FA1, FA3, FA17, HA1, HA3, HA21 - Writing algebraic expressions | Eduqas GCSE Mathematics

Algebra describes a calculation before you know the numbers. Learn to read its compact notation, distinguish expressions from mathematical statements, and write expressions and formulae from words. FA1 and HA1 supply the notation, FA3 and HA3 its vocabulary, and FA17 and HA21 connect it to situations.

Algebraic notation

A ticket costs $p$ pounds and the booking fee is £2. How can you write the cost of three tickets, with one booking fee, before knowing the ticket price?

A letter can represent a number. If each ticket costs $p$ pounds, three tickets cost $p+p+p$ , or $3\times p$ . We shorten this to $3p$ . The missing multiplication sign saves space; it does not turn the letter into a digit.

Coefficient

A coefficient is the numerical multiplier of a term. In $3p$ , the coefficient of $p$ is $3$ . In $p$ , it is $1$ ; in $-p$ , it is $-1$ .

Use these conventions to read an expression back into operations.

Algebraic notation	Meaning
$ab$	$a\times b$
$a^2$ , $a^3$	$a\times a$ , $a\times a\times a$
$a^2b$	$a\times a\times b$
$\frac{a}{b}$	$a\div b$ , with $b\ne0$
$\frac{3}{4}y$	three quarters of $y$
$3(a+b)$	three times the whole sum $a+b$

The fraction bar groups its numerator: $\frac{a+b}{4}$ means $(a+b)\div4$ . By contrast, $a+\frac{b}{4}$ divides only $b$ by $4$ . Brackets and fraction bars tell you which calculation belongs together.

Worked example 1: reading products and powers

Question. Write $5\times a\times a\times b$ in algebraic notation. Explain what its power counts.

Plan. Put the number first and replace the repeated factor $a\times a$ by a square.

\begin{aligned} 5\times a\times a\times b&=5a^2b &&\text{two factors of }a \end{aligned}

Answer. $5a^2b$ . The power $2$ applies only to $a$ , so there is one factor of $b$ .

Check. With $a=2$ and $b=3$ , the original product is $5\times2\times2\times3=60$ ; the compact form gives $5\times2^2\times3=60$ .

Change the multiplier to a fraction. Write fractional coefficients exactly rather than as rounded decimals.

Worked example 2: a fractional multiplier

Question. Write $\frac{2}{3}\times a\times a\times b$ in algebraic notation. State its coefficient.

Plan. Keep the fraction as one multiplier and shorten the repeated letters.

\frac{2}{3}\times a\times a\times b=\frac{2}{3}a^2b

The fraction multiplies the whole product $a^2b$ .

Answer. $\frac{2}{3}a^2b$ , with coefficient $\frac{2}{3}$ .

Check. For $a=3$ , $b=2$ , both forms give $\frac{2}{3}\times9\times2=12$ .

Do not confuse multiplying by a number with raising to a power.

Watch out

$3a$ means $a+a+a$ ; $a^3$ means $a\times a\times a$ . For $a=2$ , they give $6$ and $8$ , so they are different expressions.

Use the same notation with new letters.

Algebraic vocabulary

An expression names a value; a mathematical statement says something about values.

In $4x+7$ , addition joins two pieces: $4x$ and $7$ . In $4x$ , multiplication joins $4$ and $x$ . This gives two different ways to describe the structure.

Terms and factors

Terms are the pieces added or subtracted in an expression: $4x-7$ has terms $4x$ and $-7$ . Factors are pieces multiplied together: $4$ and $x$ are factors of $4x$ . The whole bracket $(x+2)$ is a factor of $3(x+2)$ .

A term can contain several factors. The sign belongs with its term: in $5a-2b+6$ , the terms are $5a$ , $-2b$ and $6$ . A term without a variable is called a constant term.

