Combining Like Terms

What Are Like Terms?

Like terms are terms that have the same variable raised to the same power.

Like terms:

• 3x and 5x (both have x)
• 7y and -2y (both have y)
• 4x² and 9x² (both have x²)

NOT like terms:

• 3x and 5y (different variables)
• 4x and 4x² (different powers)
• 7 and 7x (one has variable, one doesn't)

Constants Are Like Terms

Constants (numbers with no variables) are like terms with each other.

Like terms:

• 5, 12, -3 (all constants)

Why Combine Like Terms?

Combining simplifies expressions to make them easier to work with!

Example: 3x + 5x = ?

Think: 3 apples + 5 apples = 8 apples

So: 3x + 5x = 8x

Combining Like Terms: Addition

Add the coefficients (numbers in front) and keep the variable.

Example: 4y + 7y

Coefficients: 4 + 7 = 11

Answer: 11y

Example: 6x + 2x + 5x

Coefficients: 6 + 2 + 5 = 13

Answer: 13x

Example: 3 + 8 + 2

All constants: 3 + 8 + 2 = 13

Answer: 13

Combining Like Terms: Subtraction

Subtract the coefficients and keep the variable.

Example: 9a - 4a

Coefficients: 9 - 4 = 5

Answer: 5a

Example: 10x - x

Remember: x means 1x

Coefficients: 10 - 1 = 9

Answer: 9x

Combining with Negative Terms

Example: 5y - 3y + 7y

Think: 5 - 3 + 7 = 9

Answer: 9y

Example: 8n - 12n

Coefficients: 8 - 12 = -4

Answer: -4n

Expressions with Multiple Variables

Combine each variable separately!

Example: 4x + 3y + 2x + 5y

Step 1: Group like terms

• x terms: 4x + 2x
• y terms: 3y + 5y

Step 2: Combine each

• 4x + 2x = 6x
• 3y + 5y = 8y

Answer: 6x + 8y

Example: 7a + 2b - 3a + 4b

Group:

• a terms: 7a - 3a = 4a
• b terms: 2b + 4b = 6b

Answer: 4a + 6b

Expressions with Constants

Don't forget to combine the constants too!

Example: 5x + 3 + 2x + 7

Group:

• x terms: 5x + 2x = 7x
• Constants: 3 + 7 = 10

Answer: 7x + 10

Example: 6y - 4 + 3y + 9

Group:

• y terms: 6y + 3y = 9y
• Constants: -4 + 9 = 5

Answer: 9y + 5

Expressions with Different Powers

Keep them separate! x and x² are NOT like terms.

Example: 3x² + 5x + 2x² + 7x

Group:

• x² terms: 3x² + 2x² = 5x²
• x terms: 5x + 7x = 12x

Answer: 5x² + 12x

Cannot simplify further — they're different terms!

Order Doesn't Matter

Commutative property: You can rearrange terms to group like terms.

Example: 4 + 3x + 7 + 5x

Rearrange:

• 3x + 5x + 4 + 7
• 8x + 11

Answer: 8x + 11

Common Mistakes

❌ Combining unlike terms:

• 3x + 4y ≠ 7xy (can't combine different variables!)

❌ Forgetting the variable:

• 5x + 2x ≠ 7 (answer is 7x, not 7)

❌ Combining different powers:

• 2x + 3x² ≠ 5x² (different powers, keep separate)

❌ Losing negative signs:

• 5x - 2x ≠ 7x (answer is 3x)

Simplifying Complex Expressions

Example: 8x - 3y + 5 - 2x + 7y - 1

Step 1: Group like terms

• x terms: 8x - 2x
• y terms: -3y + 7y
• Constants: 5 - 1

Step 2: Combine each

• 8x - 2x = 6x
• -3y + 7y = 4y
• 5 - 1 = 4

Answer: 6x + 4y + 4

Real-World Applications

Perimeter: A rectangle has sides x + 3 and 2x.

Perimeter = (x + 3) + 2x + (x + 3) + 2x

Combine: x + 2x + x + 2x + 3 + 3 = 6x + 6

Shopping: You buy x notebooks at $2 each and x pencils at $1 each.

Cost = 2x + 1x = 3x

Practice