You can choose any value for [tex] x [/tex], while you must choose [tex] y=0 [/tex].
Intuitively, this makes sense since [tex] x+y=x-y [/tex] means that you want to start from number [tex] x [/tex], and you want to obtain the same result by moving left or right by [tex] y [/tex] units. This can't happen, unless you don't move at all, i.e. [tex] y=0 [/tex].
If you want to be more formal, you have
[tex] x+y = x-y \implies x-x = -y-y \implies -2y=0 \iff y = 0 [/tex]
while [tex] x [/tex] was canceled, so it can assume any value.
