The problem states that the variables $x$ and $y$ are inversely proportional. This means that their product is constant, i.e.,

$x \cdot y = k$

where $k$ is a constant.

We are given the information:

• $y = 2$ when $x = 3$.

Step 1: Find the constant $k$

Using the values provided:

$3 \cdot 2 = k$ $k = 6$

Step 2: Find the value of $y$ when $x = 9$

Now that we know $k = 6$, we can use the equation $x \cdot y = 6$ to find $y$ when $x = 9$:

$9 \cdot y = 6$ $y = \frac{6}{9} = \frac{2}{3}$

Final Answer:

When $x = 9$, $y = \frac{2}{3}$.

Would you like more details or have any questions on this? Here are some follow-up questions for you:

1. How does the concept of inverse proportionality differ from direct proportionality?
2. Can you derive a general formula for inverse proportionality?
3. What happens to the value of $y$ as $x$ increases?
4. How would you solve this problem if the relationship between $x$ and $y$ was directly proportional instead of inversely?
5. Can you think of real-world scenarios where inverse proportionality applies?

Tip: When dealing with inverse proportionality, always remember that the product of the two variables remains constant.