Ok, I am after your views on this?
Is (y-2)/x = 3 the equation of a line?
Do we need to worry that it is not defined for x=0 ?
Obviously, it can be re-arranged into a standard form, but in this form what do we conclude?
Look at this plot.
There is an unexpected 'whole' in the plot. The graph is not continuous.
Then is that a line? Or even a linear relation? What do YOU say?
I do not know of one. Whether in geometry or analysis lines are continuous. They have the property that a point is between two points on a line then that point is also on the line. In the case you present, I would say that we have the union of two half-lines: $(-\infty,2)\cup(2,\infty)$