For example, use points on a plane to represent the numerical solutions of the equation
$$
3 x+4 y=1 \text{.}
$$
Solving: By observation, an integer solution to the indeterminate equation $3x + 4y = 1$ is $\left\{\begin{array}{l}x_{0}=-1 \\ y_{0}=1\end{array}\right.$, thus all integer solutions are $\left\{\begin{array}{l}x=-1+4t, \\ y=1-3t .\end{array} \quad(t \in \mathbb{Z})\right.$

The graph is as follows:
\begin{tabular}{r|r|r|r|r|r|r}
\hline$t$ & $\cdots$ & -1 & 0 & 1 & 2 & $\cdots$ \\
\hline$x$ & $\cdots$ & -5 & -1 & 3 & 7 & $\cdots$ \\
\hline$y$ & $\cdots$ & 4 & 1 & -2 & -5 & $\cdots$ \\
\hline
\end{tabular}

It can be seen that the set of all integer solutions to the indeterminate equation $\mathrm{ax} + \mathrm{by} = \mathrm{c}$ (where $a$, $b$, and $c$ are integers, and $ab \neq 0$), geometrically, is the set of all integer points (i.e., points with integer coordinates) on the line $\mathrm{ax} + \mathrm{by} = \mathrm{c}$. This set is $M=\left\{(x, y):\left\{\begin{array}{l}x=x_{0}+bt \\ y=y_{0}-at\end{array} \quad t \in \mathbb{Z}\right\}\right.$. These integer points are uniformly distributed on the line, with the absolute difference in the x-coordinates of adjacent points $|\Delta x| = |b|$, the absolute difference in the y-coordinates $|\Delta y| = |a|$, and the distance between adjacent points $d = \sqrt{a^2 + b^2}$.

In fact, because $M=\{(x, y) : \left\{\begin{array}{l}x=x_{0}+bt \\ y=y_{0}-at,\end{array} \quad t \in \mathbb{Z}\right\}$, if $t$ takes two consecutive integer values $k$ and $k+1$, we get the adjacent points:
$$
\begin{array}{l} 
M_{1}\left(x_{0}+bk, y_{0}-ak\right), \\
M_{2}\left(x_{0}+b(k+1), y_{0}-a(k+1)\right) \\
\text{Clearly, }|\Delta x|=\mid x_{0}+b(k+1) \\
-\left(x_{0}+bk\right)|=| b |, \\
|\Delta y|=\mid y_{0}-a(k+1) \\
-\left(y_{0}-ak\right)|=|-a|=| a | \\
d=\left|M_{1} M_{2}\right| \\
=\sqrt{(\Delta x)^{2}+(\Delta y)^{2}} \\
=\sqrt{a^{2}+b^{2}} .
\end{array}
$$