Math Lecture: Simplifying Algebraic Expressions

Key Concepts

• Simplifying algebraic expressions involves using the distributive property and combining like terms.
• Distributive Property: Used to remove parentheses by distributing a factor across terms within parentheses.
• Like Terms: Terms in an expression that have the same variable raised to the same power, which can be combined by adding or subtracting their coefficients.

Example Problems

Problem 1

• Expression: 13a + 4(a + 9)
• Steps:
  1. Distribute the 4:
     • 4 * a = 4a
     • 4 * 9 = 36
     • Expression becomes 13a + 4a + 36
  2. Combine like terms:
     • 13a + 4a = 17a
     • Final Simplified Expression: 17a + 36

Problem 2

• Expression: 5(x^2 - 3) + 10 - 4x
• Steps:
  1. Distribute the 5:
     • 5 * x^2 = 5x^2
     • 5 * -3 = -15
     • Expression becomes 5x^2 - 15 + 10 - 4x
  2. Combine like terms:
     • Combine constants -15 and 10: -5
     • Final Simplified Expression: 5x^2 - 4x - 5

Problem 3

• Expression: 7(g + 3h) + 4(2g - 6h)
• Steps:
  1. Distribute the 7 and 4:
     • 7 * g = 7g
     • 7 * 3h = 21h
     • 4 * 2g = 8g
     • 4 * -6h = -24h
     • Expression becomes 7g + 21h + 8g - 24h
  2. Combine like terms:
     • Combine 7g and 8g: 15g
     • Combine 21h and -24h: -3h
     • Final Simplified Expression: 15g - 3h

Problem 4

• Expression: 18x - 10(2x - 2y + 9) - 6x
• Steps:
  1. Distribute the -10:
     • -10 * 2x = -20x
     • -10 * -2y = 20y
     • -10 * 9 = -90
     • Expression becomes 18x - 20x + 20y - 90 - 6x
  2. Combine like terms:
     • 18x - 20x - 6x = -8x
     • Final Simplified Expression: -8x + 20y - 90*

Additional Notes

• When writing expressions, typically list terms with the greatest exponent first, followed by lesser exponents, and constants last.
• Remember that variables without a written exponent have an implicit exponent of one.