C2.1, C2.5; E2.1, E2.5 - Letters, expressions and substitution | Cambridge IGCSE Mathematics

Letters let you describe a calculation before you know all its numbers. You will read algebraic notation, turn words into expressions and formulas, and find their values by substitution, including negative numbers and fractions.

Letters as generalised numbers

How can one short calculation describe the total price of three identical tickets, whatever one ticket costs? If a ticket costs $4$ dollars, the total is $3\times4$ . At $7$ dollars, it is $3\times7$ . Replace the changing price by the letter $p$ , and both calculations become $3\times p$ , written $3p$ .

The letter represents a number, rather than the word “price”. In this situation, $p$ is the price of one ticket in dollars. Writing what a letter measures, including its units, gives the expression a clear meaning.

Variable

A variable is a letter representing a number that can take different values. In one evaluation, repeated copies of the same letter have the same value. Different letters may have different values, or may happen to have the same value.

If $p=4$ , every $p$ in the calculation means $4$ . You cannot make one copy $4$ and another copy $7$ . A letter can also represent a fixed but unknown number; the context tells you how it is being used.

An expression is a number, a letter or a combination of numbers and letters joined by operations. It does not state that two quantities are equal. These conventions shorten expressions:

Written form	Meaning	If $a=3$ and $b=5$
$4a$	$4\times a$	$12$
$ab$	$a\times b$	$15$
$a^2$	$a\times a$	$9$
$2a^2$	$2\times(a\times a)$	$18$
$(2a)^2$	$(2a)\times(2a)$	$36$
$\frac{a+b}{2}$	$(a+b)\div2$	$4$

Write the number first in a product: $4a$ , rather than $a4$ . Unlike a two-digit numeral, $ab$ means multiplication, not placing digits together. The small raised number is a power or index: $a^3=a\times a\times a$ . An index applies to the expression immediately before it; brackets can make that expression larger.

Watch out

$2a$ and $a^2$ have different meanings. With $a=3$ , they are $6$ and $9$ . They happen to agree when $a=2$ , but that one agreement does not make them the same expression for every value.

The ticket expression $3p$ works for many prices because only the price changes; the instruction “multiply by three” stays fixed.

Expressions and formulas from words

Translate the operations in a sentence, keeping their order. “Five less than $n$ ” starts with $n$ and subtracts $5$ , so it is $n-5$ . “Five minus $n$ ” is $5-n$ . With $n=8$ , these give $3$ and $-3$ , so the order matters.

Brackets preserve a quantity that must be treated as one unit. “Three times the sum of $n$ and $4$ ” is $3(n+4)$ . “Four more than three times $n$ ” is $3n+4$ . Read your expression back into words to check it.

Formula

A formula is an equation showing a relationship between quantities. For example, $C=3p$ says that total cost $C$ is three times price $p$ . The equals sign means that both sides have the same value.

An expression such as $3p$ names a quantity; the formula $C=3p$ tells you how that quantity relates to another one. You can construct a formula without knowing a particular price.

Worked example 1: a fixed charge and a rate

Question. A bicycle hire costs $6$ dollars plus $4$ dollars for each hour. Write a formula for the total cost $C$ dollars for $h$ hours.

Plan. Find the charge for all the hours, then add the fixed charge once.

\begin{aligned} \text{hourly charge}&=4h && \text{four dollars per hour}\\ C&=6+4h && \text{add the fixed six dollars} \end{aligned}

Answer. $C=6+4h$ , where $h\ge0$ .

Check. For $h=3$ , the words give $6+4+4+4=18$ . The formula gives $6+4\times3=18$ .

Now keep the two rates but change which charge repeats: there is a separate fixed charge for each bicycle.

Worked example 2: the whole charge repeats

Question. Each bicycle costs $6$ dollars plus $4$ dollars per hour to hire. Write a formula for the total cost $C$ dollars of hiring $b$ bicycles for $h$ hours each.

Plan. Put one bicycle's complete charge in brackets, then multiply it by the number of bicycles.

\begin{aligned} \text{one bicycle's cost}&=6+4h && \text{fixed charge plus hours}\\ C&=b(6+4h) && \text{repeat the whole cost} \end{aligned}

Answer. $C=b(6+4h)$ , where $b$ is a positive integer and $h\ge0$ .

Check. Two bicycles hired for one hour cost $10+10=20$ dollars. The formula gives $2(6+4\times1)=20$ .

The algebra can accept many numbers, but a context may restrict them: a number of bicycles is a whole number, and a hire duration cannot be negative.

Watch out

For the second example, $C=6+4bh$ would charge the fixed $6$ dollars only once. The brackets in $b(6+4h)$ show that each bicycle has both charges.

Check the meaning of the brackets before doing any arithmetic.

Substituting positive and negative values

To substitute, replace every occurrence of a letter with its given number, then use the usual order of operations. Write a negative replacement inside brackets so its sign stays attached to it. The letter's value changes between evaluations; the expression's operations do not.

For example, if $x=-4$ , then $x^2=(-4)^2=16$ . By contrast, $-x^2=-((-4)^2)=-16$ : the minus sign outside the square is still outside. In $3x$ , substitution gives $3(-4)$ , a product.

Worked example 3: substituting a positive number

Question. Find the value of $2x^2-3x+1$ when $x=3$ .

Plan. Replace both copies of $x$ , calculate the square, then the products.

\begin{aligned} 2x^2-3x+1&=2(3)^2-3(3)+1 && \text{replace every }x\\ &=2(9)-9+1 && \text{square before multiplying}\\ &=18-9+1 && \text{complete the products}\\ &=10 && \text{add and subtract} \end{aligned}

Answer. $10$ .

Check. Calculate the three original terms separately: $18$ , $-9$ and $1$ . Their sum is $10$ .

Change only the sign of the input. The squared term stays positive, but the term $-3x$ changes sign.

Worked example 4: substituting a negative number

Question. Find the value of $2x^2-3x+1$ when $x=-3$ .

Plan. Bracket each negative replacement before calculating powers and products.

\begin{aligned} 2x^2-3x+1&=2(-3)^2-3(-3)+1 && \text{replace both copies}\\ &=2(9)+9+1 && \text{square; subtract a negative}\\ &=18+9+1 && \text{complete the product}\\ &=28 && \text{add} \end{aligned}

Answer. $28$ .

Check. The squared term and constant are unchanged from the positive-input case. Replacing $-9$ by $+9$ increases the answer by $18$ : $10+18=28$ .

A fraction is a valid input too. Keep it exact. For $P=2a+1$ , substitution of $a=\frac34$ gives

P=2\left(\frac34\right)+1=\frac32+1=\frac52.

The same method applies to a formula: substitute all the known quantities and calculate the remaining side. If a letter is in a denominator, its given value must not make that denominator zero.

Watch out

Writing $2(-3)^2=-18$ loses the brackets' meaning. The square is $(-3)(-3)=9$ , so $2(-3)^2=18$ . The expression $2\times(-3^2)$ would have a different value.

You can now answer the opening ticket question for any allowed ticket price by substituting that price into $3p$ .

Quick Recap

A letter represents a number; repeated copies use the same value within one evaluation.
$3x$ , $x^3$ and $(3x)^2$ give different instructions.
Translate words in order and use brackets to keep a whole quantity together.
Substitute into every occurrence, bracket negative inputs and keep fractions exact.

Three identical tickets cost $3p$ : one expression describes every possible ticket price $p$ .

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C2.1, C2.5; E2.1, E2.5 - Letters, expressions and substitution

Letters as generalised numbers

Expressions and formulas from words

Substituting positive and negative values

Quick Recap

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