Combining Like Terms

How do you simplify expressions with multiple variables? Combining like terms is essential for simplifying algebraic expressions and solving equations efficiently!

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What Are Terms?

A term is a number, variable, or product of numbers and variables.

Examples of terms:

• 5 (constant term)
• x (variable term)
• 3x (coefficient 3, variable x)
• -2y (coefficient -2, variable y)
• 7ab (coefficient 7, variables a and b)

Terms are separated by + or - signs

Expression: 3x + 5 - 2y + 8 Terms: 3x, 5, -2y, 8 (four terms)

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What Are Like Terms?

Like terms have the SAME variable(s) raised to the SAME power.

Like terms:

• 3x and 5x (both have x)
• 7y and -2y (both have y)
• 4 and 9 (both are constants)
• 2ab and 5ab (both have ab)

NOT like terms:

• 3x and 3y (different variables)
• x and x² (different powers)
• 2a and 2ab (different variables)

Think: Can only combine apples with apples, not apples with oranges!

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Why Combine Like Terms?

Combining like terms simplifies expressions.

Before: 3x + 2 + 5x + 7 After: 8x + 9

Simpler = easier to work with!

Benefits:

• Shorter expressions
• Easier to evaluate
• Necessary for solving equations
• Reduces chance of errors

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How to Combine Like Terms

Rule: Add or subtract the COEFFICIENTS, keep the variable part the same.

Example 1: 4x + 3x

Coefficients: 4 + 3 = 7 Variable: x

Answer: 7x

Think: 4 apples + 3 apples = 7 apples

Example 2: 8y - 5y

Coefficients: 8 - 5 = 3 Variable: y

Answer: 3y

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Combining Multiple Like Terms

Example: 2x + 5x + 3x

Add coefficients: 2 + 5 + 3 = 10

Answer: 10x

Example 2: 7a - 3a + 4a

Combine: 7 - 3 + 4 = 8

Answer: 8a

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Expressions with Different Variables

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

Step 1: Identify like terms

• x terms: 3x and 2x
• y terms: 4y and y

Step 2: Combine each group

• 3x + 2x = 5x
• 4y + y = 5y (remember y = 1y)

Answer: 5x + 5y

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Including Constants

Example: 5x + 3 + 2x + 7

Step 1: Group like terms

• x terms: 5x + 2x
• Constants: 3 + 7

Step 2: Combine

• 5x + 2x = 7x
• 3 + 7 = 10

Answer: 7x + 10

Remember: Constants (numbers alone) are like terms with each other!

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With Subtraction

Example: 8m - 3m + 5

Think of subtraction as adding a negative: 8m + (-3m) + 5

Combine m terms: 8 + (-3) = 5

Answer: 5m + 5

Example 2: 6 - 2n + 3 - 5n

Rearrange: 6 + 3 - 2n - 5n

Constants: 6 + 3 = 9 n terms: -2n - 5n = -7n

Answer: 9 - 7n (or -7n + 9)

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Negative Coefficients

Example: -4x + 7x - 2x

Combine: -4 + 7 - 2 = 1

Answer: 1x = x

Remember: When coefficient is 1, we usually just write the variable!

Example 2: 3y - 5y + y

Combine: 3 - 5 + 1 = -1

Answer: -1y = -y

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Multiple Variables in One Expression

Example: 4a + 3b - 2a + 5b - 1

Step 1: Identify groups

• a terms: 4a - 2a
• b terms: 3b + 5b
• Constants: -1

Step 2: Combine

• 4a - 2a = 2a
• 3b + 5b = 8b
• Constant: -1

Answer: 2a + 8b - 1

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More Complex Examples

Example 1: 7x + 2y - 3x + 8y - 4

Group and combine:

• x: 7x - 3x = 4x
• y: 2y + 8y = 10y
• Constant: -4

Answer: 4x + 10y - 4

Example 2: 10 - 3m + 5 + 7m - 2

Combine:

• Constants: 10 + 5 - 2 = 13
• m terms: -3m + 7m = 4m

Answer: 13 + 4m (or 4m + 13)

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When Terms Are Not Like Terms

Example: 3x + 4y + 2

Cannot combine! Different variables and a constant.

Answer: 3x + 4y + 2 (already simplified)

Example 2: 5a + 3b

Different variables, cannot combine.

Answer: 5a + 3b (already simplified)

Important: Only combine like terms! Don't combine different variables.

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Identifying Like Terms

Practice identifying:

Which are like terms with 5x?

