Given the problem where [tex]\( y \)[/tex] varies directly as [tex]\( x \)[/tex], this relationship can be expressed as:
[tex]\[ y = kx \][/tex]
where [tex]\( k \)[/tex] is the constant of variation.
We are given that [tex]\( y = 48 \)[/tex] when [tex]\( x = 6 \)[/tex]. Using this information, we can determine the value of [tex]\( k \)[/tex]. Plugging the values into the direct variation formula:
[tex]\[ 48 = k \cdot 6 \][/tex]
To solve for [tex]\( k \)[/tex], divide both sides of the equation by 6:
[tex]\[ k = \frac{48}{6} \][/tex]
Now we know the constant of variation [tex]\( k \)[/tex], and we can use it to find the value of [tex]\( y \)[/tex] when [tex]\( x = 2 \)[/tex].
Substituting [tex]\( k \)[/tex] and [tex]\( x = 2 \)[/tex] back into the direct variation formula, we get:
[tex]\[ y = k \cdot 2 \][/tex]
[tex]\[ y = \left( \frac{48}{6} \right) \cdot 2 \][/tex]
Thus, the expression that can be used to find the value of [tex]\( y \)[/tex] when [tex]\( x = 2 \)[/tex] is:
[tex]\[ y = \frac{48}{6}(2) \][/tex]
The correct answer is:
[tex]\[ y = \frac{48}{6}(2) \][/tex]
