} = 1$ , then $1 - \tfrac{1}{4} = \tfrac{3}{4}$ .
3. Step 3
  Denominator: $\tfrac{1}{2} + \tfrac{1}{4} = \tfrac{3}{4}$ .
4. Step 4
  Divide.
  $\dfrac{3/4}{3/4} = 1$
Answer
$1$
5Substitute into an expression involving $\pi$
Core• substitution, pi
Question
The volume of a cylinder is $V = \pi r^{2} h$ . Find $V$ when $r = 3$ and $h = 5$ , leaving your answer in terms of $\pi$ .
Step-by-step solution
1. Step 1
  Substitute.
  $V = \pi (3)^{2}(5)$
2. Step 2
  Power first.
  $= \pi (9)(5) = 45\pi$
Answer
$V = 45\pi$
Examiner tip
When the question says "in terms of $\pi$ ", do not convert to a decimal — $45\pi$ is the final answer.

Key Formulae — Substitution

The formulae you need to memorise for substitution on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.

Order of operations (BIDMAS / PEMDAS)
$B \to I \to DM \to AS$
$B$
Brackets first
$I$
Indices (powers and roots) next
$DM$
Division and Multiplication, left to right
$AS$
Addition and Subtraction, left to right
When to use
Always apply BIDMAS when evaluating substituted expressions — it determines what you simplify first.
Bracket-substitution rule
$\text{If } x = a, \text{ then } f(x) \to f((a))$
When to use
Always wrap the value you substitute in brackets, especially when the value is negative or a fraction. Brackets protect signs and indices.

Key Definitions and Keywords — Substitution

Definitions to memorise and the exact keywords mark schemes credit for substitution answers — sharpened from recent examiner reports for the 2026 0580 sitting.

Variable
Examiner keyword
A symbol (typically a letter) that represents an unknown number.
Example
In $3x + 2$ , the variable is $x$ .
Coefficient
Examiner keyword
The numerical factor in front of a variable.
Example
In $5x$ , the coefficient is $5$ .
Constant
A term in an expression that has no variable — just a number.
Example
In $3x + 7$ , the constant is $7$ .
Term
Examiner keyword
Each part of an algebraic expression separated by a $+$ or $-$ sign.
Example
In $4x - 3y + 5$ , the terms are $4x$ , $-3y$ and $5$ .
Substitute
Examiner keyword
Replace a variable with a given numerical value before evaluating the expression.
Evaluate
Examiner keyword
Find the numerical value of an expression after substitution.

Common Mistakes and Misconceptions — Substitution

The traps other students keep falling into on substitution questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.

✕Substituting a negative value without brackets
0580 examiner reports — recurring across multiple sittings
Why it happens
Students write $-5^{2}$ instead of $(-5)^{2}$ and end up with $-25$ instead of $25$ .
How to avoid it
Always wrap the substituted value in brackets — especially when it's negative.
✕Adding before squaring or multiplying
Why it happens
Reading left-to-right, students add adjacent terms before applying the indices/multiplications.
How to avoid it
Underline the powers and multiplications in the substituted expression first; do those before any addition or subtraction.
✕Mishandling double negatives
Why it happens
$- (-6)$ is written as $-6$ instead of $+6$ .
How to avoid it
Rewrite double signs explicitly: $-(-6) = +6$ . Then continue.
✕Converting $\pi$ to a decimal when the question asks for "in terms of $\pi$ "
0580/22 — geometry questions
Why it happens
Calculator habit — students hit the $\pi$ button to get a number.
How to avoid it
Read the instruction. "In terms of $\pi$ " means leave the answer as $k\pi$ for some number $k$ .
✕Inverting the fraction in $\dfrac{a}{b/c}$
Why it happens
When the denominator is itself a fraction, students forget to flip-and-multiply.
How to avoid it
Rewrite $\dfrac{a}{b/c}$ as $\dfrac{a \cdot c}{b}$ before substituting numerical values.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Video lesson

Short walkthrough of the concepts students most often get stuck on.

Substitution — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Substitution

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Substitute integers into a linear expression

2Substitute into expressions with squares and cubes

3Substitute into a physics-style formula

4Substitute fractions into a rational expression

5Substitute into an expression involving π\piπ

Order of operations (BIDMAS / PEMDAS)

Bracket-substitution rule

Variable

Coefficient

Constant

Term

Substitute

Evaluate

✕Substituting a negative value without brackets

✕Adding before squaring or multiplying

✕Mishandling double negatives

✕Converting π\piπ to a decimal when the question asks for "in terms of π\piπ"

✕Inverting the fraction in ab/c\dfrac{a}{b/c}b/ca​

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Substitution — frequently asked questions

5Substitute into an expression involving $\pi$

✕Converting $\pi$ to a decimal when the question asks for "in terms of $\pi$ "

✕Inverting the fraction in $\dfrac{a}{b/c}$