Algebra 1 - Unit 1, Lesson 1: Variables and Expressions

Instructor: Kirk Wyler

Key Concepts

What is an Expression?

• Expression: Combination of numbers and/or variables using operations like addition, subtraction, multiplication, division, roots, and exponents.
• Example: Numerical expressions and evaluation using the order of operations.

Order of Operations (PEMDAS)

1. Parentheses
2. Exponents
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

Exercises and Examples

Exercise 1: Numerical Expressions

Example 1a

• Expression: 3 * 2 + 7
• Evaluation:
  • Step 1: 3 * 2 = 6
  • Step 2: 6 + 7 = 13
• Correct Order: Multiplication before Addition

Example 1b

• Expression: 8 - 1/2 * 6 + 24 / 6
• Evaluation:
  • Step 1: 1/2 * 6 = 3
  • Step 2: 24 / 6 = 4
  • Step 3: 8 - 3 + 4 = 9
• Key Point: Multiplication and Division from left to right

Example 1c

• Expression: 4 * (8 - 6) - 7 * (5 - 3)
• Evaluation:
  • Step 1: Evaluate inside parentheses 8 - 6 = 2, 5 - 3 = 2
  • Step 2: 4 * 2 = 8, 7 * 2 = 14
  • Step 3: 8 - 14 = -6
• Key Point: Evaluate expressions in parentheses first

More Challenging Problems

Example 1d

• Expression: 5^2 - 4^2 + 3 / (1 - 5)
• Evaluation:
  • Step 1: 5^2 = 25, 4^2 = 16
  • Step 2: Numerator: 25 - 16 + 3 = 12
  • Step 3: Denominator: 1 - 5 = -4
  • Step 4: Division: 12 / -4 = -3
• Key Point: Evaluate numerator and denominator separately

Variables in Expressions

How to Read Variable Expressions

Exercise 2

• Example 2a: 3x - 8
  • Operations: Multiply x by 3, then subtract 8
• Example 2b: (x - 4) / 2
  • Operations: Subtract 4 from x, then divide by 2
• Example 2c: 4x^2 - 8
  • Operations: Square x, multiply result by 4, then subtract 8

Evaluating Expressions with Variables

Exercise 3

Example 3a

• Expression: 4x - 7 (with x=5)
• Evaluation: Substitute x with 5
  • Step 1: 4 * 5 = 20
  • Step 2: 20 - 7 = 13*

Example 3b

• Expression: 25 - (2/3)n (with n=12)
• Evaluation: Substitute n with 12
  • Step 1: (2/3) * 12 = 8
  • Step 2: 25 - 8 = 17*

Example 3c (with exponents and negative numbers)

• Expression: c^2 + 6c (with c=-4)
• Evaluation: Substitute c with -4
  • Step 1: (-4)^2 = 16
  • Step 2: 6 * (-4) = -24
  • Step 3: 16 - 24 = -8*

Example 3d

• Expression: (1/2)t^2 - 2t + 9 (with t=6)
• Evaluation: Substitute t with 6
  • Step 1: 6^2 = 36
  • Step 2: (1/2) * 36 = 18
  • Step 3: 2 * 6 = 12
  • Step 4: 18 - 12 + 9 = 15

Real-World Application

Example 4

• Problem: The amount of money earned by a tour company is given by 12(n - 4) where n is the number of people.
  • If n=14:
  • Evaluation:
    • Step 1: 14 - 4 = 10
    • Step 2: 12 * 10 = 120
• Result: The company makes $120 when 14 people go on the tour.*

Conclusion

• Basic understanding of expressions and order of operations is critical in algebra.
• Recognizing operations performed on variables and their correct order is key to solving algebraic expressions.

Practice: Review the exercises and ensure you understand each step.