Algebra ToolkitCambridge IGCSE Maths: Revision notes

Section 1

What does a letter represent in algebra?

A letter (or variable) stands for a generalised number — a value that can change or is currently unknown. Algebra lets us write rules that work for every number, not just one example.

$n$ could represent any number
Writing $2n$ means "double whatever $n$ is"
An expression is a collection of letters and numbers combined with operations (e.g. $3x + 5$ ) — it has no equals sign
An equation states that two expressions are equal (e.g. $3x + 5 = 11$ )
A formula connects two or more different letters/quantities (e.g. $A = lw$ )

Key termsvariableexpressionequationformula

Exam tip

Examiners often ask you to state whether something is an expression, equation or formula — check for an equals sign and how many different letters are involved.

Section 2

How do you substitute into expressions and formulas?

Substitution means replacing each letter with a given numerical value, then working out the answer using the correct order of operations (BIDMAS).

Write out the expression
Replace each letter with its given value, using brackets to avoid sign errors
Calculate using BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction)

For example, if $a = 3$ and $b = -2$ , then $2a^2 - b = 2(3)^2 - (-2) = 18 + 2 = 20$ .

Key termssubstitutionBIDMAS

Common mistake

When substituting a negative number, always put it in brackets first — e.g. for $b=-2$ , write $(-2)^2$ , not $-2^2$ , otherwise the negative sign is easy to lose or misapply.

Example

If $P = 2l + 2w$ and $l = 7, w = 4$ , then $P = 2(7) + 2(4) = 14 + 8 = 22$ .

Section 3

How do you simplify by collecting like terms?

Like terms contain exactly the same letter(s) raised to the same power (e.g. $3x$ and $5x$ are like terms; $3x$ and $3x^2$ are not). To simplify, add or subtract the coefficients of like terms, keeping unlike terms separate.

For example: $2a + 3b + 5a - 9b = (2a+5a) + (3b-9b) = 7a - 6b$

Only combine terms with identical letter parts
Keep track of the sign directly in front of each term
The result cannot usually be simplified further once all like terms are combined

Key termslike termscoefficient

Common mistake

A common error is combining $x$ and $x^2$ terms — these are never like terms and must stay separate, e.g. $x^2 + 3x$ cannot be simplified further.

Section 4

How do you construct expressions, equations and formulas from words?

Translate a worded description into algebra one phrase at a time, defining a letter for the unknown quantity first.

Phrase	Algebra
"2 more than $n$ "	$n + 2$
"3 less than $n$ "	$n - 3$
"double $n$ "	$2n$
"the sum of $n$ costs $\pounds5$ each"	$5n$

For a formula, identify how the quantities relate — e.g. "total cost is a $\pounds3$ call-out fee plus $\pounds5$ per hour, $h$ " gives $C = 3 + 5h$ .

Key termsconstruct

Exam tip

Define your letter explicitly ("let $n$ = the number of items") before writing the expression — this earns method marks even if the final expression has an error.

Must Know

A letter represents a generalised or unknown number
Expressions have no equals sign; equations and formulas do
Substitute by replacing letters with values, using brackets for negatives, then apply BIDMAS
Only like terms (same letter, same power) can be collected together
Build expressions from words by translating each phrase in order, defining letters first

That's the notes covered.

Carry on to the next subtopic.