Combining Like Terms: Steps and Examples

Combining like terms means adding or subtracting terms that have the same variable part. Like terms have the same variables raised to the same powers, such as $4x$ and $-x$ . To combine them, add their coefficients and keep the variable part: $4x + 3 - x + 5 = 3x + 8$ .

Before this lesson: You should know what the terms of an expression are and how to add and subtract negative numbers.

Extra practice: The Algebra 1 Unit 1 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.

What Are Like Terms?

Like terms have the same variable part. That means the same letters with the same exponents. Only the coefficients can be different.

Think of each variable part as a kind of item. 3 apples plus 2 apples is 5 apples. But 3 apples plus 2 pears is still just 3 apples and 2 pears.

Terms	Like terms?	Why
$5x$ and $-2x$	Yes	Both have $x$ .
$3y^2$ and $7y^2$	Yes	Both have $y^2$ .
$4ab$ and $-ba$	Yes	$ab$ and $ba$ are the same, because order doesn’t matter when you multiply.
$9$ and $-4$	Yes	Constants are like terms with each other.
$6x$ and $6x^2$	No	$x$ and $x^2$ are different variable parts.
$2x$ and $2y$	No	The letters are different.

How to Combine Like Terms

Follow these steps to simplify an expression:

Find the like terms. Include the sign in front of each term.
Group the like terms together.
Add the coefficients. Keep the variable part the same.
Write the answer with the highest exponent first and the constant last.

The expression 4x + 3 − x + 5. The x-terms 4x and −x combine to 3x, and the constants 3 and 5 combine to 8, so the answer is 3x + 8. — Combine the x-terms and the constants separately: 4x − x = 3x and 3 + 5 = 8.

Here is the rule in symbols. It works for any variable part, not just x.

ax + bx = (a + b)x

This rule comes from the distributive property, which you’ll learn next.

Worked Examples

Example 1: One variable and constants

Simplify $4x + 3 - x + 5$ .

Step 1. Find the like terms. The $x$ -terms are $4x$ and $-x$ . The constants are 3 and 5.

4x + 3 - x + 5 = 4x - x + 3 + 5

Step 2. Move the like terms next to each other. Each term keeps its sign.

4x - x + 3 + 5 = 3x + 3 + 5

Step 3. Combine the $x$ -terms. Remember that $-x$ means $-1x$ , so $4 - 1 = 3$ .

3x + 3 + 5 = 3x + 8

Step 4. Combine the constants.

Answer: $3x + 8$

Example 2: Two variables

Simplify $7a - 2b - 3a + 6b$ .

Step 1. Find the like terms. The $a$ -terms are $7a$ and $-3a$ . The $b$ -terms are $-2b$ and $6b$ .

7a - 2b - 3a + 6b = 7a - 3a - 2b + 6b

Step 2. Move the like terms next to each other.

7a - 3a - 2b + 6b = 4a - 2b + 6b

Step 3. Combine the $a$ -terms: $7 - 3 = 4$ .

4a - 2b + 6b = 4a + 4b

Step 4. Combine the $b$ -terms: $-2 + 6 = 4$ . (See adding and subtracting integers.)

Answer: $4a + 4b$

Example 3: Terms with exponents

Simplify $5x^2 + 3x - 2x^2 - 8x + 1$ .

Step 1. Find the $x^2$ -terms, the $x$ -terms and the constant. $x^2$ and $x$ are not like terms.

5x^2 + 3x - 2x^2 - 8x + 1 = 5x^2 - 2x^2 + 3x - 8x + 1

Step 2. Move the like terms next to each other.

5x^2 - 2x^2 + 3x - 8x + 1 = 3x^2 + 3x - 8x + 1

Step 3. Combine the $x^2$ -terms. The exponent stays the same.

3x^2 + 3x - 8x + 1 = 3x^2 - 5x + 1

Step 4. Combine the $x$ -terms: $3 - 8 = -5$ .

Answer: $3x^2 - 5x + 1$

Example 4: Fraction coefficients

Simplify $\frac{1}{2}y + \frac{3}{4} - \frac{1}{4}y + 2$ .

\frac{1}{2}y - \frac{1}{4}y + \frac{3}{4} + 2

Step 1. Move the like terms next to each other.

\frac{2}{4}y - \frac{1}{4}y + \frac{3}{4} + 2

Step 2. Use a common denominator of 4 for the $y$ -terms: $\frac{1}{2} = \frac{2}{4}$ . (See adding and subtracting fractions.)

\frac{1}{4}y + \frac{3}{4} + 2

Step 3. Combine the $y$ -terms.

\frac{1}{4}y + \frac{3}{4} + \frac{8}{4}

Step 4. Write 2 as $\frac{8}{4}$ .

\frac{1}{4}y + \frac{11}{4}

Step 5. Add the constants.

Answer: $\frac{1}{4}y + \frac{11}{4}$ . Need a refresher? See fractions.

Example 5: Word problem

A triangle has sides of $2x + 1$ , $3x - 4$ and $x + 6$ centimeters. Write a simplified expression for its perimeter. Then find the perimeter when $x = 5$ .

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

Step 1. The perimeter is the sum of the three sides.

P = 2x + 1 + 3x - 4 + x + 6

Step 2. Remove the parentheses. Each term keeps its sign.

