is 1/x-y the same thing as 1/x - 1/y?

No, 1/(x - y) is not the same as (1/x) - (1/y).

When x = 2 and y = 3 :
1/(x - y) = 1/(2 - 3) = -1
(1/x) - (1/y) = (1/2) - (1/3) = 1/6

Hence, 1/(x - y) ≠ (1/x) - (1/y)

No, it is [1/x] - y

1/x - y ≠ 1/x - 1/y
try x=1, y=2, -1≠1/2

if you meant 1/(x-y)
1/(x-y) ≠ 1/x - 1/y
try x=2, y=1
1≠-1/2

So no.

No.

as typed you have 1 / x - y/1 and 1 / [ x- y ] cannot be rewitten