All Things Algebra Unit 3 Homework 2 Answer Key

All Things Algebra Unit 3 Homework 2 Answer Key: A Comprehensive Guide

Finding the correct answers for your math homework can be a struggle, especially when dealing with complex topics like those covered in Unit 3 of All Things Algebra. This comprehensive guide will delve into the potential problems found in Homework 2 of this unit, providing detailed explanations and solutions to help you master the concepts. We'll cover a range of topics, from simplifying expressions to solving equations, ensuring a thorough understanding of the material. Remember, understanding why an answer is correct is more important than just having the answer itself.

Note: Since I don't have access to the specific questions in your All Things Algebra Unit 3 Homework 2, I will provide examples of common problems encountered in this unit and explain the steps to solve them. You can apply these methods to your specific assignment. If you have specific problems you're struggling with, please provide them, and I will do my best to assist you.

Section 1: Simplifying Algebraic Expressions

This section usually focuses on combining like terms and applying the order of operations (PEMDAS/BODMAS). Let's tackle some examples:

Example 1: Simplify the expression: 3x + 2y - x + 5y

Solution:

1. Identify like terms: We have terms with 'x' and terms with 'y'.
2. Combine like terms: 3x - x = 2x and 2y + 5y = 7y
3. Simplified expression: 2x + 7y

Example 2: Simplify: 2(3x + 4) - 5x

Solution:

1. Distribute: Multiply 2 by each term inside the parentheses: 2 * 3x = 6x and 2 * 4 = 8.
2. Rewrite the expression: 6x + 8 - 5x
3. Combine like terms: 6x - 5x = x
4. Simplified expression: x + 8

Example 3: Simplify: 4x² + 6x - 2x² + 3x + 5

Solution:

1. Identify like terms: We have x² terms and x terms.
2. Combine like terms: 4x² - 2x² = 2x² and 6x + 3x = 9x
3. Simplified expression: 2x² + 9x + 5

Section 2: Solving Linear Equations

This section typically involves finding the value of the variable that makes the equation true. Here are some common types of linear equations and their solutions:

Example 4: Solve for x: x + 7 = 12

Solution:

1. Isolate the variable: Subtract 7 from both sides of the equation: x + 7 - 7 = 12 - 7
2. Simplify: x = 5

Example 5: Solve for y: 3y - 6 = 15

Solution:

1. Add 6 to both sides: 3y - 6 + 6 = 15 + 6
2. Simplify: 3y = 21
3. Divide both sides by 3: 3y / 3 = 21 / 3
4. Simplify: y = 7

Example 6: Solve for z: (z/4) + 2 = 5

Solution:

1. Subtract 2 from both sides: (z/4) + 2 - 2 = 5 - 2
2. Simplify: z/4 = 3
3. Multiply both sides by 4: 4 * (z/4) = 3 * 4
4. Simplify: z = 12

Example 7: Solve for x: 2x + 5 = 3x - 2

Solution:

1. Subtract 2x from both sides: 2x + 5 - 2x = 3x - 2 - 2x
2. Simplify: 5 = x - 2
3. Add 2 to both sides: 5 + 2 = x - 2 + 2
4. Simplify: x = 7

Section 3: Solving Inequalities

Solving inequalities involves finding the range of values for the variable that satisfy the inequality. The process is similar to solving equations, with one key difference: when multiplying or dividing by a negative number, you must reverse the inequality sign.

Example 8: Solve for x: x + 3 > 7

Solution:

1. Subtract 3 from both sides: x + 3 - 3 > 7 - 3
2. Simplify: x > 4

Example 9: Solve for y: -2y ≤ 10

Solution:

1. Divide both sides by -2: -2y / -2 ≥ 10 / -2 (Note the reversed inequality sign)
2. Simplify: y ≥ -5

Example 10: Solve for z: 3z - 5 < 16

Solution:

1. Add 5 to both sides: 3z - 5 + 5 < 16 + 5
2. Simplify: 3z < 21
3. Divide both sides by 3: 3z / 3 < 21 / 3
4. Simplify: z < 7

Section 4: Word Problems

This section requires translating real-world scenarios into algebraic equations and then solving them. Here's an example:

Example 11: John is three years older than his sister Mary. The sum of their ages is 25. How old is John?

Solution:

1. Define variables: Let J represent John's age and M represent Mary's age.
2. Write equations: J = M + 3 (John is 3 years older) and J + M = 25 (Sum of their ages is 25)
3. Substitute: Substitute the first equation into the second: (M + 3) + M = 25
4. Simplify: 2M + 3 = 25
5. Solve for M: 2M = 22 => M = 11
6. Find J: J = M + 3 = 11 + 3 = 14
7. Answer: John is 14 years old.

Section 5: Advanced Topics (Potentially included in Unit 3)

This section might include more advanced topics depending on the curriculum. These could include:

• Systems of Equations: Solving for multiple variables using methods like substitution or elimination.
• Quadratic Equations: Equations involving x². Solutions might involve factoring, the quadratic formula, or completing the square.
• Exponents and Radicals: Working with powers and roots.

Remember to always show your work, clearly stating each step of your solution. This helps you understand the process and identify any errors you may have made. Practice is key to mastering algebra; the more problems you work through, the more confident you'll become. Don't hesitate to seek help from your teacher, classmates, or online resources if you're struggling with any particular concept. Good luck!