Algebraic Expressions: Variables, Terms, Coefficients and Like Terms

Q: What is the difference between an expression and an equation?

An expression is a phrase like 3x + 5: it has a value but makes no claim. An equation states that two expressions are equal, like 3x + 5 = 11, and can be solved.

Q: Are $xy$ and $yx$ like terms?

Yes. Multiplication can be done in any order, so xy = yx, and 3xy + 2yx = 5xy.

The vocabulary of algebra in plain words: variables, terms, coefficients and constants. How to translate words into expressions, evaluate them safely, and simplify by collecting like terms.

3 min read Supports 4 lessons

Algebra starts with one idea: use a letter to stand for a number. The letter might be a number we do not know yet, or one that can change. Either way it is called a variable.

Once numbers have names, one calculation can describe many situations at once. If a taxi charges £3 plus £2 per mile, the fare for $m$ miles is $2m + 3$ , whatever $m$ turns out to be.

The vocabulary

\underbrace{4x^2}_{\text{term}} \;-\; \underbrace{7x}_{\text{term}} \;+\; \underbrace{5}_{\text{constant}}

An expression made of three terms.

A term is a single piece separated by $+$ or $-$ signs. The number multiplying the variable part is its coefficient: in $-7x$ the coefficient is $-7$ . A term with no variable, like $5$ , is a constant.

Algebra has some shorthand of its own. $3x$ means $3 \times x$ ; $\frac{x}{2}$ means $x \div 2$ ; $x^2$ means $x \times x$ ; and $xy$ means $x \times y$ .

Words into algebra

Worked example

Write an expression for five less than twice a number $n$.

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"Twice a number" is $2n$ .

Watch out

" $5$ less than $x$ " is $x - 5$ , not $5 - x$ . Read carefully: the thing you start with comes first.

Evaluating

To evaluate an expression, substitute a value for each variable and calculate, following the order of operations.

$2x^2 - 3y \quad\text{when } x = -3,\; y = 4$
Substitute each value.

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Tip

Brackets protect you. Without them, $2 \cdot -3^2$ invites the mistake $-3^2 = -9$ .

Like terms

Like terms have exactly the same variable part, including powers. $3x$ and $-5x$ are like terms. $3x$ and $3x^2$ are not, and neither are $3x$ and $3y$ .

Combine like terms by adding their coefficients. Think of $x$ as a unit, like apples: $4$ apples minus $2$ apples is $2$ apples. You cannot merge apples with pears.

$4x + 7 - 2x + 3$
Spot the like terms: the $x$ terms and the constants.

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Worked example

Simplify $3a^2 - 2a + 5a^2 + 7a$ .

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Group like terms.
$(3a^2 + 5a^2) + (-2a + 7a)$

Watch out

$x + x = 2x$ , but $x \times x = x^2$ . Adding like terms changes the coefficient; multiplying changes the power.

Key idea

Simplifying never changes the value of an expression. $4x + 7 - 2x + 3$ and $2x + 10$ give the same answer for every $x$ . That is a good way to check your work.

Common questions

What is the difference between an expression and an equation?: An expression is a phrase like $3x + 5$ : it has a value but makes no claim. An equation states that two expressions are equal, like $3x + 5 = 11$ , and can be solved.
Are $xy$ and $yx$ like terms?: Yes. Multiplication can be done in any order, so $xy = yx$ , and $3xy + 2yx = 5xy$ .

Practise algebraic expressions

The Variables & Expressions lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

What is the difference between an expression and an equation?

An expression is a phrase like $3x + 5$ : it has a value but makes no claim. An equation states that two expressions are equal, like $3x + 5 = 11$ , and can be solved.

Are $xy$ and $yx$ like terms?

Yes. Multiplication can be done in any order, so $xy = yx$ , and $3xy + 2yx = 5xy$ .