Combining Like Terms & Distributive Property

Simplifying algebraic expressions is a fundamental skill you will use in every area of mathematics. Two of the most important tools for simplification are combining like terms and the distributive property.

What Are Like Terms?

Like terms are terms that have the same variable(s) raised to the same power(s). Only the coefficients can differ.

Like Terms	Not Like Terms
3x and 7x	3x and 3x² (different powers)
-2y² and 5y²	4x and 4y (different variables)
8 and -3 (both constants)	2xy and 2x (different variable sets)

Combining Like Terms

To combine like terms, add or subtract their coefficients while keeping the variable part unchanged.

Worked Example 1: Basic Combining

Simplify: 5x + 3y - 2x + 7y

Group like terms: (5x - 2x) + (3y + 7y)
Combine coefficients: 3x + 10y

The Distributive Property

The distributive property states:

a(b + c) = ab + ac

You multiply the factor outside the parentheses by each term inside.

Worked Example 2: Distributing

Expand: 4(2x - 3)

Multiply 4 by the first term: 4 × 2x = 8x
Multiply 4 by the second term: 4 × (-3) = -12
Result: 8x - 12

Worked Example 3: Distribute Then Combine

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

Distribute the 3: 6x + 12
Distribute the -2: -2x + 10 (note: -2 times -5 is +10)
Combine: 6x + 12 - 2x + 10
Group like terms: (6x - 2x) + (12 + 10) = 4x + 22

Common Mistake: Forgetting to Distribute the Negative

In -(3x - 7), the negative sign distributes to BOTH terms: -3x + 7. A very common error is writing -3x - 7 by forgetting to change the sign of the second term.

Subtraction is Adding the Opposite

Think of 5x - 2x as 5x + (-2x). This makes it clear that you add the coefficients: 5 + (-2) = 3, giving 3x.

Worked Example 4: Complex Expression

Simplify: 2(3x² - x + 4) + 5x - 3(x² + 2)

Distribute 2: 6x² - 2x + 8
The 5x stays: + 5x
Distribute -3: -3x² - 6
Combine all: 6x² - 2x + 8 + 5x - 3x² - 6
Group: (6x² - 3x²) + (-2x + 5x) + (8 - 6)
Simplify: 3x² + 3x + 2

Practice Problems

1. Simplify: 8a + 3b - 5a + b

Show Solution

(8a - 5a) + (3b + b) = 3a + 4b

2. Expand: -5(2x - 7)

Show Solution

-5(2x) + (-5)(-7) = -10x + 35

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

Show Solution

4x + 12 - 2x + 2 = 2x + 14

4. Simplify: 7y - 3(2y - 4) + y

Show Solution

7y - 6y + 12 + y = 2y + 12

5. Simplify: -(x² - 3x + 2) + 4x² - x

Show Solution

-x² + 3x - 2 + 4x² - x = 3x² + 2x - 2

Lesson Summary

Like terms share the same variable(s) and exponent(s). Combine them by adding coefficients.
The distributive property: a(b + c) = ab + ac. Multiply by every term inside the parentheses.
Watch out for distributing negative signs -- they affect every term.
Strategy: distribute first, then combine like terms.