• 3x ✓ (same variable)
• -2x ✓ (same variable)
• 5y ✗ (different variable)
• x ✓ (same as 1x)
• 5 ✗ (constant, no variable)

Which are like terms with 7?

• -3 ✓ (both constants)
• 10 ✓ (both constants)
• 7x ✗ (has variable)

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Combining in Equations

Example: Solve 3x + 5x = 24

Step 1: Combine like terms 8x = 24

Step 2: Solve x = 3

Combining made it a one-step equation!

Example 2: Solve 4y - y + 6 = 15

Step 1: Combine like terms 3y + 6 = 15

Step 2: Solve (two-step) 3y = 9 y = 3

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Distributive Property Then Combine

Example: 2(x + 3) + 3(x + 1)

Step 1: Distribute 2x + 6 + 3x + 3

Step 2: Combine like terms 5x + 9

Answer: 5x + 9

Example 2: 5(2a - 1) - 3(a + 2)

Step 1: Distribute 10a - 5 - 3a - 6

Step 2: Combine 7a - 11

Answer: 7a - 11

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Rearranging Before Combining

Example: 5 + 3x - 2 + 7x

Rearrange to group like terms: 3x + 7x + 5 - 2

Combine: 10x + 3

Tip: Grouping like terms helps avoid mistakes!

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Real-World Applications

Perimeter: Rectangle with length (2x + 3) and width (x + 5)

Perimeter = 2(length) + 2(width) = 2(2x + 3) + 2(x + 5) = 4x + 6 + 2x + 10 = 6x + 16

Shopping: Buy 3 shirts at $x each and 5 more at$x each Total: 3x + 5x = 8x

Total shirts: 8 shirts at $x each

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Order Doesn't Matter

Commutative Property: Can rearrange terms

5x + 3y = 3y + 5x (same thing!)

Standard form: Usually write in alphabetical order

• 5x + 3y (not 3y + 5x)
• Constants at the end: 2x + 5 (not 5 + 2x)

But mathematically equivalent!

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Common Mistakes to Avoid

❌ Mistake 1: Combining unlike terms

• Wrong: 3x + 4y = 7xy
• Right: 3x + 4y (cannot combine)

❌ Mistake 2: Forgetting coefficient of 1

• Wrong: 5x + x = 5x
• Right: 5x + x = 5x + 1x = 6x

❌ Mistake 3: Sign errors

• Wrong: 8x - 5x = 13x
• Right: 8x - 5x = 3x

❌ Mistake 4: Combining different powers

• Wrong: x + x² = x³
• Right: x + x² (cannot combine)

❌ Mistake 5: Changing the variable

• Wrong: 3x + 2x = 5 (lost the x!)
• Right: 3x + 2x = 5x

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Problem-Solving Strategy

To simplify expressions:

1. Identify all like terms
2. Group like terms together
3. Add or subtract coefficients
4. Keep variable part the same
5. Write in standard form

To solve equations:

1. Distribute if needed
2. Combine like terms on each side
3. Use inverse operations to solve
4. Check your answer

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Quick Reference

Like Terms: Same variable(s) and power(s)

Combining:

• Add/subtract coefficients only
• Keep variable part unchanged

Examples:

• 4x + 3x = 7x
• 8y - 5y = 3y
• 3 + 7 = 10
• 2a + 3b cannot combine

Steps:

1. Identify like terms
2. Group them
3. Combine coefficients
4. Simplify

Remember: Only like terms can be combined!

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Practice Tips

Tip 1: Circle or underline like terms

• Helps visually group them
• Reduces errors

Tip 2: Use different colors

• One color for x terms
• Another for y terms
• Another for constants

Tip 3: Write coefficients clearly

• Remember x = 1x
• Don't forget negative signs!

Tip 4: Check by substituting

• Pick a value for variables
• Evaluate before and after combining
• Should get same result

Tip 5: Practice identifying

• Before combining, make sure terms are actually like
• "Same variable and power?"

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Summary

Like terms have the same variable(s) and power(s):

• 5x and 3x are like terms
• 2y and 7z are NOT like terms

Combining like terms:

• Add or subtract coefficients
• Keep variable part the same
• Simplifies expressions

Process:

1. Identify like terms
2. Group them together
3. Combine coefficients
4. Write simplified expression

Applications:

• Simplifying expressions
• Solving equations more easily
• Working with formulas
• Real-world problem solving

Key skill: Recognizing which terms can be combined is essential for all future algebra!

Mastering combining like terms makes algebra much easier and is used in every equation you'll solve!