Variables and expressions

What a variable is, how to read and write algebraic expressions, and how to evaluate and simplify them.

Algebra is arithmetic with a twist: some of the numbers are not known yet, or can change. Instead of a number we write a letter, and the letter stands in for it. This lesson gives you the vocabulary you will use in every later lesson.

Variables

A variable is a letter (usually $x$ , $y$ , $n$ or $t$ ) that stands for a number. It is "variable" because it can take different values in different situations.

Why bother? Because a letter lets us describe a whole family of situations at once. A taxi charges a fixed fee of 3 plus 2 per kilometre. For 5 km the price is $3 + 2 \cdot 5 = 13$ , for 8 km it is $3 + 2 \cdot 8 = 19$ . With a variable $k$ for the distance, one formula covers every trip:

$price = 3 + 2 k$

Expressions

An algebraic expression combines numbers, variables and operations (+, −, ×, ÷, powers). Some examples:

$3 x + 5$
$a^{2} - 4 b$
$\frac{n + 1}{2}$

An expression is not a sentence that is true or false; it is a recipe that gives a number once the variables have values. (An equation, with an equals sign, comes in the next lesson.)

Some vocabulary, using $7 x^{2} - 3 x + 4$ :

Word	Meaning	Here
Term	A piece separated by + or −	$7 x^{2}$ , $- 3 x$ , $4$
Coefficient	The number multiplying the variable	$7$ and $- 3$
Constant	A term with no variable	$4$

Multiplication is usually written without a sign: $3 x$ means $3 \cdot x$ , and $ab$ means $a \cdot b$ .

Evaluating an expression

To evaluate an expression, replace each variable by its value and compute, respecting the order of operations: brackets, then powers, then multiplication and division, then addition and subtraction.

Example 1. Evaluate $2 x^{2} - 3 x + 1$ for $x = 4$ .

$2 \cdot 4^{2} - 3 \cdot 4 + 1 = 2 \cdot 16 - 12 + 1 = 32 - 12 + 1 = 21.$

Example 2. Evaluate $\frac{a + b}{2}$ for $a = 7$ , $b = 11$ . This is the average of two numbers:

$\frac{7 + 11}{2} = \frac{18}{2} = 9.$

Careful with negatives. For $x = - 3$ , the expression $x^{2}$ is $(- 3)^{2} = 9$ , not $- 9$ . Put the value in brackets when you substitute.

Simplifying: combining like terms

Like terms have exactly the same variable part, so they can be added together: $3 x + 5 x = 8 x$ , because three of something plus five of the same thing is eight of it. But $3 x + 5 x^{2}$ cannot be combined, since $x$ and $x^{2}$ are different things.

Example 3. Simplify $4 a + 3 b - a + 2 b$ .

Group like terms: $(4 a - a) + (3 b + 2 b) = 3 a + 5 b$ .

The distributive property

Multiplying a bracket distributes the factor over each term:

$a (b + c) = ab + a c .$

Example 4. Simplify $3 (x + 4) - 2 x$ .

$3 x + 12 - 2 x = x + 12.$

Check by trying a value: for $x = 1$ , the original is $3 \cdot 5 - 2 = 13$ and the result is $1 + 12 = 13$ . Substituting a test value is a quick way to catch mistakes.

Translating words into expressions

In words	Expression
Five more than a number	$n + 5$
Twice a number, minus three	$2 n - 3$
The square of a number, divided by four	$\frac{n ^{2}}{4}$
Three consecutive integers, starting at $n$	$n, n + 1, n + 2$

Common mistakes

Writing $2 + 3 x = 5 x$ . The terms $2$ and $3 x$ are not alike; the expression stays as it is.
Forgetting to distribute to every term: $2 (x + 3) = 2 x + 6$ , not $2 x + 3$ .
Treating $- x^{2}$ and $(- x)^{2}$ as equal. The first is $- (x^{2})$ ; the second is $x^{2}$ .

Practice

Evaluate $5 - 2 x$ for $x = 3$ and for $x = - 3$ .
Simplify $7 y - 4 + 2 y + 9$ .
Simplify $2 (3 a - 1) + 4 (a + 2)$ .
Write an expression for "the perimeter of a rectangle whose length is $3$ more than its width $w$ ".

Answers

$- 1$ and $11$ .
$9 y + 5$ .
$6 a - 2 + 4 a + 8 = 10 a + 6$ .
$2 w + 2 (w + 3) = 4 w + 6$ .

Next: now that we can write expressions, we can ask when two of them are equal: Solving linear equations.