The given equation can be rewritten as $$((y-10)x =1+10y\implies x = \frac{10y+1}{y-10}=\frac{10(y-10)+101}{y-10}=10+\frac{101}{y-10}$$

Because $x$ is an integer, $y-10$ must divide $101$, implying $y-10 =\pm 1, \pm 101$. Hence: $$y= 9, 11, -91, 111\implies (x,y) = (9, -91), (-91, 9), (111, 11), (11, 111)$$