Let's go through the simplification step by step to identify which properties are used:
1. Distributive Property:
The expression [tex]\( 2 \cdot (x+5) + 7x \)[/tex] is simplified by distributing the 2 through the parentheses:
[tex]\[
2 \cdot (x+5) + 7x = (2x + 10) + 7x
\][/tex]
This step uses the distributive property.
2. Commutative Property of Addition:
Next, the expression [tex]\( (2x + 10) + 7x \)[/tex] is reordered to [tex]\( (10 + 2x) + 7x \)[/tex]:
[tex]\[
(2x + 10) + 7x = (10 + 2x) + 7x
\][/tex]
This uses the commutative property of addition.
3. Associative Property of Addition:
Then, [tex]\( (10 + 2x) + 7x \)[/tex] is regrouped to [tex]\( 10 + (2x + 7x) \)[/tex]:
[tex]\[
(10 + 2x) + 7x = 10 + (2x + 7x)
\][/tex]
This step uses the associative property of addition.
4. Simplifying within Parentheses:
Combine the like terms within the parentheses:
[tex]\[
10 + (2x + 7x) = 10 + 9x
\][/tex]
5. Commutative Property of Addition:
Finally, [tex]\( 10 + 9x \)[/tex] is reordered to [tex]\( 9x + 10 \)[/tex]:
[tex]\[
10 + 9x = 9x + 10
\][/tex]
This again uses the commutative property of addition.
From the analysis, we can see that:
- The distributive property is used.
- The commutative property of addition is used.
- The associative property of addition is used.
However, the commutative property of multiplication is not used at any point during the simplification of this expression.
