To determine the value of [tex]\( y \)[/tex] when [tex]\( x = 9 \)[/tex] given that [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are directly proportional and that [tex]\( y = 2 \)[/tex] when [tex]\( x = 3 \)[/tex], we follow these steps:
1. Understand Direct Proportionality: When two variables are directly proportional, their relationship can be written as:
[tex]\[
y = kx
\][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.
2. Determine the Constant of Proportionality: Using the given values [tex]\( x = 3 \)[/tex] and [tex]\( y = 2 \)[/tex], we can find [tex]\( k \)[/tex]:
[tex]\[
2 = k \cdot 3
\][/tex]
Solving for [tex]\( k \)[/tex] gives:
[tex]\[
k = \frac{2}{3}
\][/tex]
3. Use the Proportionality Constant: Next, we use this constant [tex]\( k \)[/tex] to find the new value of [tex]\( y \)[/tex] when [tex]\( x = 9 \)[/tex].
4. Calculate the New Value of [tex]\( y \)[/tex]:
[tex]\[
y = \left(\frac{2}{3}\right) \cdot 9
\][/tex]
Simplify the multiplication:
[tex]\[
y = \frac{2 \cdot 9}{3} = \frac{18}{3} = 6
\][/tex]
Therefore, when [tex]\( x = 9 \)[/tex], the value of [tex]\( y \)[/tex] is [tex]\( 6 \)[/tex]. So the correct answer is:
[tex]\[
\boxed{6}
\][/tex]
