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Combining Like Terms

What are Like Terms?

Like terms are terms that have the same variables raised to the same powers. They may have different coefficients (the numbers in front of the variables).

Examples of Like Terms

Like Terms

  • 3x and 5x
  • 2y² and -7y²
  • 4xy and 9xy
  • 6 and 10 (constants)

Unlike Terms

  • 3x and 5y (different variables)
  • 2x and 3x² (different powers)
  • 4xy and 5x²y (different powers)
  • 7 and 2x (constant and variable)

How to Combine Like Terms

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

Example 1: Combining Terms with One Variable

3x + 5x
= (3 + 5)x
= 8x

We add the coefficients (3 and 5) and keep the variable (x) the same.

Example 2: Combining Terms with Subtraction

7y - 2y
= (7 - 2)y
= 5y

Example 3: Combining Terms with Multiple Variables

4xy + 3xy
= (4 + 3)xy
= 7xy

Example 4: Combining Constants

9 + 5
= 14

Constants (numbers without variables) are also like terms and can be combined.

Simplifying Expressions by Combining Like Terms

When simplifying algebraic expressions, we identify and combine like terms to make the expression more compact.

Example 1: Simple Expression

3x + 4y + 2x - y
= 3x + 2x + 4y - y
= 5x + 3y

We grouped like terms together: the x terms (3x + 2x) and the y terms (4y - y).

Example 2: Expression with Constants

5x + 7 + 3x - 2
= 5x + 3x + 7 - 2
= 8x + 5

Example 3: More Complex Expression

2x² + 3xy + 4x² - xy + 5
= 2x² + 4x² + 3xy - xy + 5
= 6x² + 2xy + 5

We combined the x² terms (2x² + 4x²), the xy terms (3xy - xy), and kept the constant (5).

Using the Distributive Property with Like Terms

Often, we need to use the distributive property before we can combine like terms.

Example 1: Distributing and Then Combining

3(2x + 4) + 5x
= 6x + 12 + 5x
= 6x + 5x + 12
= 11x + 12

First, we applied the distributive property to 3(2x + 4), then combined like terms.

Example 2: Multiple Distributions

2(3x - 1) - 4(x + 2)
= 6x - 2 - 4x - 8
= 6x - 4x - 2 - 8
= 2x - 10

Related Topics

Practice Combining Like Terms

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Key Concepts

  • 1
    Like terms have the same variables raised to the same powers.
  • 2
    To combine like terms, add or subtract their coefficients while keeping the variable part the same.
  • 3
    Constants (numbers without variables) are also like terms and can be combined.
  • 4
    Often, you need to use the distributive property before combining like terms.
  • 5
    Combining like terms is a key step in simplifying algebraic expressions.