Unit 1 Algebra Basics Homework 5 Evaluating Expressions Answer Key

Unit 1 Algebra Basics Homework 5: Evaluating Expressions – Answer Key & Comprehensive Guide

Algebra can feel daunting at first, but mastering the basics is key to unlocking more advanced concepts. This comprehensive guide tackles Unit 1, Homework 5, focusing on evaluating algebraic expressions. We'll walk through the core concepts, provide example problems with detailed solutions, and offer strategies to improve your understanding and problem-solving skills. Think of this as your ultimate answer key and study guide, rolled into one!

Understanding Algebraic Expressions

Before diving into the homework, let's solidify our understanding of algebraic expressions. An algebraic expression is a combination of variables (like x, y, z), constants (numbers), and mathematical operations (+, -, ×, ÷). Evaluating an expression means substituting given values for the variables and simplifying the expression to find a numerical result.

Key Concepts to Remember:

• Order of Operations (PEMDAS/BODMAS): Remember this acronym religiously! It dictates the order in which operations should be performed:
  • Parentheses (Brackets)
  • Exponents (Orders)
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
• Substitution: Carefully replace the variables in the expression with their assigned values. Use parentheses to avoid confusion, especially when dealing with negative numbers or multiplication.
• Simplification: After substitution, perform the calculations according to the order of operations to arrive at the final answer.

Example Problems and Solutions

Let's work through some examples that mirror the types of problems you might find in your homework. These examples will serve as a detailed answer key, providing a step-by-step approach to each problem.

Problem 1: Evaluate the expression 3x + 5y - 2z when x = 2, y = -1, and z = 4.

Solution:

1. Substitution: Replace x, y, and z with their respective values: 3(2) + 5(-1) - 2(4)
2. Multiplication: Perform the multiplications first: 6 + (-5) - 8
3. Addition and Subtraction: Work from left to right: 1 - 8 = -7

Therefore, the answer is -7.

Problem 2: Evaluate (2a - b)² + c when a = 3, b = 4, and c = -2.

Solution:

1. Substitution: Substitute the values: (2(3) - 4)² + (-2)
2. Parentheses (Innermost First): Solve the expression inside the parentheses: (6 - 4)² + (-2) which simplifies to 2² + (-2)
3. Exponents: Evaluate the exponent: 4 + (-2)
4. Addition: Add the numbers: 2

Therefore, the answer is 2.

Problem 3: Evaluate (x/y) * z - w when x = 12, y = 3, z = 5, and w = 2.

Solution:

1. Substitution: Substitute the given values: (12/3) * 5 - 2
2. Division: Perform the division: 4 * 5 - 2
3. Multiplication: Perform the multiplication: 20 - 2
4. Subtraction: Perform the subtraction: 18

Therefore, the answer is 18.

Problem 4: Evaluate √(a² + b²) - c when a = 6, b = 8, and c = 5.

Solution:

1. Substitution: Substitute the values: √(6² + 8²) - 5
2. Exponents: Evaluate the exponents: √(36 + 64) - 5
3. Addition: Add the numbers inside the square root: √100 - 5
4. Square Root: Calculate the square root: 10 - 5
5. Subtraction: Subtract the numbers: 5

Therefore, the answer is 5.

Problem 5 (More Challenging): Evaluate (2x + y)/(z - 3w) + 4x when x = 2, y = -1, z = 10, w = 1.

Solution:

1. Substitution: Substitute values: (2(2) + (-1))/(10 - 3(1)) + 4(2)
2. Parentheses (Numerator): Simplify the numerator: (4 + (-1)) = 3
3. Parentheses (Denominator): Simplify the denominator: (10 - 3) = 7
4. Division: Perform the division: 3/7
5. Multiplication: Perform the multiplication: 4(2) = 8
6. Addition: Add the results: 3/7 + 8 (Convert 8 to a fraction with denominator 7: 56/7)
7. Addition of Fractions: Add the fractions: 59/7

Therefore, the answer is 59/7 (or approximately 8.43).

Strategies for Success

Here are some tips to master evaluating algebraic expressions and excel in your homework:

• Practice Regularly: The more you practice, the more comfortable you'll become with the process. Work through as many problems as possible, even beyond the assigned homework.
• Break Down Complex Problems: If a problem seems overwhelming, break it down into smaller, more manageable steps. Focus on one operation at a time.
• Check Your Work: Always double-check your calculations to avoid careless errors. It's easy to make mistakes, especially when dealing with negative numbers or fractions.
• Use a Calculator Wisely: Calculators can be helpful, but they shouldn't replace your understanding of the underlying concepts. Use them for complex calculations but make sure you understand each step.
• Seek Help When Needed: Don't hesitate to ask your teacher, classmates, or tutor for help if you are struggling with a particular concept or problem. There are also many online resources available to support your learning.
• Understand the "Why": Don't just memorize the steps; strive to understand why you're performing each operation. This deeper understanding will make it easier to solve new and more complex problems in the future.
• Visual Aids: Consider using visual aids such as diagrams or number lines to help you visualize the problem and track your progress.
• Identify Common Mistakes: Keep a note of the types of mistakes you tend to make and actively work to avoid them in the future.

Beyond the Homework: Expanding Your Knowledge

Mastering evaluating algebraic expressions is a foundational step in algebra. Once you feel confident with this unit, you can move on to more advanced topics such as:

• Solving Equations: Using algebraic manipulation to find the value of an unknown variable.
• Inequalities: Working with expressions that involve greater than or less than signs.
• Graphing Equations and Inequalities: Representing algebraic relationships visually.
• Systems of Equations: Solving multiple equations simultaneously.
• Polynomial Operations: Working with expressions that contain multiple terms with variables raised to different powers.

By consistently practicing and building a solid understanding of the basics, you'll be well-equipped to tackle the challenges of higher-level algebra and beyond. Remember, algebra is a journey of learning and understanding. Embrace the process, stay persistent, and celebrate your progress! Good luck with your homework!