Notes on Simplifying Algebraic Expressions

Introduction

• Focus on simplifying algebraic expressions.
• Key techniques: combining like terms and using the distributive property.

Combining Like Terms

• Definition: Like terms share the same variables raised to the same powers.
• Goal: Rewrite an expression to make it simpler and easier to understand.

Example 1: Combine 9x + 3x

• Terms: 9x and 3x (like terms)
• Combine coefficients: 9 + 3 = 12
• Simplified expression: 12x

Example 2: Combine 8G + 7 + 5G + 2

• Like terms: 8G, 5G (combine to get 13G) and constants: 7, 2 (combine to get 9)
• Final simplified expression: 13G + 9

Example 3: Combine 6y^2 + 10y + 2y^2 + 3y + y

• Like terms:
  • y^2 terms: 6y^2, 2y^2 → combine to 8y^2
  • y terms: 10y, 3y, y → combine to 14y
• Final expression: 8y^2 + 14y

Example 4: Combine 7x + 2y - 4x + 2y

• Like terms: 7x, -4x (combine to get 3x) and 2y, 2y (combine to get 4y)
• Final expression: 3x + 4y

Alternate Approach

• By rewriting subtraction as adding the opposite can clarify the expression.
  • Example: 7x + 2y - 4x + 2y becomes 7x + 2y + (-4x) + 2y.

Distributive Property

• Definition: Distributing a value across terms inside parentheses.
• This is applicable for both addition and subtraction.

Example: Use the Distributive Property

• Expression: 2(5 + 3)
  • Using order of operations: 2 * (5 + 3) = 2 * 8 = 16
  • Using distributive property: 25 + 23 = 10 + 6 = 16

Algebraic Example: 8(2n + 6)

• Distribute 8: 8 * 2n + 8 * 6 → 16n + 48

Additional Examples with Distributive Property

1. Example 1: 7(a - 9) -> 7a - 63
2. Example 2: 10(-5x - 4y) -> -50x - 40y

Combining Techniques

• When faced with parentheses, use the distributive property first, then combine like terms.

Example 1: Combine 4 + 2(n + 6)

• Distribute 2: 2n + 12 → then combine with 4 → 2n + 16

More Complex Example: 13a + 4(a + 9)

• Distributing 4 gives: 13a + 4a + 36 → Combine to give: 17a + 36

Final Thoughts

• Reminders on writing expressions: order of powers, like terms next to each other, constant terms last.

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• These strategies can help simplify complex algebraic expressions effectively and help in problem-solving.