It is meanwhile routine on this platform that questions come up on the use of, or problems (i.e. "unexpected behavior") with, Subscript. Sanders description of those as a "typographical construct" is just perfect!

Because in the beginning I had problems myself in proper understanding, I would like to give a little explanation:

If you have a variable (say a) and you make an assignment, that means that the variable gets an "ownvalue":

a = 3;
OwnValues[a]
(*  Out:  {HoldPattern[a]\[RuleDelayed]3}  *)

Whereas when you make the assignments:

there are no OwnValues at all! The point rather is that in a way you changed the properties of Subscript:

This is pretty much the same as if you write:

f[1] = 3;

There is no problem in doing this, but one would not want to call f[1] a "variable".

Not really a problem, more of a misconception. One should see Subscript merely as a typographical construct, nothing else. If one adheres to those confines generally there is no `problem'...

If you use your variables for real computations (as opposed to math typesetting) the I would consider labeling variables differently, as @Bill Simpson recommended too. The easiest:

Array[a, {3, 3}]

{{a[1, 1], a[1, 2], a[1, 3]}, {a[2, 1], a[2, 2], a[2, 3]}, {a[3, 1], a[3, 2], a[3, 3]}}

then an example of computation is:

Solve[a[1, 2]^3 - a[1, 2] + 1 == 0, a[1, 2]];
TraditionalForm[Column[%]]

I think the main problem is that Subscript[x,1] or so does not have Symbol as a Head, while a normal variable has, so it is not recognised and breaks everything... I think changing it will be (very) bad idea, as it will break everything. But introducing a new function IndexedVariable or so could work....

Thanks, will look into it. Didnt know there would be such problems with subscripts.

Thank you so much for answering so quickly. Your answer was the one I needed!