Combining Like Terms

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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.

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.

TermsLike terms?Why
5x and -2xYesBoth have x.
3y^2 and 7y^2YesBoth have y^2.
4ab and -baYesab and ba are the same, because order doesn’t matter when you multiply.
9 and -4YesConstants are like terms with each other.
6x and 6x^2Nox and x^2 are different variable parts.
2x and 2yNoThe letters are different.

How to Combine Like Terms

Follow these steps to simplify an expression:

  1. Find the like terms. Include the sign in front of each term.
  2. Group the like terms together.
  3. Add the coefficients. Keep the variable part the same.
  4. 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. The like terms are 4x and -x, and the constants 3 and 5.

Step 2. Group them. Remember that -x means -1x.

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

Answer: 3x + 8

Example 2: Two variables

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

Step 1. The a-terms are 7a and -3a. The b-terms are -2b and 6b.

Step 2. Add the coefficients of each group.

\begin{aligned}
7a - 2b - 3a + 6b &= (7a - 3a) + (-2b + 6b) \\
&= 4a + 4b
\end{aligned}

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.

Step 2. Combine each group. The exponents stay the same.

\begin{aligned}
5x^2 + 3x - 2x^2 - 8x + 1 &= (5x^2 - 2x^2) + (3x - 8x) + 1 \\
&= 3x^2 - 5x + 1
\end{aligned}

Answer: 3x^2 - 5x + 1

Example 4: Fraction coefficients

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

Step 1. Combine the y-terms. Use a common denominator of 4: \frac{1}{2} = \frac{2}{4}.

Step 2. Combine the constants. Write 2 as \frac{8}{4}.

\begin{aligned}
\tfrac{1}{2}y - \tfrac{1}{4}y &= \tfrac{2}{4}y - \tfrac{1}{4}y = \tfrac{1}{4}y \\
\tfrac{3}{4} + 2 &= \tfrac{3}{4} + \tfrac{8}{4} = \tfrac{11}{4}
\end{aligned}

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.

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

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

Step 2. Combine the x-terms and the constants.

\begin{aligned}
P &= (2x + 3x + x) + (1 - 4 + 6) \\
&= 6x + 3
\end{aligned}

Step 3. Replace x with 5.

6(5) + 3 = 30 + 3 = 33

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

Common Mistakes

MistakeWhat it looks likeHow to fix it
Combining unlike terms3x + 4 = 7x3x and 4 are not like terms. The answer stays 3x + 4.
Adding the exponents2x^2 + 3x^2 = 5x^4Only add the coefficients: 5x^2.
Mixing x and x²4x^2 + 3x = 7x^3They are not like terms, so you can’t combine them.
Dropping a minus sign5x - 7 - 2x becomes 7x - 7The 2x is negative: 5x - 2x = 3x.
Forgetting the hidden 16x + x = 6xx 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.

  1. Which terms are like terms with 6m? 6m^2, -m, 6n, 0.5m
  2. 9x + 4x
  3. 3y - 8 + 2y + 10
  4. -6p + 2q + p - 7q
  5. x + x + x - 4
  6. 4a^2 - a + 3a^2 + 5a - 2
  7. 2.5t + 1.2 - 0.5t + 3.8
  8. \frac{2}{3}k + \frac{1}{6}k
  9. 8xy - 3x + 2yx + x
  10. 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
  1. -m and 0.5m. 6m^2 has a different exponent, and 6n has a different letter.
  2. 13x
  3. (3y + 2y) + (-8 + 10) = 5y + 2
  4. (-6p + p) + (2q - 7q) = -5p - 5q
  5. 3x - 4
  6. (4a^2 + 3a^2) + (-a + 5a) - 2 = 7a^2 + 4a - 2
  7. (2.5t - 0.5t) + (1.2 + 3.8) = 2t + 5
  8. \frac{4}{6}k + \frac{1}{6}k = \frac{5}{6}k
  9. xy and yx are like terms: (8xy + 2xy) + (-3x + x) = 10xy - 2x
  10. (3w + 2) + w + (3w + 2) + w = 8w + 4 feet. When w = 4: 8(4) + 4 = 36, so the perimeter is 36 feet.

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

Back to the Algebra 1 Index

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