Here's a possible solution for x and y.
a/b=(xy-1)/(x+y), so one solution gives us two equations:
(1) a=xy-1
(2) b=x+y
y=b-x from (2). Substitute in (1):
a=x(b-x)-1, a=bx-x2-1, x2-bx+a+1=0,
x=(b±√(b2-4a-4))/2 and y=b-x=(b∓√(b2-4a-4))/2, that is:
(x,y)=((b+√(b2-4a-4))/2,(b-√(b2-4a-4))/2) or ((b-√(b2-4a-4))/2,(b+√(b2-4a-4))/2).
This solution finds x and y in terms of a and b only.