1/x+1/y=6??? xy!=0...
a) x=y, if x=y=1/3 possible, otherwise not possible. Insufficient
b) (x+y)=2/3. eqn: (x+y)/xy=6 or xy=1/9. Can be possible or not possible.
(x=1/6,y=1/2 -> not possible; x=y=1/3 possible)
Insufficient

a&b) x+y = 2/3 and x=y, thus x=y=1/3. Sufficient
IMO C

2) x+y= 2/3
why is B not sufficient? can't u just flip it?

NO, you definitely cannot do anything with the sum of two fractions other than adding them together.
This is a pretty common mistake, don't make it.

All you can do with

1/x + 1/y is put them over a common denominator: y/xy + x/xy = (x+y)/(xy).

From there, you can go through the statements. As cans showed, you need 2 different pieces of information to find the value of an expression like (x+y)/(xy), so the answer is C.