Overview
This lecture introduces simplifying algebraic expressions by combining like terms and applying the distributive property, with step-by-step examples and strategies.
Combining Like Terms
- Like terms have the same variable(s) raised to the same power(s).
- To combine like terms, add or subtract the coefficients while keeping the variable part the same.
- Constants (numbers without variables) are also considered like terms with each other.
- Rearranging expressions to group like terms together can simplify combining.
- Always include the sign in front of each term when combining.
Examples of Combining Like Terms
- 9x + 3x = 12x
- 8G + 5G + 7 + 2 = 13G + 9
- 6y² + 2y² + 10y + 3y + y = 8y² + 14y
- 7x + 2y - 4x + 2y = 3x + 4y
- 9C + 8C + 6D + 5D + 2D = 17C + 13D
- -10a + 9a + 9a + 2ab + ab - 8 = 8a + 3ab - 8
Using the Distributive Property
- Distributive property: a(b + c) = ab + ac (applies with addition or subtraction in the parentheses).
- To remove parentheses, multiply the factor outside by each term inside.
- The property holds for both numbers and variables.
Examples of the Distributive Property
- 2(5 + 3) = 2×5 + 2×3 = 10 + 6 = 16
- 8(2n + 6) = 16n + 48
- 7(a - 9) = 7a - 63
- 10(-5x - 4y) = -50x - 40y
- 9(3C + 2) = 27C + 18
- 4(-6y - 5) = -24y - 20
- -2(-8x - 7y) = 16x + 14y
- 15(3G + H) = 45G + 15H
Combining Distributive Property & Like Terms
- Apply the distributive property first to remove parentheses.
- After distributing, combine any like terms to simplify further.
- Examples:
- 4 + 2(n + 6) = 2n + 16
- 6(3x + 5) - 9x = 9x + 30
- 23 + 3(8y - 10) = 24y - 7
- -8(C - D) - 5D = 8C + 3D
- 13a + 4(a + 9) = 17a + 36
- 5(x² - 3) + 10 - 4x = 5x² - 4x - 5
- 7(G + 3H) + 4(2G - 6H) = 15G - 3H
- 18x - 10(2x - 2y + 9) - 6x = -8x + 20y - 90
Key Terms & Definitions
- Like terms — Terms with the same variable(s) raised to the same power(s).
- Coefficient — The numerical part multiplied by the variable in a term.
- Constant — A term without a variable.
- Distributive property — Multiplying a term outside parentheses by each term inside.
Action Items / Next Steps
- Practice simplifying expressions by combining like terms and using the distributive property.
- Review example problems and attempt similar exercises for mastery.