Question

Statement Reason
2(x+3) = 10 given
2x+6=10 which property?
(2x+6) + -6 = 10+-6
(2x + (6+-6) = 10 + -6
2x+ 0 = 10 + -6
2x + 0 =4
2x = 4
2x/2 = 4/2
1x=2
x=2
Answers:
Associative property of additionAssociative property of addition
B Commutative property of additionCommutative property of addition
C Multiplicative property of equalityMultiplicative property of equality
D Distributive property of multiplication over additio

Answer 1

Let's analyze the steps in the proof based on the statement and the reasons provided:

1. **Statement**: \( 2(x + 3) = 10 \)
**Reason**: Given

2. **Statement**: \( 2x + 6 = 10 \)
**Reason**: **Distributive property of multiplication over addition**
This step applies the distributive property, which states that \( a(b + c) = ab + ac \).

3. **Statement**: \( (2x + 6) + -6 = 10 + -6 \)
**Reason**: **Addition property of equality**
You can add the same value (in this case, -6) to both sides of an equation.

4. **Statement**: \( 2x + (6 + -6) = 10 + -6 \)
**Reason**: **Associative property of addition**
This states that the way you group the numbers when adding does not change the sum.

5. **Statement**: \( 2x + 0 = 10 + -6 \)
**Reason**:
From the previous step, since \( 6 + (-6) = 0 \), we simplify accordingly.

6. **Statement**: \( 2x + 0 = 4 \)
**Reason**: Direct calculation, since \( 10 + (-6) = 4 \).

7. **Statement**: \( 2x = 4 \)
**Reason**: Subtracting 0 does not change the value.

8. **Statement**: \( \frac{2x}{2} = \frac{4}{2} \)
**Reason**: **Multiplicative property of equality**
This states that if two expressions are equal, you can divide both by the same nonzero number without changing the equality.

9. **Statement**: \( 1x = 2 \)
**Reason**: Simplifying \( \frac{2x}{2} \) gives \( x \).

10. **Statement**: \( x = 2 \)
**Reason**: Direct statement of final solution.

**Summary of Answers Corresponding to Reasons**:
- \( 2(x + 3) = 10 \): Given
- \( 2x + 6 = 10 \): **D** (Distributive property of multiplication over addition)
- \( (2x + 6) + -6 = 10 + -6 \): **C** (Multiplicative property of equality)
- \( (2x + (6 + -6)) = 10 + -6 \): **A** (Associative property of addition)
- \( 2x + 0 = 10 + -6 \): Logic
- \( 2x + 0 = 4 \): Calculation
- \( 2x = 4 \): Logic
- \( \frac{2x}{2} = \frac{4}{2} \): **C** (Multiplicative property of equality)
- \( 1x = 2 \): Simplification
- \( x = 2 \): Result

So the correct answer for each step can be summarized as:
1. Given
2. D
3. C
4. A
5. (not needed, but logical step)
6. Calculation
7. (not needed, but logical step)
8. C
9. (not needed, simplification)
10. Result

Would you like to clarify anything else about this proof?