P = 2x + 3x + x + 1 - 4 + 6

Step 3. Move the like terms next to each other.

P = 6x + 1 - 4 + 6

Step 4. Combine the $x$ -terms: $2 + 3 + 1 = 6$ .

P = 6x + 3

Step 5. Combine the constants: $1 - 4 + 6 = 3$ .

P = 6(5) + 3

Step 6. Replace $x$ with 5.

P = 30 + 3

Step 7. Multiply.

P = 33

Step 8. Add.

Answer: The perimeter is $6x + 3$ centimeters. When $x = 5$ , it is 33 centimeters.

Common Mistakes

Mistake	What it looks like	How to fix it
Combining unlike terms	$3x + 4 = 7x$	$3x$ and 4 are not like terms. The answer stays $3x + 4$ .
Adding the exponents	$2x^2 + 3x^2 = 5x^4$	Only add the coefficients: $5x^2$ .
Mixing x and x²	$4x^2 + 3x = 7x^3$	They are not like terms, so you can’t combine them.
Dropping a minus sign	$5x - 7 - 2x$ becomes $7x - 7$	The $2x$ is negative: $5x - 2x = 3x$ .
Forgetting the hidden 1	$6x + x = 6x$	$x$ means $1x$ , so $6x + x = 7x$ .

Quick check: Pick a number for the variable. Put it into the original expression and into your answer. Both should give the same value.

Combining Like Terms Before Solving

Many equations have like terms on one side. Combine them first, and the equation gets much easier.

\begin{aligned} 3x + 2x - 4 &= 16 \\ 5x - 4 &= 16 \\ 5x &= 20 \\ x &= 4 \end{aligned}

You’ll practice this in multi-step equations.

Practice Problems

Simplify each expression by combining like terms.

Which terms are like terms with $6m$ ? $6m^2$ , $-m$ , $6n$ , $0.5m$
$9x + 4x$
$3y - 8 + 2y + 10$
$-6p + 2q + p - 7q$
$x + x + x - 4$
$4a^2 - a + 3a^2 + 5a - 2$
$2.5t + 1.2 - 0.5t + 3.8$
$\frac{2}{3}k + \frac{1}{6}k$
$8xy - 3x + 2yx + x$
A rectangle has a length of $3w + 2$ feet and a width of $w$ feet. Add the four sides to write a simplified expression for the perimeter. Then find the perimeter when $w = 4$ .

Answers

Show answers

Like terms have the same variable with the same exponent. $6m$ has $m$ to the first power
$6m^2$ has a different exponent, so it is not a like term
$6n$ has a different letter, so it is not a like term
The like terms are $-m$ and $0.5m$

9x + 4x = 13x

Add the coefficients, 9 + 4 = 13.

3y + 2y - 8 + 10

Move like terms together.

5y - 8 + 10

Combine the y-terms.

5y + 2

Combine the constants.

-6p + p + 2q - 7q

Move like terms together.

-5p + 2q - 7q

Combine the p-terms, −6 + 1 = −5.

-5p - 5q

Combine the q-terms, 2 − 7 = −5 (see adding and subtracting integers)

3x - 4

Each x is 1x, so add 1 + 1 + 1 = 3.

4a^2 + 3a^2 - a + 5a - 2

Move like terms together.

7a^2 - a + 5a - 2

Combine the a² terms.

7a^2 + 4a - 2

Combine the a-terms, −1 + 5 = 4.

2.5t - 0.5t + 1.2 + 3.8

Move like terms together.

2t + 1.2 + 3.8

Combine the t-terms.

2t + 5

Combine the constants.

\frac{4}{6}k + \frac{1}{6}k

Use a common denominator of 6.

\frac{5}{6}k

Add the numerators.

8xy + 2xy - 3x + x

$yx$ is the same as $xy$ , so they are like terms. Move like terms together.

10xy - 3x + x

Combine the xy-terms.

10xy - 2x

Combine the x-terms, −3 + 1 = −2.

10.

(3w + 2) + w + (3w + 2) + w

Add the four sides.

3w + 2 + w + 3w + 2 + w

Remove the parentheses.

3w + w + 3w + w + 2 + 2

Move like terms together.

8w + 2 + 2

Combine the w-terms.

8w + 4

Combine the constants.

8(4) + 4

Replace w with 4.

32 + 4

Multiply.

36

Add.

The perimeter is 36 feet.

Want more practice? The Algebra 1 Unit 1 Practice Pack has 26 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

Combining Like Terms FAQ

What are like terms?

Like terms have the same variables with the same exponents, such as $3x^2$ and $-x^2$ . Constants, like 4 and $-9$ , are also like terms.

What is the rule for combining like terms?

Add or subtract the coefficients and keep the variable part the same. Never change the exponents.

Can you combine x and x squared?

No. $x$ and $x^2$ have different exponents, so they are not like terms. $x^2 + x$ is already simplified.

Are xy and yx like terms?

Yes. Multiplication can happen in any order, so $xy$ and $yx$ are the same variable part.

Summary

Like terms have the same letters with the same exponents.
To combine like terms, add the coefficients and keep the variable part.
Each term keeps the sign in front of it.
A variable alone, like $x$ , has a coefficient of 1.
Unlike terms, such as $x$ and $x^2$ , stay separate.

Previous lesson: Algebraic Expressions

Next lesson: Distributive Property

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