$4 \cdot 121$ Solve the system of equations
$$
\left\{\begin{array}{l}
2^{\lg x}+3^{\lg y}=5, \\
2^{\lg x} \cdot 3^{\lg y}=4 .
\end{array}\right.
$$
[Sol] Let $2^{\lg x}, 3^{\lg y}$ be the roots of the equation $a^{2}-5 a+4=0$, then we have $\left\{\begin{array}{l}2^{\lg x}=4, \\ 3^{\lg y}=1 ;\end{array} \quad\right.$ or $\quad\left\{\begin{array}{l}2^{\lg x}=1, \\ 3^{\lg y}=4 .\end{array}\right.$

Therefore, $\left\{\begin{array}{l}x_{1}=100, \\ y_{1}=1 ;\end{array} \quad\left\{\begin{array}{l}x_{2}=1, \\ y_{2}=10\end{array} \log _{3}^{4}\right.\right.$.
Upon verification, they are all solutions to the original system of equations.