Let's analyze the steps in the proof based on the statement and the reasons provided:
-
Statement: \( 2(x + 3) = 10 \)
Reason: Given -
Statement: \( 2x + 6 = 10 \)
Reason: Distributive property of multiplication over addition
This step applies the distributive property, which states that \( a(b + c) = ab + ac \). -
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. -
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. -
Statement: \( 2x + 0 = 10 + -6 \)
Reason: From the previous step, since \( 6 + (-6) = 0 \), we simplify accordingly. -
Statement: \( 2x + 0 = 4 \)
Reason: Direct calculation, since \( 10 + (-6) = 4 \). -
Statement: \( 2x = 4 \)
Reason: Subtracting 0 does not change the value. -
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. -
Statement: \( 1x = 2 \)
Reason: Simplifying \( \frac{2x}{2} \) gives \( x \). -
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:
- Given
- D
- C
- A
- (not needed, but logical step)
- Calculation
- (not needed, but logical step)
- C
- (not needed, simplification)
- Result
Would you like to clarify anything else about this proof?