Question

How do you end up with 0=0

Answer 1

I don't know what these other people are talking about, this is in fact a system of two equations.

Your first step or two looks correct.
y = x-6

2x - 2(y) = 12
2x - 2(x-6) = 12

Distributing the -2,

2x - 2x + 12 = 12

Combining like-terms,

12 = 12

The x's completely fell out of the equation,
and we end up with a true result.
That means that every x satisfies the system.

The system is called "coincident".
Yes, they are essentially the same line.
That doesn't mean it's not a system though.

If you want to get 0 = 0,
you would subtract 12 from both sides as a final step.

Answer 2

Ok so first plug in y for in the equation 2x-2y=12
Therefore 2x - 2(x -6)=12
Then distribute the two to the x and the -6 
2x-2x+12=12
The 2x's cancel leaving you with 
12=12
From there you could subtract the 12 from both sides to get 0=0, I assume