Find the minimum distance from any lattice point to the line $y = \frac{5}{3} x + \frac{12}{7}.$
(5/3)x + 12/7 = y
5/3 is a terminating decimal while 12/7 is not......so it will not pass through any lattice points
I also believe that there is no minimum distance to any lattice point.....
The minimum distances will approach zero but they never converge to any particular value