Four, (10 points) Find the number of integer pairs $(x, y)$ that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
2 x \geqslant 3 y, \\
3 x \geqslant 4 y, \\
5 x-7 y \leqslant 20
\end{array}\right.
$$
Four, make the variable substitution $\left\{\begin{array}{l}u=2 x-3 y, \\ v=3 x-4 y .\end{array}\right.$
Obviously, the integer pairs $(x, y)$ correspond to integer pairs $(u, v)$, and vice versa.
In the new coordinate system, the inequalities have the form
$$
\left\{\begin{array}{l}
u \geqslant 0, \\
v \geqslant 0, \\
u+v \leqslant 20 .
\end{array}\right.
$$

It is only necessary to find all integer pairs $(u, v)$ that satisfy the system of inequalities.

All integer pairs $(u, v)$ that satisfy the conditions correspond one-to-one with all integer points (points with integer coordinates) inside and on the sides of $\triangle A O B$ in the Cartesian coordinate system $u O v$.

As shown in Figure 7, there are 21 integer points on the hypotenuse $A B$ of $\triangle A B O$, and $\triangle A B O$ and $\triangle A B C$ form a square $A O B C$ with a side length of 20, which contains a total of 441 integer points. Removing the points on the diagonal $A B$ from the square, we get that the "half" of the square contains 210 integer points, i.e., the number of integer points in $\triangle A B O$ is $21+210=231$.