Question 3 Given $x, y \in(0,1)$, and $3x+7y$, $5x+y$ are both integers. Then there are $\qquad$ pairs of $(x, y)$.

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【Analysis】Let $\left\{\begin{array}{l}3 x+7 y=u, \\ 5 x+y=v .\end{array}\right.$
We can first determine the integer values of $u$ and $v$ based on $x, y \in(0,1)$.
Solution Let $\left\{\begin{array}{l}3 x+7 y=u, \\ 5 x+y=v,\end{array}\right.$ where the possible values of $u$ and $v$ are $u \in\{1,2, \cdots, 9\}, v \in\{1,2, \cdots, 5\}$.
Thus, $\left\{\begin{array}{l}x=\frac{7 v-u}{32}, \\ y=\frac{5 u-3 v}{32} .\end{array}\right.$
Since $x, y \in(0,1)$, we have
$$
\left\{\begin{array}{l}
0<7 v-u<32, \\
0<5 u-3 v<32 .
\end{array}\right.
$$

Upon verification, when $v=1$, $u=1,2,3,4,5,6$;
when $v=2$, $u=2,3,4,5,6,7$;
when $v=3$, $u=2,3,4,5,6,7,8$;
when $v=4$, $u=3,4,5,6,7,8$;
when $v=5$, $u=4,5,6,7,8,9$.
In summary, there are 31 pairs of $(u, v)$ that satisfy the conditions, which means there are 31 pairs of $(x, y)$ that satisfy the conditions.