Introduction

When solving problems it is sometimes necessary to translate words, a table of data, or a physical model into expressions. There are two types of expressions: variable and numerical.

Variable expressions have at least one variable (sometimes called an unknown) in them. Numerical expressions involve only numbers. Here are examples of each:

Variable expressions

Numerical expressions

8 + m

g - 17

$n-12+3$

8 + 13

29 - 17

$25-12+3$

Here is a table of key words that indicate addition and subtraction.

Operation	Verbal expressions (key words)	Variable expression
$\large~+$	add 3 to a number a number plus 3 the sum of a number and 3 3 more than a number a number increased by 3	$\large~n+3$
$\large~-$	subtract 12 from a number a number minus 12 the difference of a number and 12 12 less than a number a number decreased by 12	$\large~x-12$

Example 1

Anglea and Bill are siblings. When Angela was 5 years old, her brother was 8 years old. When Angela was 6 years old, her brother was 9 years old. If Angela is n years old, how old is Bill?

To find Bill's age, begin by placing their ages in a table...

When Angela's age is...	Bill's age is...
5	8
6	9
7	10
12
n	n+3

Notice that Bill is always 3 years older than Angela. So when Angela is 12 years old Bill will be 15 years old.

When Angela is n years old, Bill will be (n+3) years old.

Example 2

Alan is 8 years old.

Question: How old will he be in 5 years' time? Answer: 8 + 5
Question: How old will he be in x years? Answer: 8 + x

Example 3

Question: Suppose that you buy something that costs b dollars. Suppose that you pay with a $10 dollar note. Write an expression for the change that you get. If needed, this diagram might help you understand the situation.

Answer: 10 - b

EXPLORE!

Move the red point up and down the number line and observe how the blue point moves in relation.
What variable expression describes the blue point, if we consider the red point as $\large~x$ ?