Task 2. Determine all pairs of integers $x$ and $y$ for which

$$
x y - 7 x - y = 3
$$
Solution. The equation $x y-7 x-y=3$ can be written as

$$
x y-y=3+7 x \text{, i.e., } y(x-1)=7 x+3
$$

For $x=1$, we get the equation $y \cdot 0=10$ which has no solutions. For $x \neq 1$, we have $y=\frac{7 x+3}{x-1}=7+\frac{10}{x-1}$. So $y$ is an integer if $\frac{10}{x-1}$ is an integer, which means $x-1 \in\{1,-1,2,-2,5,-5,10,-10\}$, so $x \in\{2,0,3,-1,-4,6,11,-9\}$. Therefore, the integer solutions of the equation are $(2,17),(0,-3),(3,12),(-1,2),(6,9),(-4,5),(11,8)$, $(-9,6)$.