U2L2 More Algebraic Proofs Practice Key (1) - Free Printable

Problem Analysis and Solution

The provided worksheet involves two main sections:
1. Identifying Properties of Equality: Match each statement with the appropriate property of equality.
2. Completing Proofs Using Properties of Equality: Fill in the missing steps and reasons for given proofs.

Let's solve each section step by step.

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Section 1: Identifying Properties of Equality

#### Given Statements and Matching Properties:
1. If \( a = 2b \), then \( a - c = 2b - c \)
- Property: Subtraction Property of Equality
- Reason: Subtracting the same value (\( c \)) from both sides of an equation maintains equality.
- Answer: B

2. \( x = x \)
- Property: Reflexive Property
- Reason: Any quantity is equal to itself.
- Answer: G

3. \( 3(p - 7) = 3p - 21 \)
- Property: Distributive Property
- Reason: Distributing the multiplication over addition/subtraction.
- Answer: E

4. If \( -7k = -42 \), then \( k = 6 \)
- Property: Division Property of Equality
- Reason: Dividing both sides of an equation by the same nonzero number maintains equality.
- Answer: D

5. If \( m + n = 15 \) and \( n = 2 \), then \( m + 2 = 15 \)
- Property: Substitution Property
- Reason: Replacing a variable with its equivalent value in an equation.
- Answer: F

6. If \( \frac{x}{4} = -5 \), then \( x = -20 \)
- Property: Multiplication Property of Equality
- Reason: Multiplying both sides of an equation by the same number maintains equality.
- Answer: C

7. If \( w^2 = 2x \) and \( 2x = y \), then \( w^2 = y \)
- Property: Transitive Property
- Reason: If \( a = b \) and \( b = c \), then \( a = c \).
- Answer: I

8. If \( c - 9 = -1 \), then \( c = 8 \)
- Property: Addition Property of Equality
- Reason: Adding the same value to both sides of an equation maintains equality.
- Answer: A

9. If \( n = -3 \), then \( -3 = n \)
- Property: Symmetric Property
- Reason: If \( a = b \), then \( b = a \).
- Answer: H

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Section 2: Completing Proofs Using Properties of Equality

#### Proof 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 |
| \( x = 7 \) | Division Property |

- Step-by-Step Explanation:
1. Start with the given equation: \( -8(x - 3) = -32 \).
2. Apply the Distributive Property: Distribute \(-8\) over \( (x - 3) \) to get \( -8x + 24 = -32 \).
3. Use the Subtraction Property of Equality: Subtract 24 from both sides to isolate the term with \( x \): \( -8x = -56 \).
4. Use the Division Property of Equality: Divide both sides by \(-8\) to solve for \( x \): \( x = 7 \).

#### Proof 11: Given \( -16 = \frac{m}{5} - 18 \); Prove \( m = 10 \)

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

- Step-by-Step Explanation:
1. Start with the given equation: \( -16 = \frac{m}{5} - 18 \).
2. Use the Addition Property of Equality: Add 18 to both sides to isolate the term with \( m \): \( 2 = \frac{m}{5} \).
3. Use the Multiplication Property of Equality: Multiply both sides by 5 to solve for \( m \): \( 10 = m \).
4. Use the Symmetric Property of Equality: Rewrite the equation as \( m = 10 \).

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Final Answer:

\[
\boxed{
\text{Section 1: } B, G, E, D, F, C, I, A, H
}
\]
\[
\boxed{
\text{Section 2: Proofs are completed as shown above.}
}
\]