Example 3. Solve the system of equations
$$
\left\{\begin{array}{l}
\frac{x y}{x+y}=\frac{1}{9}, \\
\frac{y z}{y+z}=\frac{1}{11}, \\
\frac{x z}{x+z}=\frac{1}{12} .
\end{array}\right.
$$
Obviously, $x, y, z$ are all non-zero. Therefore, we can take the reciprocal of both sides of each equation and convert them into partial fractions, obtaining
$$
\left\{\begin{array}{l}
\frac{1}{x}+\frac{1}{y}=9 \\
\frac{1}{y}+\frac{1}{z}=11 \\
\frac{1}{z}+\frac{1}{x}=12
\end{array}\right.
$$
$$
\text { (1) }+ \text { (2) }+ \text { (3) gives }-\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=16 \text {. }
$$

Subtracting (1), (2), and (3) from (4) respectively, we get
$$
x=\frac{1}{5}, \quad y=\frac{1}{4}, \quad z=\frac{1}{7} .
$$