Overview
This lecture explains how to simplify algebraic expressions by combining like terms and using the distributive property, including step-by-step examples for both skills.
Combining Like Terms
- Like terms have the same variable(s) to the same exponent(s); constants are also like terms.
- Combine like terms by adding or subtracting their coefficients.
- Rearranging expressions to group like terms can make combining easier.
- Always bring the sign in front of a term when moving or combining terms.
- Example: 9x + 3x = 12x; 8G + 5G + 7 + 2 = 13G + 9.
Strategies for Combining Like Terms
- You can rewrite subtraction as “adding the opposite” to clarify negative terms.
- Expressions are typically written with variables in order of highest exponent to lowest, and constants last.
- When combining variables with the same exponent, use alphabetical order if possible.
The Distributive Property
- The distributive property removes parentheses: a(b + c) = ab + ac.
- Multiply the outside term by each term inside the parentheses.
- Works with both addition and subtraction inside parentheses.
- Distributing does not change the value of the expression.
- Example: 8(2n + 6) = 16n + 48; 7(a - 9) = 7a - 63.
Combining Like Terms with Distributive Property
- When both are needed, distribute first, then combine like terms.
- Example: 4 + 2(n + 6) → 4 + 2n + 12 → 2n + 16.
- Distribute negatives carefully: -2(-8x - 7y) = 16x + 14y.
More Complex Examples
- Simplify by distributing, then grouping and combining like terms.
- Combine all coefficients for the same variable/exponent; add/subtract constants.
- Example: 13a + 4(a + 9) = 13a + 4a + 36 = 17a + 36.
- For expressions with various variables and exponents, keep order: highest exponents first, then variables, then constants.
Key Terms & Definitions
- Like Terms — Terms with the same variable(s) and exponent(s).
- Coefficient — The numerical part multiplying the variable.
- Constant Term — A term without variables.
- Distributive Property — For any a, b, c: a(b + c) = ab + ac.
Action Items / Next Steps
- Practice simplifying expressions by identifying like terms and applying the distributive property.
- Review notes on rearranging terms for clarity.
- Complete assigned homework on expression simplification as directed.