Online Mathematica, pros and cons, linear algebra problem

I apologize in advance if this question is irrelevant to this website.

I would like to use Mathematica to solve a system of linear equations with lots of unknowns(729 unknowns), the unknowns are tensor components of curvature tensors arising from a differential geometry problem.

I would like to buy Mathematica for this purpose and I have to decide between buying it online or installing the desktop version on a PC. I m thinking of buying the online version. I have the following questions:

  1. What are the advantages and disadvantages of the desktop version over the online version ? For example, are there mathematical or programming functionalities which are available only on the desktop version and not in the online version ?

  2. I assume that if I buy the online version, then I will get a username and a password to access an online version of mathematica from any computer. (Just like how one can type latex on from an online account using any PC). Is my assumption correct ?

3)Does Mathematica provide a user friendly way for solving linear simultaneous equations with lots of unknowns ? Let me elaborate with an example: Say I want to solve the simultaneous equations $ x=2y+a, y-3x=7x+2$ for $ x,y$ . I would like a software where I can just type: $ x=2y+a, y-3x=7x+2$ and ask the software to solve for $ x,y$ and just give me the solution symbolically in terms of parameter $ a$ instead of me having to rearrange terms so that the equations become $ x-2y=a, y-10x=2$ and then write it in matrix form, then ask it to make a matrix inversion. The difference I am talking about might seem silly in this example but it will not be silly in my original problem where I have 700 unknowns. If this feature exists in Mathematica, it will save me a lot of time.

Thank you,