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

1 answer

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?