Algebra Toolkit Notes

Cambridge IGCSE Maths: Revision notes

Key facts

A variable is a letter standing for a number: $2n$ means "double $n$ ".
An expression has no equals sign; an equation states two expressions are equal; a formula links several letters.
Substitute with brackets, then follow BIDMAS.
Like terms have exactly the same letters and powers: $3x$ and $5x$ , but not $3x$ and $3x^2$ .
Define a letter for the unknown before writing an expression from words.

Letters and types

A letter stands for a number that can change or is unknown; algebra writes rules that work for every number.

$n$ could represent any number. Check for an equals sign and the number of different letters to decide what you have: examiners often ask you to state whether something is an expression, equation or formula.

The equation 3x + 5 = 11 is true only where the line y = 3x + 5 meets y = 11, at x = 2

Expression

No equals sign
$3x + 5$

Equation

Two expressions equal
$3x + 5 = 11$

Formula

Links different quantities
$A = lw$

Which of these is a formula?

Substitution

Replace each letter with its value, using brackets, then follow BIDMAS.

Write the expression, replace each letter with its given value in brackets, then calculate. For a negative value write $(-2)^2$ , not $-2^2$ . If $P = 2l + 2w$ and $l = 7$ , $w = 4$ , then $P = 14 + 8 = 22$ .

1
Write it out
the expression or formula
2
Replace each letter
use brackets for negatives
3
Calculate
follow BIDMAS

Substituting into an expression

Worked example

If $a = 3$ and $b = -2$ , find $2a^2 - b$ .

1
Substitute with brackets: $2(3)^2 - (-2)$ .
2
Indices first: $2 \times 9 - (-2) = 18 + 2$ .

If $x = -3$ , what is $x^2$ ?

Collecting like terms

Add or subtract the coefficients of terms with identical letter parts.

Like terms have exactly the same letters raised to the same powers. Keep track of the sign in front of each term. $x^2 + 3x$ cannot be simplified because $x$ and $x^2$ are never like terms.

Like terms add: 3x + 5x = 8x, so the lines y = 3x and y = 5x sum to y = 8x (dashed)

Like terms

$3x$ and $5x$
$2ab$ and $7ab$

Unlike terms

$3x$ and $3x^2$
$2a$ and $2b$

Worked example

Simplify $2a + 3b + 5a - 9b$ .

1
Group the $a$ terms: $2a + 5a = 7a$ .
2
Group the $b$ terms: $3b - 9b = -6b$ .

Simplify $4x + 3 + 2x - 1$ .

Algebra from words

Translate one phrase at a time, defining a letter for the unknown first.

"Total cost is a £3 call-out fee plus £5 per hour, $h$ " gives $C = 3 + 5h$ . Say "let $n$ = the number of items" before writing the expression: this earns method marks even if the final expression has an error.

The formula C = 3 + 5h: a 4-hour job costs £23

Phrase

2 more than $n$
3 less than $n$
Double $n$
$n$ items at £5 each

Algebra

$n + 2$
$n - 3$
$2n$
$5n$

Worked example

Using $C = 3 + 5h$ , find the cost of a 4-hour job.

1
Substitute $h = 4$ : $C = 3 + 5(4)$ .
2
$C = 3 + 20$ .

"5 more than three times $n$ " is...

Try an exam question

(a) Simplify $4a + 3b - a + 5b$ . (b) Using $C = 3 + 5h$ , where $C$ is the cost in pounds and $h$ is the hours worked, find the cost of a 6-hour job.

[4 marks]

That's the notes covered.

Carry on to the next subtopic.