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.
