Which property is not used to simplify the following expression?

[tex]\[
\begin{aligned}
(x+5) \cdot 2+7 & = 2(x+5) + 7 \\
& = (2x + 10) + 7 \\
& = 2x + (10 + 7) \\
& = 2x + 17
\end{aligned}
\][/tex]

A. Associative property of addition
B. Distributive property
C. Commutative property of multiplication
D. Commutative property of addition



Answer :

Let's analyze the step-by-step simplification of the expression [tex]\((x+5) \cdot 2 + 7\)[/tex]:

1. Distributive Property:
The first step involves applying the distributive property to expand [tex]\((x+5) \cdot 2\)[/tex] into [tex]\(2(x+5)\)[/tex]:
[tex]\[ (x+5) \cdot 2 = 2(x+5) \][/tex]

[tex]\[ 2(x+5) + 7 \][/tex]

2. Distributive Property:
Next, we distribute the 2 over the terms inside the parentheses:
[tex]\[ 2(x+5) = 2x + 2 \cdot 5 = 2x + 10 \][/tex]

[tex]\[ 2x + 10 + 7 \][/tex]

3. Associative Property of Addition:
We now group [tex]\(10\)[/tex] and [tex]\(7\)[/tex] together using the associative property of addition:
[tex]\[ 2x + (10 + 7) \][/tex]

4. Addition:
We then simplify [tex]\(10 + 7\)[/tex] to get:
[tex]\[ 2x + 17 \][/tex]

Now, let’s examine the properties used in each step:

- Distributive Property: Used to break down [tex]\(2(x+5)\)[/tex] into [tex]\(2x + 10\)[/tex].
- Associative Property of Addition: Used to regroup and combine [tex]\(10 + 7\)[/tex].
- Commutative Property of Addition: Although reordering of terms doesn't explicitly occur, addition can be commutative, such as combining [tex]\(10 + 7\)[/tex] (where order doesn't matter).

The property that is NOT used in this simplification process is:

- Commutative Property of Multiplication: This property states that the order in which you multiply numbers does not change the product (i.e., [tex]\(a \cdot b = b \cdot a\)[/tex]). This property is not employed in any of the steps described above for the given expression.

Thus, the property not used to simplify the expression [tex]\((x+5) \cdot 2 + 7\)[/tex] is the commutative property of multiplication.