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Simplify expressions by combining like terms
Learn step-by-step with practice exercises built right in.
How do you simplify expressions with multiple variables? Combining like terms is essential for simplifying algebraic expressions and solving equations efficiently!
A term is a number, variable, or product of numbers and variables.
Examples of terms:
Terms are separated by + or - signs
Expression: 3x + 5 - 2y + 8 Terms: 3x, 5, -2y, 8 (four terms)
Like terms have the SAME variable(s) raised to the SAME power.
Like terms:
Simplify: 3x + 5x
Step 1: Identify like terms. Both terms have the variable x.
Step 2: Add the coefficients. 3 + 5 = 8
Step 3: Keep the variable. 8x
Answer: 3x + 5x = 8x
Simplify: 7y - 2y + 4y
Avoid these 3 frequent errors
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NOT like terms:
Think: Can only combine apples with apples, not apples with oranges!
Combining like terms simplifies expressions.
Before: 3x + 2 + 5x + 7 After: 8x + 9
Simpler = easier to work with!
Benefits:
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
Example: 2x + 5x + 3x
Add coefficients: 2 + 5 + 3 = 10
Answer: 10x
Example 2: 7a - 3a + 4a
Combine: 7 - 3 + 4 = 8
Answer: 8a
Example: 3x + 4y + 2x + y
Step 1: Identify like terms
Step 2: Combine each group
Answer: 5x + 5y
Example: 5x + 3 + 2x + 7
Step 1: Group like terms
Step 2: Combine
Answer: 7x + 10
Remember: Constants (numbers alone) are like terms with each other!
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)
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
Example: 4a + 3b - 2a + 5b - 1
Step 1: Identify groups
Step 2: Combine
Answer: 2a + 8b - 1
Example 1: 7x + 2y - 3x + 8y - 4
Group and combine:
Answer: 4x + 10y - 4
Example 2: 10 - 3m + 5 + 7m - 2
Combine:
Answer: 13 + 4m (or 4m + 13)
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.
Practice identifying:
Which are like terms with 5x?
Which are like terms with 7?
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
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
Example: 5 + 3x - 2 + 7x
Rearrange to group like terms: 3x + 7x + 5 - 2
Combine: 10x + 3
Tip: Grouping like terms helps avoid mistakes!
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 Total: 3x + 5x = 8x
Total shirts: 8 shirts at $x each
Commutative Property: Can rearrange terms
5x + 3y = 3y + 5x (same thing!)
Standard form: Usually write in alphabetical order
But mathematically equivalent!
❌ Mistake 1: Combining unlike terms
❌ Mistake 2: Forgetting coefficient of 1
❌ Mistake 3: Sign errors
❌ Mistake 4: Combining different powers
❌ Mistake 5: Changing the variable
To simplify expressions:
To solve equations:
Like Terms: Same variable(s) and power(s)
Combining:
Examples:
Steps:
Remember: Only like terms can be combined!
Tip 1: Circle or underline like terms
Tip 2: Use different colors
Tip 3: Write coefficients clearly
Tip 4: Check by substituting
Tip 5: Practice identifying
Like terms have the same variable(s) and power(s):
Combining like terms:
Process:
Applications:
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!
Step 1: All terms have the variable y (like terms).
Step 2: Combine the coefficients. 7 - 2 + 4 = 9
Step 3: Attach the variable. 9y
Answer: 9y
Simplify: 4x + 3 + 2x - 5
Step 1: Identify like terms. Variable terms: 4x and 2x Constant terms: 3 and -5
Step 2: Combine variable terms. 4x + 2x = 6x
Step 3: Combine constant terms. 3 + (-5) = 3 - 5 = -2
Step 4: Write the final expression. 6x - 2
Answer: 6x - 2
Simplify: 5a + 3b - 2a + 7b
Step 1: Identify like terms. a terms: 5a and -2a b terms: 3b and 7b
Step 2: Combine a terms. 5a - 2a = 3a
Step 3: Combine b terms. 3b + 7b = 10b
Step 4: Write the final expression. 3a + 10b
Answer: 3a + 10b
Simplify: 2x² + 5x - 3 + 4x² - 2x + 8
Step 1: Identify like terms. x² terms: 2x² and 4x² x terms: 5x and -2x Constant terms: -3 and 8
Step 2: Combine x² terms. 2x² + 4x² = 6x²
Step 3: Combine x terms. 5x - 2x = 3x
Step 4: Combine constant terms. -3 + 8 = 5
Step 5: Write in standard form (highest degree first). 6x² + 3x + 5
Answer: 6x² + 3x + 5