Example 3 Find the integer solutions of the equation $205 x+93 y=7$.

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
Explanation: First, write $\frac{205}{93}$ in the form of a continued fraction:
$$
\begin{aligned}
\frac{205}{93} & =2+\frac{19}{93}, \frac{93}{19}=4+\frac{17}{19}, \\
\frac{19}{17} & =1+\frac{2}{17}, \frac{17}{2}=8+\frac{1}{2}, \\
\text { we get } \quad \frac{205}{93} & =2+\frac{1}{4}+\frac{1}{1}+\frac{1}{8}+\frac{1}{2}=[2,4,1,8,2] .
\end{aligned}
$$

Next, find its convergents, thus obtaining Table 3.
Table 3
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline$a_{n}$ & & & 2 & 4 & 1 & 8 & 2 \\
\hline$p_{n}$ & 0 & 1 & 2 & 9 & 11 & 97 & 205 \\
\hline$q_{n}$ & 1 & 0 & 1 & 4 & 5 & 44 & 93 \\
\hline
\end{tabular}

Thus, $205 \times 44 - 93 \times 97 = (-1)^{5} = -1$.
Multiplying both sides of the above equation by -7, we get
$$
205 \times (-44 \times 7) + 93 \times (97 \times 7) = 7 \text{. }
$$

Therefore, $x = -308, y = 679$ is a set of integer solutions to the equation
$$
205 x + 93 y = 7
$$

The general solution is
$$
x = -308 - 93 t, y = 679 + 205 t (t \in \mathbf{Z}) \text{. }
$$

Similarly, the equation $205 x - 93 y = 7$ has the solution
$$
x = -308, y = -679 \text{. }
$$