Name	Example	What it tells you
Expression	$3n+2$	A calculation or value, with no equality or inequality sign
Equation	$3n+2=14$	Two expressions are equal; here this is true when $n=4$
Identity	$n+n\equiv2n$	The expressions are equal for every allowed value of the variable
Formula	$C=3p+2$	A rule connecting quantities, here total cost and ticket price
Inequality	$n+2<10$	One expression is smaller or greater, possibly allowing equality

An identity uses $\equiv$ to emphasise “true for every allowed value”. You also meet identities written with $=$ . An equation and a formula both use equality; “formula” describes how an equation is used to connect quantities. These categories need not be mutually exclusive.

For example, $x+4=9$ holds only when $x=5$ , whereas $x+4\equiv4+x$ holds for every $x$ because changing the order of addition preserves the sum. Trying a few values can disprove an identity if one fails; a few successful trials alone cannot establish it for every value.

An inequality uses $<,>,\le$ or $\ge$ . For $n+2\le10$ , equality is allowed: $n=8$ satisfies it. A bare expression such as $n+2$ makes no claim that could be true or false.

Key idea

Look at the operations and the relation sign. Terms are joined by addition or subtraction; factors by multiplication. An expression has no relation sign, while equations, identities and inequalities make statements.

Choose the description that fits both the sign and the values allowed.

Writing expressions

To translate words into algebra, choose what each letter means and preserve the order of the operations.

Start with a number $n$ . “Add 5, then double the result” gives $2(n+5)$ , because the whole result of the addition is doubled. “Double the number, then add 5” gives $2n+5$ . At $n=3$ , these are $16$ and $11$ . The same words in a different order can describe a different calculation.

Words such as “less than” need care. “Four less than $n$ ” is $n-4$ , while “ $n$ less than four” is $4-n$ . Read your expression as instructions to check the order.

Worked example 3: one fee for the whole purchase

Question. Tickets cost $p$ pounds each. There is one £2 booking fee for the purchase. Write an expression for the cost of three tickets.

Plan. Calculate the ticket cost, then add the fee once.

\begin{aligned} \text{ticket cost}&=3p &&\text{three equal prices}\\ \text{total cost}&=3p+2 &&\text{one booking fee} \end{aligned}

Answer. $3p+2$ pounds. If $C$ is the total cost in pounds, a formula is $C=3p+2$ .

Check. At £5 per ticket, three tickets cost £15 and the fee makes £17. The expression gives $3\times5+2=17$ .

Now change the fee condition: the £2 charge applies to each ticket. The brackets must show that change.

Worked example 4: a fee on each ticket

Question. Each ticket costs $p$ pounds plus a £2 charge per ticket. Write an expression for the cost of three tickets.

Plan. Group the complete cost of one ticket, then multiply by three.

\begin{aligned} \text{one ticket with its charge}&=p+2\\ \text{total cost}&=3(p+2) &&\text{three complete costs} \end{aligned}

Answer. $3(p+2)$ pounds, or $C=3(p+2)$ as a formula for the total cost $C$ in pounds.

Check. At $p=5$ , each ticket costs £7 including its charge, so the total is £21. The expression gives $3(5+2)=21$ .

Division can also apply to a whole result: “add 6 to $n$ , then halve” is $\frac{n+6}{2}$ . The fraction bar performs the same grouping job as brackets.

Watch out

If asked for an expression, write $n-4$ , not $n=n-4$ . The added equality wrongly says that the starting number equals the number four less than itself. A formula needs a separately defined output, such as $r=n-4$ .

The opening ticket question has one fee for the whole purchase, so its answer is $3p+2$ . Decide what is charged per item before choosing brackets.

Quick Recap

A missing multiplication sign, a power, a fraction bar and brackets each have a precise meaning.
Terms are added or subtracted; factors are multiplied.
Expressions name values; equations, identities and inequalities state relationships. A formula connects defined quantities.
Write the operations in their stated order and check them using a simple number.

Three tickets with one £2 booking fee cost $3p+2$ pounds.

Last Accessed

Number

Algebra

Ratio and proportion

Geometry and measures

Higher graphs

Probability

Statistics

FA1, FA3, FA17, HA1, HA3, HA21 - Writing algebraic expressions

Algebraic notation

Algebraic vocabulary

Writing expressions

Quick Recap

Learn faster with Nova