Simplifying Expressions: Combining Like Terms and Using the Distributive Property

Warm-up: Match the Like Terms

To simplify expressions, you often need to combine like terms. Two terms are like terms if they have the same variable part: same variable(s) with the same exponent(s). The number in front is the coefficient, and coefficients can be combined.

What counts as the “variable part”?

• 7x has variable part x
• -3x has variable part x
• 5x^2 has variable part x^2
• 2ab has variable part ab
• -ab has variable part ab

Matching activity (pair the like terms)

Pair each term in Column A with all like terms in Column B.

Tip: Like terms must match the entire variable part. For example, x and x^2 are not like terms.

Guided Practice: Combining Like Terms

When you combine like terms, you add or subtract their coefficients and keep the same variable part.

Example 1: 3x + 5x - x

Step 1: Identify like terms. All three terms have variable part x, so they are like terms.

Continue in our app.

• Listen to the audio with the screen off.
• Earn a certificate upon completion.
• Over 5000 courses for you to explore!

Or continue reading below...

Download the app

Step 2: Combine coefficients.

3x + 5x - x = (3 + 5 - 1)x = 7x

Why “- x” becomes “-1x”: If no coefficient is shown, the coefficient is 1. So -x means -1x.

Example 2: Mix of like and unlike terms

4x + 2y - 3x + 5y

Group like terms (mentally or by rearranging):

(4x - 3x) + (2y + 5y) = 1x + 7y = x + 7y

Example 3: Constants combine with constants

8 + 3x - 2 + x = (8 - 2) + (3x + x) = 6 + 4x

The Distributive Property (Including Negatives)

The distributive property lets you multiply a number by every term inside parentheses.

Core idea

a(b + c) = ab + ac

This is often the key step before you can combine like terms, because parentheses can “hide” like terms.

Guided Example 1: 2(x + 6)

Step 1: Multiply 2 by each term inside.

2(x + 6) = 2·x + 2·6 = 2x + 12

Guided Example 2 (with a negative): -3(2x - 4)

Step 1: Multiply -3 by each term inside.

-3(2x - 4) = (-3)·(2x) + (-3)·(-4)

Step 2: Simplify each product carefully.

= -6x + 12

Sign reminder: A negative times a negative is positive, so (-3)(-4) = +12.

Distributing a minus sign (common shortcut)

If you see -(...), it means -1 times everything inside.

-(x - 5 + 2x) = -x + 5 - 2x

Multi-step Simplification: Distribute + Combine Like Terms

Many expressions require both distributing and combining like terms. Use this checklist each time.

Simplification checklist

• Distribute (if there is a number or negative sign outside parentheses)
• Remove parentheses (after distributing, rewrite without parentheses)
• Combine like terms (add/subtract coefficients of matching variable parts)
• Write in standard form (typically variable terms first, then constants; combine fully)

Problem 1: 4(x + 2) + 3x

Distribute:

4(x + 2) + 3x = (4x + 8) + 3x

Remove parentheses and combine like terms:

4x + 8 + 3x = 7x + 8

Problem 2: -2(3x - 5) + x - 4

Distribute:

-2(3x - 5) + x - 4 = (-6x + 10) + x - 4

Remove parentheses:

-6x + 10 + x - 4

Combine like terms:

(-6x + x) + (10 - 4) = -5x + 6

Problem 3: 5(2x - 1) - 3(x + 4)

Distribute to each set of parentheses:

5(2x - 1) - 3(x + 4) = (10x - 5) + (-3x - 12)

Remove parentheses:

10x - 5 - 3x - 12

Combine like terms:

(10x - 3x) + (-5 - 12) = 7x - 17

Problem 4: -3(2x - 4) + 2(x + 6) - x

Distribute:

-3(2x - 4) + 2(x + 6) - x = (-6x + 12) + (2x + 12) - x

Remove parentheses:

-6x + 12 + 2x + 12 - x

Combine like terms:

(-6x + 2x - x) + (12 + 12) = (-5x) + 24 = -5x + 24

Common Pitfalls (and How to Avoid Them)

Pitfall 1: Combining unlike terms

You can only combine terms with the same variable part.

• Not allowed: x + x^2 (variable parts x and x^2 do not match)
• Correct: keep as x + x^2 or write in standard form as x^2 + x

Similarly, 2x and 2 are not like terms, and ab and a are not like terms.

Pitfall 2: Sign errors when combining

When subtracting a term, you are adding a negative coefficient.

6x - 9x = (6 - 9)x = -3x

A helpful habit is to rewrite subtraction as addition:

6x - 9x = 6x + (-9x)

Pitfall 3: Forgetting to distribute to every term

Distribute to each term inside the parentheses.

• Incorrect: 2(x + 6) = 2x + 6
• Correct: 2(x + 6) = 2x + 12

With negatives, distribute the negative to every term:

• Incorrect: -(x - 4) = -x - 4
• Correct: -(x - 4) = -x + 4

Quick self-check while simplifying

• Did I multiply the outside number by every term inside the parentheses?
• Did I keep track of negative signs during distribution?
• Did I only combine terms with matching variable parts (same variable and exponent)?
• Is my final expression fully simplified and written in standard form?