Certainly! The solution can be broken down step-by-step and identified with the appropriate mathematical property used at each stage:
1. Starting with the expression:
[tex]\[
6 + 5(x + 7)
\][/tex]
2. From [tex]\(6 + 5(x + 7)\)[/tex] to [tex]\(6 + (5x + 35)\)[/tex]:
This step involves the Distributive Property, which states that [tex]\(a(b + c) = ab + ac\)[/tex]. Here, [tex]\(5\)[/tex] distributes over [tex]\(x + 7\)[/tex]:
[tex]\[
6 + 5(x + 7) \rightarrow 6 + (5x + 35)
\][/tex]
3. From [tex]\(6 + (5x + 35)\)[/tex] to [tex]\(6 + (35 + 5x)\)[/tex]:
This step uses the Commutative Property of Addition, which states that [tex]\(a + b = b + a\)[/tex]. Here, [tex]\(5x + 35\)[/tex] is rearranged as [tex]\(35 + 5x\)[/tex]:
[tex]\[
6 + (5x + 35) \rightarrow 6 + (35 + 5x)
\][/tex]
4. From [tex]\(6 + (35 + 5x)\)[/tex] to [tex]\((6 + 35) + 5x\)[/tex]:
This step applies the Associative Property of Addition, which states that [tex]\((a + b) + c = a + (b + c)\)[/tex]. Here, the grouping of terms changes:
[tex]\[
6 + (35 + 5x) \rightarrow (6 + 35) + 5x
\][/tex]
5. From [tex]\((6 + 35) + 5x\)[/tex] to [tex]\(41 + 5x\)[/tex]:
This step involves Simplification, where the arithmetic within the parentheses is solved:
[tex]\[
(6 + 35) + 5x \rightarrow 41 + 5x
\][/tex]
6. From [tex]\(41 + 5x\)[/tex] to [tex]\(5x + 41\)[/tex]:
This step again involves the Commutative Property of Addition, which allows the order of addition to be switched:
[tex]\[
41 + 5x \rightarrow 5x + 41
\][/tex]
Putting it all together, the properties used are:
[tex]\[
\begin{aligned}
6+5 & (x+7) \\
& =6+(5 x+35) \quad \text{Distributive Property} \\
& =6+(35+5 x) \quad \text{Commutative Property of Addition} \\
& =(6+35)+5 x \quad \text{Associative Property of Addition} \\
& =41+5 x \quad \text{Simplification} \\
& =5 x+41 \quad \text{Commutative Property of Addition}
\end{aligned}
\][/tex]
Therefore, the correct properties for each step are:
[tex]\[
\begin{aligned}
6 + 5 & (x + 7) \\
& = 6 + (5x + 35) \quad \text{Distributive Property} \\
& = 6 + (35 + 5x) \quad \text{Commutative Property of Addition} \\
& = (6 + 35) + 5x \quad \text{Associative Property of Addition} \\
& = 41 + 5x \quad \text{Simplification} \\
& = 5x + 41 \quad \text{Commutative Property of Addition}
\end{aligned}
\][/tex]
Now placing the respective properties next to the given steps:
1. Distributive Property
2. Commutative Property of Addition
3. Associative Property of Addition
4. Simplification
5. Commutative Property of Addition
