The problem states that x varies directly as the product of u and v and inversely as their sum. This means that x is proportional to \(\frac{u \cdot v}{u + v}\). We can write this relationship as an equation: \(x = k \cdot \frac{u \cdot v}{u + v}\), where k is a constant of proportionality.

We are given that x = 3 when u = 3 and v = 1. We can use this information to find the value of k:

\(3 = k \cdot \frac{3 \cdot 1}{3 + 1}\)

\(3 = k \cdot \frac{3}{4}\)

\(k = 4\)

Now that we know the value of k, we can use the equation \(x = k \cdot \frac{u \cdot v}{u + v}\) to find the value of x when u = 3 and v = 3:

\(x = 4 \cdot \frac{3 \cdot 3}{3 + 3}\)

\(x = 4 \cdot \frac{9}{6}\)

\(x = 6\)