Math Core

Long expressions are hard to work with. $5x + 3 - 2x + 7 - x$ has five terms, but it means the same thing as $2x + 10$ , which has only two. Combining like terms is how you make that happen, and it's a step you'll use in almost every equation in Algebra 1.

What makes terms "like"

Definition

Like terms

Like terms have exactly the same variables raised to exactly the same powers. Only their coefficients may differ. All constant terms are like terms with each other.

like terms	not like terms	reason
$6k$ and $-k$	$6k$ and $6$	$6$ has no variable
$2x^2$ and $9x^2$	$2x^2$ and $2x$	different powers of $x$
$-3ab$ and $ab$	$3ab$ and $3a$	$3a$ has no $b$
$4$ and $-\tfrac{1}{2}$	$5m$ and $5n$	different variables

The order of the letters doesn't matter: $ba$ and $ab$ are like terms, because multiplication is commutative.

Why it works

Combining like terms is the distributive property run backward. For example,

7y + 4y = (7 + 4)y = 11y.

You can also picture it: $7$ bags of $y$ marbles plus $4$ more bags is $11$ bags. But $7y + 4$ is seven bags plus four loose marbles. You can't write that as a single term.

Combining like terms

Add the coefficients of like terms and keep the variable part exactly the same. Terms that aren't alike stay separate.

The process

Read each term with the sign in front of it. In $9 - 4x$ , the $x$ -term is $-4x$ .
Use the commutative and associative properties to group like terms together.
Add the coefficients within each group. Remember that $x$ means $1x$ and $-x$ means $-1x$ .
Write the result in a standard order: higher powers first, constant last.

Worked example: Integers and a hidden coefficient

Simplify $-5a + 8 - a + 3a - 12$ .

The $a$ -terms are $-5a$ , $-a$ and $3a$ . The constants are $8$ and $-12$ .

\begin{aligned} -5a + 8 - a + 3a - 12 &= (-5a - a + 3a) + (8 - 12) \\ &= (-5 - 1 + 3)a + (-4) \\ &= -3a - 4 \end{aligned}

Worked example: More than one variable part

Simplify $4x^2 - 3x + 6 - x^2 + 7x - 2$ .

Sort the terms into three groups:

$x^2$ -terms: $4x^2 - x^2 = 3x^2$
$x$ -terms: $-3x + 7x = 4x$
constants: $6 - 2 = 4$

The result is $3x^2 + 4x + 4$ . Notice that $3x^2$ and $4x$ stay separate, because the powers are different.

Worked example: Subtracting an expression

Simplify $(3m - 2n) - (5m - 6n + 1)$ .

Subtracting a group means subtracting every term in it, so each sign inside the second group flips:

\begin{aligned} (3m - 2n) - (5m - 6n + 1) &= 3m - 2n - 5m + 6n - 1 \\ &= (3m - 5m) + (-2n + 6n) - 1 \\ &= -2m + 4n - 1 \end{aligned}

Common mistake

Three mistakes show up again and again:

Losing a sign. In $12 - 5b + 2b$ , the $b$ -terms are $-5b$ and $2b$ , which give $-3b$ . The answer is $12 - 3b$ , not $12 - 7b$ .
Mixing powers. $3x^2 + 2x$ does not simplify. It is not $5x^2$ or $5x^3$ .
Flipping only the first sign when you subtract a group. In $-(5m - 6n + 1)$ , all three signs change.

Tip

Check your answer by substituting a simple value, like $a = 2$ , into both the original and the simplified expression. In the first example: $-10 + 8 - 2 + 6 - 12 = -10$ , and $-3(2) - 4 = -10$ . They match.

Practice

Practice 1

Which pair are like terms?

Practice 2

Simplify $9a - 4 - 11a + 10$ .

Practice 3

Simplify $3x^2 + 5x - x^2 - 8x + 2$ . What is the coefficient of $x$ in the result?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Simplify $4m + 3n - m + 5n - 2$ .

Practice 5

Simplify $(6k - 5) - (-2k + 4)$ .

Practice 6

Simplify $\dfrac{2}{3}y + 4 - \dfrac{1}{6}y$ . What is the coefficient of $y$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

A rectangle has length $3w + 2$ and width $w$ . Adding all four sides, its perimeter simplifies to $aw + 4$ . What is $a$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

Simplify $5p - 3 - 2p + 7 - 3p + p$ , then evaluate it when $p = -6$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.