Student's completed algebra homework focusing on properties of equality and logical proofs.

Let's go through the homework step by step and correctly solve and explain each part. The assignment is about properties of equality and algebraic proofs.

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✔ Part 1: Name the property of equality that justifies each statement.

We are given statements and must choose the correct property from the list (A–I).

#### 1. If $ a = 2b $, then $ a - c = 2b - c $

- We are subtracting $ c $ from both sides.
- This is the Subtraction Property of Equality.
- ✔ Answer: B

#### 2. $ x = x $

- A number is equal to itself.
- This is the Reflexive Property of Equality.
- ✔ Answer: G

#### 3. $ 3(p - 7) = 3p - 21 $

- Distributing the 3 across $ p - 7 $.
- This is the Distributive Property.
- ✔ Answer: E

#### 4. If $ -7k = -42 $, then $ k = 6 $

- Dividing both sides by $-7$ to solve for $k$.
- This is the Division Property of Equality.
- ✔ Answer: D

#### 5. If $ m + n = 15 $ and $ n = 2 $, then $ m + 2 = 15 $

- Replacing $ n $ with $ 2 $ in the first equation.
- This is the Substitution Property.
- ✔ Answer: F

#### 6. If $ \frac{x}{4} = -5 $, then $ x = -20 $

- Multiplying both sides by 4.
- This is the Multiplication Property of Equality.
- ✔ Answer: C

#### 7. If $ w^2 = 2x $ and $ 2x = y $, then $ w^2 = y $

- Two things equal to the same thing are equal to each other.
- This is the Transitive Property.
- ✔ Answer: I

#### 8. If $ c - 9 = -1 $, then $ c = 8 $

- Adding 9 to both sides.
- This is the Addition Property of Equality.
- ✔ Answer: A

#### 9. If $ n = -3 $, then $ -3 = n $

- The order of equality can be reversed.
- This is the Symmetric Property.
- ✔ Answer: H

✔ Summary of Answers (1–9):
1. B
2. G
3. E
4. D
5. F
6. C
7. I
8. A
9. H

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✔ Part 2: Complete each proof using properties of equality.

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#### Problem 10: Given: $ -8(x - 5) = -32 $, Prove: $ x = 7 $

| Statements | Reasons |
|----------|--------|
| $ -8(x - 5) = -32 $ | Given |
| $ -8x + 40 = -32 $ | Distributive Property |
| $ -8x = -72 $ | Subtraction Property of Equality (subtract 40 from both sides) |
| $ x = 9 $ | Division Property of Equality (divide both sides by -8) |

Wait — but the student wrote $ x = 7 $. That’s incorrect!

Let’s check:
$$
-8(x - 5) = -32 \\
x - 5 = 4 \quad \text{(dividing both sides by -8)} \\
x = 9
$$

So the conclusion should be $ x = 9 $, not $ x = 7 $.

But the problem says "Prove: $ x = 7 $" — that’s a mistake.

✘ So either the given or the conclusion is wrong.

Let’s double-check:

If $ x = 7 $, then:
$$
-8(7 - 5) = -8(2) = -16 \neq -32
$$

So $ x = 7 $ is not correct.

Therefore, either the problem has a typo, or the goal is to find the correct value.

But since it says “Prove: $ x = 7 $”, and it’s false, we must assume there's an error.

But perhaps the original equation was meant to be:
$$
-8(x - 5) = -56 \Rightarrow x = 7?
$$
Check:
$$
-8(7 - 5) = -8(2) = -16 \neq -56
$$

No.

Try:
$$
-8(x - 5) = -32 \Rightarrow x = 9
$$

So unless the problem is wrong, the correct answer is $ x = 9 $.

But let's assume the student made a mistake in the proof.

Here’s the corrected version:

| Statements | Reasons |
|----------|--------|
| $ -8(x - 5) = -32 $ | Given |
| $ -8x + 40 = -32 $ | Distributive Property |
| $ -8x = -72 $ | Subtraction Property of Equality (subtract 40 from both sides) |
| $ x = 9 $ | Division Property of Equality (divide both sides by -8) |

✔ So the final answer should be $ x = 9 $, not $ x = 7 $.

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#### Problem 11: Given: $ -16 = \frac{m}{5} - 18 $, Prove: $ m = 10 $

Let’s work through this step by step.

| Statements | Reasons |
|----------|--------|
| $ -16 = \frac{m}{5} - 18 $ | Given |
| $ -16 + 18 = \frac{m}{5} $ | Addition Property of Equality (add 18 to both sides) |
| $ 2 = \frac{m}{5} $ | Simplify |
| $ 10 = m $ | Multiplication Property of Equality (multiply both sides by 5) |
| $ m = 10 $ | Symmetric Property of Equality (reverse the order) |

Now fill in the table:

| Statements | Reasons |
|----------|--------|
| $ -16 = \frac{m}{5} - 18 $ | Given |
| $ 2 = \frac{m}{5} $ | Addition Property of Equality |
| $ 10 = m $ | Multiplication Property of Equality |
| $ m = 10 $ | Symmetric Property of Equality |

✔ Final Answer: $ m = 10 $, which matches the goal.

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✔ Final Notes:

- The first proof (Problem 10) has a mistake: the conclusion $ x = 7 $ is incorrect based on the given equation.
- Correct solution: $ x = 9 $
- The second proof (Problem 11) is correct and well-structured.

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✔ Corrected Answers Summary:

#### Part 1: Properties
1. B
2. G
3. E
4. D
5. F
6. C
7. I
8. A
9. H

#### Part 2: Proof 10 (Corrected)
| Statements | Reasons |
|----------|--------|
| $ -8(x - 5) = -32 $ | Given |
| $ -8x + 40 = -32 $ | Distributive Property |
| $ -8x = -72 $ | Subtraction Property of Equality |
| $ x = 9 $ | Division Property of Equality |

> ⚠️ Note: The statement "Prove: $ x = 7 $" is false. It should be $ x = 9 $.

#### Part 2: Proof 11 (Completed)
| Statements | Reasons |
|----------|--------|
| $ -16 = \frac{m}{5} - 18 $ | Given |
| $ 2 = \frac{m}{5} $ | Addition Property of Equality |
| $ 10 = m $ | Multiplication Property of Equality |
| $ m = 10 $ | Symmetric Property of Equality |

✔ All steps are justified.

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✔ Final Recommendation:

The student likely made a small error in Problem 10. Double-check the original problem. If the given equation is truly $ -8(x - 5) = -32 $, then $ x = 9 $, not $ x = 7 $. Otherwise, the rest of the work is excellent!