To solve this problem, we need to use the concept of direct variation. When we say that [tex]\(y\)[/tex] varies directly as [tex]\(x\)[/tex], we mean that there is a constant [tex]\(k\)[/tex] such that:
[tex]\[ y = kx \][/tex]
Given the information, [tex]\( y = 48 \)[/tex] when [tex]\( x = 6 \)[/tex]. We can use this to find the constant [tex]\(k\)[/tex]:
[tex]\[
48 = k \cdot 6 \implies k = \frac{48}{6}
\][/tex]
Now, we need to find the value of [tex]\(y\)[/tex] when [tex]\( x = 23 \)[/tex]. We use the expression of direct variation again, substituting [tex]\(k\)[/tex] and [tex]\( x = 23 \)[/tex] into our equation:
[tex]\[
y = k \cdot 23
\][/tex]
Since we already determined that [tex]\( k = \frac{48}{6} \)[/tex], we can substitute this into the equation:
[tex]\[
y = \left( \frac{48}{6} \right) \cdot 23
\][/tex]
Thus, the expression that shows how to find [tex]\( y \)[/tex] when [tex]\( x = 23 \)[/tex] is:
[tex]\[
y = \frac{48}{6} \cdot 23
\][/tex]
From the options given, the closest form of this expression is:
[tex]\[
y = \frac{48}{6}(2)
\][/tex]
However, be cautious with the options given because there seems to be a discrepancy in variables and choice logic. The correct theoretical framework is shown above, yet none of the options perfectly outline the situation for [tex]\( x = 23 \)[/tex] due to a potential misprint in the answer options.
