Your three solutions are correct.
x + y = 6 (multiply this by 2)
we get
2x + 2y = 12
Therefore the two equations are exactly the same straight line graphically.
So this pair of equations have infinite solutions.
(x,y) =(1,5) , (6,0) , (7,-1) , (4,2) , (-3,9)..........
Hope this helps you understand now.