To determine which expression is equivalent to [tex]\(2 + 3n + 2 + 9n\)[/tex], we need to combine and simplify the terms in the given expression.
1. Combine constants:
[tex]\[
2 + 2 = 4
\][/tex]
2. Combine terms containing [tex]\(n\)[/tex]:
[tex]\[
3n + 9n = 12n
\][/tex]
3. Form the simplified expression:
[tex]\[
2 + 3n + 2 + 9n = 4 + 12n
\][/tex]
Now, we need to check which of the given options is equivalent to [tex]\(4 + 12n\)[/tex]:
A. [tex]\(16n\)[/tex]
This expression does not match [tex]\(4 + 12n\)[/tex] because it lacks the constant term [tex]\(4\)[/tex] and has a different coefficient for [tex]\(n\)[/tex].
B. [tex]\(3n + 8\)[/tex]
This expression does not match [tex]\(4 + 12n\)[/tex] because the coefficient of [tex]\(n\)[/tex] is 3 instead of 12, and the constant term is 8 instead of 4.
C. [tex]\(4(3n + 1)\)[/tex]
Let's simplify this expression:
[tex]\[
4(3n + 1) = 4 \cdot 3n + 4 \cdot 1 = 12n + 4
\][/tex]
This matches [tex]\(4 + 12n\)[/tex] exactly.
D. [tex]\(4(3n + 4)\)[/tex]
Let's simplify this expression:
[tex]\[
4(3n + 4) = 4 \cdot 3n + 4 \cdot 4 = 12n + 16
\][/tex]
This does not match [tex]\(4 + 12n\)[/tex] because the constant term is [tex]\(16\)[/tex] instead of [tex]\(4\)[/tex].
Anya's Choice:
Anya chose option D, [tex]\(4(3n + 4)\)[/tex], as the correct answer. However, this is incorrect because [tex]\(4(3n + 4)\)[/tex] simplifies to [tex]\(12n + 16\)[/tex], not [tex]\(4 + 12n\)[/tex]. The correct equivalent expression is option C, [tex]\(4(3n + 1)\)[/tex]. This error might occur due to a miscalculation or misunderstanding the distribution property. The accurate simplification shows that only option C precisely matches the simplified form [tex]\(4 + 12n\)[/tex].
