https://file.aiquickdraw.com/mathbot/file/172993994337088lc7jgz.jpg

Let's solve the problem step-by-step.

### Given Information
- \( y \) varies inversely with \( x \).
- When \( x = 3 \), \( y = 8 \).

### Step 1: Writing the Inverse Variation Equation
Since \( y \) varies inversely with \( x \), we have:
\[
y = \frac{k}{x}
\]
where \( k \) is the constant of variation.

To find \( k \), substitute \( x = 3 \) and \( y = 8 \):
\[
8 = \frac{k}{3}
\]
Multiplying both sides by 3:
\[
k = 24
\]

Thus, the inverse variation equation is:
\[
y = \frac{24}{x}
\]

### Step 2: Finding \( y \) when \( x = 6 \)
Now, substitute \( x = 6 \) into the equation:
\[
y = \frac{24}{6} = 4
\]

### Summary
- **(a)** The equation that relates \( x \) and \( y \) is \( y = \frac{24}{x} \).
- **(b)** When \( x = 6 \), \( y = 4 \).

Would you like a further breakdown of these steps, or have any other questions?

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#### Follow-up Questions:
1. What is an inverse variation, and how does it differ from direct variation?
2. How would the equation change if \( y \) varied directly with \( x \) instead?
3. How can we confirm that a relation is indeed an inverse variation?
4. What happens to \( y \) as \( x \) increases in an inverse variation?
5. How can we use the inverse variation concept in real-world applications?

#### Tip:
In inverse variation, as one variable increases, the other decreases, keeping the product constant. This is essential in understanding real-world relationships like speed and travel time.

Suppose that y varies inversely with x, and y = 8 when x = 3. (a) Write an inverse variation equation that relates x and y. (b) Find y when x = 6.

Learn how to solve an inverse variation problem where y varies inversely with x, given y = 8 when x = 3. Find the equation relating x and y, and determine y when x = 6. Suitable for students in Grades 9-10.

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