763. Find all solutions of the equation

$$
5 x-7 y=3
$$

in integers $x$ and $y$.
$\triangle$ Express from the equation the unknown whose coefficient is smaller in absolute value - in this case $x$:

$$
5 x=7 y+3, \quad x=\frac{7 y+3}{5}
$$

In the numerator of the obtained fraction, $7 y$ can be split into two terms, one of which is divisible by 5 for any integer $y$, and the coefficient of the other is less than 5: $7 y=5 y+2 y$. Then divide the numerator of the fraction term by term by the denominator:

$$
x=\frac{5 y+(2 y+3)}{5}=y+\frac{2 y+3}{5}
$$

The fraction $\frac{2 y+3}{5}$ must be equal to an integer. Let

$$
\frac{2 y+3}{5}=z
$$

where $z$ is an integer. Then

$$
2 y+3=5 z
$$

We have obtained a new first-degree equation with two unknowns, but with smaller coefficients in absolute value.

From the last equation, express the unknown whose coefficient is smaller in absolute value, in this case $y$, and perform similar transformations:

$$
2 y=5 z-3, \quad y=\frac{5 z-3}{2}=\frac{6 z-z-3}{2}=3 z-\frac{z+3}{2}
$$

For how long should such a procedure be continued? In general, until an equation is obtained in which the coefficient of one of the unknowns is 1 or -1. In this case, it is already over. The fraction $\frac{z+3}{2}$ must be an integer. Denote it by $t$:

$$
\frac{z+3}{2}=t, \quad z+3=2 t, \quad z=2 t-3
$$

where, recall, $t$ is an integer.

Now express $y$ and $x$ in terms of $t$:

$$
\begin{gathered}
y=3 z-\frac{z+3}{2}=3(2 t-3)-t=5 t-9 \\
x=y+\frac{2 y+3}{5}=y+z=(5 t-9)+(2 t-3)=7 t-12
\end{gathered}
$$

We have obtained the formulas

$$
x=7 t-12, \quad y=5 t-9
$$

Here $t$ is an integer. But is $t$ any integer? To answer this question, substitute the expressions for $x$ and $y$ into the left side of the original equation:

$$
5 x-7 y=5(7 t-12)-7(5 t-9)=35 t-60-35 t+63=3
$$

Therefore, these formulas, where $t$ is any integer, give the set of all solutions of the equation in integers.

By assigning $t$, for example, the values 0, 1, and 2, we obtain particular solutions of the equation: $(-12, -9), (-5, -4), (2, 1)$.

Answer: $x=7 t-12, y=5 t-9$, where $t$ is any integer.