1MA1 · A1–A3AlgebraFoundation & HigherGrades 1–4

Algebraic Notation & Substitution

Q: What is a coefficient?

The number multiplying a letter in a term. In -7x^2 the coefficient is -7.

Q: Are ab and ba like terms?

Yes. Multiplication order does not matter, so 3ab + 2ba = 5ab.

Algebra uses letters to stand for numbers so that one statement can describe every case at once. Before solving anything, you need the vocabulary — term, expression, equation, formula, identity — and two basic skills: tidying expressions by collecting like terms, and substituting numbers into formulae without slipping on negatives or order of operations.

40 min Algebraic Expressions

The big picture

Every later algebra topic is built on this one. An exam question that says "show that", "write an expression" or "use the formula" is testing whether you read algebraic notation precisely: that $3x^{2}$ means $3 \times x \times x$ and not $(3x)^{2}$ , and that $ab$ means $a \times b$ .

Edexcel regularly asks candidates to identify which of a list is an expression, equation, formula or identity. It sounds like vocabulary, but getting it right shows you understand what each kind of statement claims.

What you'll learn

Use and interpret algebraic notation, including powers and fractions
Distinguish expressions, equations, formulae and identities
Simplify expressions by collecting like terms
Substitute positive and negative values into expressions and formulae

The language of algebra

A $\textbf{term}$ is a single number, letter or product, such as $5$ , $x$ or $4ab^{2}$ . An $\textbf{expression}$ is a collection of terms with no equals sign: $3x + 2y - 7$ .

An $\textbf{equation}$ is true only for particular values: $2x + 1 = 9$ is true only when $x = 4$ .

A $\textbf{formula}$ links different quantities: $A = \pi r^{2}$ . An $\textbf{identity}$ is true for every value, written with $\equiv$ : $2(x + 3) \equiv 2x + 6$ .

Example 1Classify: (a)

v = u + at

(b)

5x - 3 = 12

(c)

4(y - 1) \equiv 4y - 4

(d)

7p^{2} - p

1(a) links quantities: formula.

2(b) true for one value: equation.

3(c) true for all values: identity.

4(d) no equals sign: expression.

Answer(a) formula (b) equation (c) identity (d) expression

Notation conventions

Multiplication signs are omitted: $3 \times a \times b$ is written $3ab$ , with the number first and letters usually in alphabetical order.

Powers apply only to what they touch: $3x^{2} = 3 \times x^{2}$ , while $(3x)^{2} = 9x^{2}$ .

Division is written as a fraction: $a \div 4 = \dfrac{a}{4}$ .

Why this works

The difference between $3x^{2}$ and $(3x)^{2}$ is exactly the order of operations: powers before multiplication unless brackets say otherwise. When $x = 2$ , one is $12$ and the other is $36$ .

Simplifying by collecting like terms

$\textbf{Like terms}$ have exactly the same letters raised to the same powers. $4x$ and $-9x$ are like terms; $4x$ and $4x^{2}$ are not.

Add or subtract the coefficients of like terms, keeping each term’s sign attached to it.

When multiplying terms, multiply numbers and combine letters using powers: $3a \times 5a^{2}b = 15a^{3}b$ .

Example 1Simplify

5x^{2} - 3x + 2 - 2x^{2} + 7x - 9

x^{2}

terms:

5x^{2} - 2x^{2} = 3x^{2}

x

terms:

-3x + 7x = 4x

3Numbers:

2 - 9 = -7

Answer

3x^{2} + 4x - 7

Your turnSame method, new numbers — try it before moving on.

Simplify $4a^2 + 3a - a^2 - 7a + 5$ .

Your answer

Type powers with ^, e.g. 3x^2 - 4x.

Show worked solution

Solution

1 $a^2$ terms: $4a^2 - a^2 = 3a^2$ .
2 $a$ terms: $3a - 7a = -4a$ . The constant is $+5$ .

Answer: $3a^2 - 4a + 5$

Tip — Circle or underline each group of like terms, sign included, before combining. Dropping a minus sign is the usual slip.

Substitution

Replace each letter with its value, in brackets, then follow the order of operations.

