Pre-Algebra Notes on Evaluating Algebraic Expressions

Pre-Algebra Study Notes

Chapter 6: Evaluating Algebraic Expressions

Section 6.4: Introduction

• Focus on simplifying algebraic expressions using the correct order of operations.

Simplification Examples

1. Prepare to Simplify Values:

   • Expression: $2 + 32 imes 4 - 5$

2. Brackets in Expressions:

   • Expression: $(2 + 32)4 - 5$

3. Parentheses for Clarity:

   • Expression: $2 + 32(4 - 5)$

Lesson Opener: Initial Evaluations

• Evaluations to complete: 3, 3, -7 (included as placeholders for specific calculations).

Example Evaluations Using Variables

• Variable Assignments: Evaluate expressions by substituting given values.

1. For x = -2 and y = 3:
   1. Evaluate: $2x^2y$
   2. Evaluate: $2y - x + 4$
   3. Evaluate: $x^2 - y^2$
   • Calculated results: 24, 12, -5 (numbers are results of respective calculations).

Detailed Example Evaluations

Example 1: Evaluating a Simple Expression

• Expression: $2xy$
• Values: Let x = 3, y = 1.
• Calculation: $2 imes 3 imes 1 = 6$

Example 2: Complex Evaluation

• Expression: $\frac{y}{x} + z$
• Given values: x = -3, y = 4, z = 7.
• Calculation Steps:
  • Evaluate: $\frac{4}{-3} + 7$
  • Step 1: Approximate $rac{4}{-3} o -rac{4}{3}$
  • Step 2: Add z = 7 (convert to fraction for easier addition): $-\frac{4}{3} + \frac{21}{3} = \frac{17}{3}$

Example 3: Evaluate with Decimals

• Expression: $5x - 12y$
• Values: Let x = 0.4, y = -2.05.
• Calculation:
  • Step 1: $5 imes 0.4 = 2$
  • Step 2: $12 imes -2.05 = -24.6$
  • Result: $2 + 24.6 = 26.6$

Skill Check Exercises

• Evaluate the following expressions based on provided inputs:
  1. $f + e$ when e = 15 and f = 9 → Result: 24
  2. $ab + 8$ when a = 9 and b = -18 → Result: -166

Additional Evaluations for Practice

1. Evaluate: $5x - 7y$ when x = 3 and y = -3.
2. Evaluate: $12a + 3b$ when a = 2 and b = -5.
   • Resulting evaluations: 9, 65.

Follow-up Examples

Example 4: Evaluation with Powers

• Expression: $4x^2y$
• Values: Let x = 3, y = -1.
• Calculation Sequence:
  • Step 1: Calculate $4(3^2)(-1)$
  • Step 2: $(3^2 = 9) o 4 imes 9 imes -1 = -36$

Example 5: Multi-term Expression

• Expression: $-3x - 2xy + 9y$
• Values: x = 2, y = 1.
• Calculation:
  • Step 1: Substitute: $-3(2) - 2(2)(1) + 9(1)$
  • Results calculated: 70.5

Example 6: Complex Expression Handling

• Expression: $-(4y - 3x)$
• Values: x = 6, y = 2.
• Sequence:
  • Step 1: Substitute values: $-(4(2) - 3(6))$
  • Evaluate: $- (8 - 18) = -(-10) = 10$

Final Evaluations with Multiple Variables

• Evaluate the following when x = -3, y = 1, z = 3.7, and w = 0.05:
  1. Evaluate: $x^2 + y^2$
  2. Evaluate: $\frac{w}{z} + 4y$
  3. Evaluate: $5z - 0.1w$
  • Skill check results shown as application of evaluations.

Continuing Exercises

• Skills check for different variable combinations to solidify understanding of evaluating algebraic expressions.

Assignment

• Additional evaluations and exercises to reinforce learning concepts.