Variables and Expressions: Letters That Stand for Numbers

A letter standing in for a number

A variable is a letter used in place of a number.

$$3n + 5$$

The $n$ is standing in for something. It is used in two different situations:

• The value is unknown. You are trying to work out what it is.
• The value varies. It changes from case to case, which is where the name comes from.

A taxi charging $5$ dollars plus $3$ per mile costs $3n + 5$ dollars for $n$ miles. One expression covers every possible journey.

Miles ($n$)	Cost
$2$	$11$
$5$	$20$
$10$	$35$

Writing out a separate sum for each distance would never end. One expression with a letter in it covers all of them at once.

Expressions and equations

	Has an equals sign?	Example
expression	no	$3n + 5$
equation	yes	$3n + 5 = 20$

An expression is a quantity — something to evaluate. An equation is a claim that two quantities are equal — something to solve.

$$3n + 5 \quad \text{means "the value of this"}$$ $$3n + 5 = 20 \quad \text{asks "which } n \text{ makes this true?"}$$

Why $3n$ means multiplication

Two symbols written next to each other are being multiplied:

$$3n = 3 \times n \qquad ab = a \times b \qquad 5(x + 2) = 5 \times (x + 2)$$

The $\times$ sign is dropped because it is nearly identical to the letter $x$, which algebra uses constantly. In an expression like $3 \times x + 2 \times x$ the crosses and the letters blur together, and $3x + 2x$ does not.

A few related conventions follow the same logic:

Written	Means	Note
$3n$	$3 \times n$	number first
$\tfrac{n}{4}$	$n \div 4$	a fraction bar, not $\div$
$n^2$	$n \times n$	not $2n$
$n$	$1 \times n$	a lone letter has coefficient $1$

Writing an expression from words

Name the unknown, then translate phrase by phrase.

Words	Expression
five more than a number	$n + 5$
five less than a number	$n - 5$
a number decreased by five	$n - 5$
twice a number	$2n$
a number divided by five	$\tfrac{n}{5}$
twice a number, plus five	$2n + 5$
twice the sum of a number and five	$2(n + 5)$

The last two rows are the pair worth studying. “Twice a number plus five” doubles only the number. “Twice the sum” doubles everything, so the sum needs brackets around it.

Subtraction has a trap of its own. “Five less than a number” is $n - 5$, not $5 - n$ — the five is being taken away from the number, so the number comes first even though it is written second.

Why letters are worth the trouble

An expression with a letter in it says something a list of numbers cannot: it records the rule, not the results.

"$3n + 5$" tells you the taxi charges a flat $5$ plus $3$ a mile. That is visible in the expression itself — the $5$ standing alone is the fixed charge and the $3$ attached to $n$ is the rate. A table of fares hides that structure and runs out at the last row.

Because the rule is written down, it can be worked with. It can be evaluated at any distance, rearranged, compared against a different company’s rule, or set equal to a budget and solved. None of that is available from a list of prices, and all of it comes from the one letter.

Worked examples

Common mistakes

Practice problems