Math Core

An expression like $4x + 7 + 2x - 3$ has four terms, but it describes something simpler. Combining like terms tidies an expression into its shortest form, which makes it easier to evaluate, compare and, in the next unit, solve.

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	why	not like terms	why not
$5x$ and $-2x$	both $x$	$5x$ and $5y$	different variables
$3a^2$ and $a^2$	both $a^2$	$3a^2$ and $3a$	different powers
$4xy$ and $-9yx$	$yx = xy$	$4xy$ and $4x$	$4x$ has no $y$
$8$ and $-\tfrac{1}{2}$	both constants	$8$ and $8x$	one has a variable

Notice that $4xy$ and $-9yx$ are like terms: by the commutative property, $yx$ is the same as $xy$ .

Combining uses the distributive property

Why does $4x + 2x$ equal $6x$ ? Read the distributive property backwards:

4x + 2x = (4 + 2)x = 6x.

In words: $4$ of something plus $2$ of the same thing makes $6$ of it. That's why you can only combine like terms. $4x + 2y$ is $4$ of one thing and $2$ of another, and it can't be shortened.

Combining like terms

To combine like terms, add their coefficients and keep the variable part exactly the same.

ax + bx = (a + b)x

The exponents do not change: $3x^2 + 5x^2 = 8x^2$ , not $8x^4$ .

Common mistake

Two common slip-ups:

Changing the exponent. $x + x = 2x$ , not $x^2$ . (It's $x \cdot x$ that equals $x^2$ .)
Losing a sign. Each term owns the sign in front of it. In $9 - 4x + 2x$ , the $x$ -terms are $-4x$ and $+2x$ , which combine to $-2x$ .

Step by step

A reliable routine:

Identify groups of like terms. Circling or underlining each group helps; bring each term's sign with it.
Use the commutative property to put like terms next to each other.
Add the coefficients in each group.
By convention, write the answer with the highest power first and the constant last.

Worked example: Two groups

Simplify $7x + 4 - 2x - 9$ .

\begin{aligned} 7x + 4 - 2x - 9 &= 7x - 2x + 4 - 9 && \text{group like terms} \\ &= (7 - 2)x + (4 - 9) && \text{add coefficients} \\ &= 5x - 5 \end{aligned}

Worked example: Different powers

Simplify $3n^2 - 5n + n^2 + 8n - 6$ .

There are three groups: $n^2$ -terms $3n^2$ and $n^2$ (coefficient $1$ ); $n$ -terms $-5n$ and $8n$ ; and the constant $-6$ .

(3 + 1)n^2 + (-5 + 8)n - 6 = 4n^2 + 3n - 6.

Distribute first, then combine

When an expression has parentheses you can't simplify inside, distribute first. Then combine.

Worked example: Distribute, then combine

Simplify $5(y + 3) - 2(y - 4)$ .

Watch the second product carefully: the $-2$ multiplies both $y$ and $-4$ .

\begin{aligned} 5(y + 3) - 2(y - 4) &= 5y + 15 - 2y + 8 && \text{distribute; } (-2)(-4) = 8 \\ &= (5y - 2y) + (15 + 8) && \text{group like terms} \\ &= 3y + 23 \end{aligned}

Tip

Check your simplified answer by substituting a number, such as $y = 1$ , into both forms. Original: $5(4) - 2(-3) = 20 + 6 = 26$ . Simplified: $3(1) + 23 = 26$ . They match.

Worked example: A perimeter

A triangle has sides of length $2x + 1$ , $3x - 4$ and $x + 6$ . Write a simplified expression for its perimeter.

Add all three sides and combine:

(2x + 1) + (3x - 4) + (x + 6) = (2x + 3x + x) + (1 - 4 + 6) = 6x + 3.

Practice

Practice 1

Which pair are like terms?

Practice 2

Simplify $8x - 3 + x + 10$ .

Practice 3

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

Practice 4

After simplifying $6 - 4y + 3y - 3y$ , what is the coefficient of $y$ ?

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

Practice 5

Simplify $3(x + 4) + 2(x - 1)$ .

Practice 6

Simplify $4(k - 2) - 3(k - 5)$ to the form $k + c$ . Then evaluate it when $k = 23$ .

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

Practice 7

A rectangle has length $3w + 2$ and width $2w - 1$ . Its perimeter is $2(3w + 2) + 2(2w - 1)$ . When this is simplified, what is the coefficient of $w$ ?

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

Practice 8

Simplify $2(3m - 1) - (m + 4) + 5m$ .