Algebraic Expressions

Writing, simplifying, and evaluating algebraic expressions using variables and like terms.

Variables and Expressions

A variable is a letter that stands for a number (e.g. $x$ , $n$ , $a$ ).

An algebraic expression combines numbers, variables, and operations.

Word phrase	Expression
3 more than $y$	$y + 3$
twice $x$	$2x$
half of $n$	$\dfrac{n}{2}$
$a$ squared	$a^2$
5 less than $k$	$k - 5$

Parts of an Expression

In $3x + 2y - 7$ :

Terms: $3x$ , $2y$ , $-7$
Coefficients: 3, 2
Constant: $-7$
Variables: $x$ , $y$

Like Terms

Like terms have the same variable(s) and powers. Only like terms can be combined.

$\underbrace{3x + 5x}_{\text{like}} = 8x$ ✓

$3x + 5y$ — cannot simplify (different variables)

Simplify: 4a + 3b − 2a + 7b

Group like terms: $(4a - 2a) + (3b + 7b) = 2a + 10b$

Evaluating Expressions

Substitute the given value for the variable.

Find the value of 3x − 4 when x = 5

$3(5) - 4 = 15 - 4 = 11$

Evaluate 2a² + b when a = 3, b = 1

$2(9) + 1 = 18 + 1 = 19$

Real world: mobile data tariff

A phone plan charges a monthly fee of €8 plus €0.02 per MB of data used. The total bill can be written as $C = 8 + 0.02d$ , where $d$ is MB used. If a customer uses 500 MB: $C = 8 + 0.02 \times 500 = 18\text{ EUR}$ . Algebraic expressions let consumers predict costs before exceeding their data limit.

Real world: perimeter of a fenced field

A rectangular field has length $l$ metres and width $w$ metres. The perimeter expression $P = 2l + 2w$ helps a farmer quickly calculate how much fencing is needed for any field size. If $l = 50$ and $w = 30$ : $P = 2(50) + 2(30) = 160\text{ m}$ .

Real world: pay-per-hour work

A builder earns €15 per hour. After $h$ hours the total earnings are $E = 15h$ . If a week's work includes 5 days of different hours, substituting each day's hours and summing the expressions gives the weekly pay. Algebraic expressions are the foundation of every payroll system.

Tip When substituting into an expression, replace the variable with the value in brackets to avoid sign errors: write

3(x) - 4

rather than

3x - 4

before substituting

x = -2

Practice Problems

Simplify: 6m + 4n − 2m − n

Show answer

4m + 3n.

Evaluate 5x + 2 when x = 4.

Show answer

5(4) + 2 = 22.

Write an expression for the perimeter of a rectangle with length l and width w, then evaluate it when l = 7 and w = 3.

Show answer

P = 2(l + w) = 2(7 + 3) = 20.