Brackets matter most with negatives: if $x = -3$ , then $x^{2} = (-3)^{2} = 9$ but $-x^{2} = -(3^{2}) = -9$ .

For a formula, substitute and then answer with units where relevant.

Example 1Given

a = -2

and

b = 5

, find the value of

3a^{2} - 4ab + \dfrac{b}{a}

3a^{2} = 3 \times (-2)^{2} = 12

-4ab = -4 \times (-2) \times 5 = 40

\dfrac{b}{a} = \dfrac{5}{-2} = -2.5

4Total:

12 + 40 - 2.5

Answer

49.5

Your turnSame method, new numbers — try it before moving on.

Given $a = -3$ and $b = 2$ , find the value of $2a^2 - 5b + ab$ .

Your answer

Show hint

$a^2 = (-3)^2 = 9$ .

Show worked solution

Solution

1 $2a^2 = 18$ , $5b = 10$ , $ab = -6$ .
2 $18 - 10 - 6 = 2$ .

Answer: $2$

Example 2Use

s = ut + \tfrac12at^{2}

to find

s

when

u = 4

a = -1.5

and

t = 6

ut = 24

\tfrac12 \times (-1.5) \times 36 = -27

s = 24 - 27

Answer

s = -3

Think like an examiner

Substitute with brackets and show each term’s value before combining.
Use the correct symbol: $=$ for equations and formulae, $\equiv$ for identities.
Leave simplified expressions with like terms fully collected.

Notation

3x^{2} = 3 \times x \times x

Power applies to x only.

(3x)^{2} = 9x^{2}

Power applies to the whole bracket.

2(x + 3) \equiv 2x + 6

An identity: true for all x.

Watch out for these

Combining

3x + 2y

into

5xy

Unlike terms cannot be combined.

Evaluating

x^{2}

for

x = -4

-16

Use brackets:

(-4)^{2} = 16

Writing

a \times a \times a

3a

It is

a^{3}

Calling

3(x + 1) = 3x + 3

an equation.

It is true for all

x

, so it is an identity.

Stretch yourself

The perimeter of a rectangle with length $(3x + 2)$ and width $(x - 1)$ is $P$ . Write a simplified expression for $P$ , then find the area when $P = 42$ .

Hint — Perimeter adds all four sides. Solve $P = 42$ for $x$ , then multiply length by width.

Reveal solution

Solution

P = 2(3x + 2) + 2(x - 1) = 6x + 4 + 2x - 2 = 8x + 2

. Setting

8x + 2 = 42

gives

x = 5

. The length is

17

and the width is

4

, so the area is

68

square units.

\blacksquare

Questions students ask

What is a coefficient?

The number multiplying a letter in a term. In

-7x^{2}

the coefficient is

-7

Are

ab

and

ba

like terms?

Yes. Multiplication order does not matter, so

3ab + 2ba = 5ab

How do I know if something is an identity?

Simplify both sides. If they become exactly the same expression, it is true for every value and is an identity.

Key takeaways

Terms, expressions, equations, formulae and identities have distinct meanings.
Omit multiplication signs; powers apply only to what they touch.
Collect like terms, keeping each term’s sign.
Substitute in brackets, especially negative values.
Follow the order of operations after substituting.

Exam-style questions

Practise it the way it is examined

3 original questions · 7 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 23 marks

(a)Simplify $3m \times 4n \times m$ .[1]
Show mark scheme (a)
- B1 $12m^2n$
Answer: $12m^2n$
(b)Simplify $7p - 2q - 3p + 5q$ .[2]
Show mark scheme (b)
- B1One correct term
- B1 $4p + 3q$
Answer: $4p + 3q$

Question 2Grade 42 marks

$T = 3a^2 - 2b$ . Work out the value of $T$ when $a = -4$ and $b = -5$ .[2]
Show mark scheme
- M1Correct substitution: $3 \times 16 - 2 \times (-5)$
- A1 $58$
Answer: $58$

Question 3Grade 42 marks

The formula $F = \frac{9}{5}C + 32$ converts degrees Celsius to degrees Fahrenheit. Find $C$ when $F = 77$ .[2]
Show mark scheme
- M1 $77 - 32 = 45$ or $\frac{9}{5}C = 45$
- A1 $C = 25$
Answer: $25$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

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