Solve the following system of equations:

$$
\begin{aligned}
& \frac{1}{\sqrt{1-x^{2}}}+\frac{1}{\sqrt{1-y^{2}}}=\frac{35}{12} \\
& \frac{x}{\sqrt{1-x^{2}}}-\frac{y}{\sqrt{1-y^{2}}}=\frac{7}{12}
\end{aligned}
$$
Due to the square root expressions in the denominator, $|x|<1$. Then our system of equations is:

$$
a+b=\frac{7}{2}, \quad \frac{1}{a}+\frac{1}{b}=\frac{7}{3}
$$

Express $b$ from the first equation: $b=\frac{7}{2}-a$. Substituting this into the second equation:

$$
\frac{1}{a}+\frac{1}{b}=\frac{b+a}{a b}=\frac{\frac{7}{2}}{a\left(\frac{7}{2}-a\right)}=\frac{7}{3}
$$

This results in the following quadratic equation:

$$
a^{2}-\frac{7}{2} a+\frac{3}{2}=0
$$

The roots are $a_{1}=3$ and $a_{2}=\frac{1}{2}$, with the corresponding values of $b$: $b_{1}=\frac{1}{2}$ and $b_{2}=3$.

Substituting these back and expressing the original variables, we get the solutions

$x_{1}=\frac{4}{5}, y_{1}=\frac{3}{5}$ and $x_{2}=-\frac{3}{5}, y_{2}=-\frac{4}{5}$, which satisfy the constraints and the system of equations.