Example 1 Find all solutions to $10 x_{1}-7 x_{2}=17$.

保留源文本的换行和格式，直接输出翻译结果如下：

Example 1 Find all solutions to $10 x_{1}-7 x_{2}=17$.

Keep the line breaks and format of the original text, and output the translation result directly as above.
It is easy to see that $(10,7)=1$, so the equation has a solution. By inspection, we can find that $x_{1,0}=$ $1, x_{2,0}=-1$ is a particular solution. Therefore, all solutions are
$$x_{1}=1-7 t, \quad x_{2}=-1-10 t, \quad t=0, \pm 1, \pm 2, \cdots$$