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?
Well of course we must always be concerned if the expression is not defined somewhere.
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?
In that case you must tell us what definition allows for a line to have a dis-continuity.
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: $(-\infty,2)\cup(2,\infty)$