I'm trying to find out if the system have a unique solution, no solution or infinitely many solutions

Alright so this is what you do.
You need to isolate one variable so that it is y = or x =
Lets go with y = for the first one

-x +3y = -6
x - 3y = 6

Isolate y = by adding x for the first one and then subtracting x from the second one.
3y = -6 + x
-3y = 6 - x

You would then divide the second one by -3 and divide by 3 in the first one.

y = (-6 +x) / 3
y = (6 -x) / -3
Since this is what y = you can put them together.

-6+x) / 3 = 6-x)/-3 (I put one parenthesis for convenience)

This equation is now set and you aren't done yet.
You can take out -1 from each side. This makes it no solution since you must always multiply by -1.
(6+x) / 3 = -1( (6+x) / 3)

Try the second one on your own!