Student's algebra homework assignment focusing on algebraic proofs and properties of equality.

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 value 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 $
- This is the Division Property of Equality.
- ✔ Answer: D

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

- Substituting $ n = 2 $ into 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 $

- Since $ w^2 = 2x $ and $ 2x = y $, we can say $ w^2 = y $
- 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 $

- Reversing the order of equality
- This is the Symmetric Property.
- ✔ Answer: H

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✔ So far, all your answers in Part 1 are correct!

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✔ Part 2: Complete each proof using properties of equality

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#### Problem 10: Given: $ -8(x - 3) = -32 $, Prove: $ x = 7 $

| Statements | Reasons |
|-----------|--------|
| $ -8(x - 3) = -32 $ | Given |
| $ -8x + 24 = -32 $ | Distributive Property |
| $ -8x = -56 $ | Subtraction Property of Equality (subtract 24 from both sides) |
| $ x = 7 $ | Division Property of Equality (divide both sides by -8) |

> ✔ Your reasoning is mostly correct, but let’s fix the last step:
> - From $ -8x = -56 $, divide both sides by -8 → $ x = 7 $
> - So, reason is Division Property of Equality, not "Division Property of Equality" (you wrote it twice). Also, the second line is correct.

But wait — you wrote:

> $ -8x = -8 $ ← ✘ This is incorrect!

You said $ -8x = -8 $, but that’s wrong.

Let’s redo this correctly:

- $ -8x + 24 = -32 $
- Subtract 24: $ -8x = -32 - 24 = -56 $
- Then divide: $ x = (-56)/(-8) = 7 $

So, correct table:

| Statements | Reasons |
|-----------|--------|
| $ -8(x - 3) = -32 $ | Given |
| $ -8x + 24 = -32 $ | Distributive Property |
| $ -8x = -56 $ | Subtraction Property of Equality |
| $ x = 7 $ | Division Property of Equality |

✔ You had a mistake in arithmetic in the third row. It should be $ -8x = -56 $, not $ -8 $. That’s a critical error.

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

Let’s complete the proof step by step.

We’re given:

$$
-16 = \frac{m}{5} - 18
$$

We want to prove $ m = 10 $

| 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 order) |

Now fill in the table as per your format:

| 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 |

✔ So the missing steps are:

- Third row: $ 10 = m $ → Reason: Multiplication Property of Equality
- Fourth row: $ m = 10 $ → Reason: Symmetric Property of Equality

> Note: You might not need the last row if you stop at $ 10 = m $, since that implies $ m = 10 $. But if they want the conclusion written as $ m = 10 $, then use Symmetric Property.

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✔ Final Answers Summary

#### Part 1: Properties

1. B – Subtraction Property
2. G – Reflexive Property
3. E – Distributive Property
4. D – Division Property
5. F – Substitution Property
6. C – Multiplication Property
7. I – Transitive Property
8. A – Addition Property
9. H – Symmetric Property

✔ All correct!

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#### Part 2: Proofs

Problem 10: Fix the mistake

| Statements | Reasons |
|-----------|--------|
| $ -8(x - 3) = -32 $ | Given |
| $ -8x + 24 = -32 $ | Distributive Property |
| $ -8x = -56 $ | Subtraction Property of Equality |
| $ x = 7 $ | Division Property of Equality |

✘ You wrote $ -8x = -8 $, which is incorrect.

Problem 11: Fill in the rest

| 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 |

✔ You were on the right track, but missed the final two steps.

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📝 Final Notes

- Always double-check arithmetic in algebraic proofs.
- The order of operations and properties matter.
- Use Symmetric Property when flipping equations (e.g., $ a = b $ → $ b = a $).
- Use Substitution when replacing a variable with its equivalent.
- Transitive Property: If $ a = b $ and $ b = c $, then $ a = c $

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✔ Final Answer:

- Part 1: All answers are correct.
- Part 2:
- Problem 10: Corrected the mistake: $ -8x = -56 $, not $ -8 $
- Problem 11: Add rows:
- $ 10 = m $ → Multiplication Property of Equality
- $ m = 10 $ → Symmetric Property of Equality

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