You have supplied only one equation so let’s assume the second expression is equal to a.
From the first equation, e^(xy)=-1/x and from the second equation e^(xy)=(a-1)/y.
So -1/x=(a-1)/y. Cross-multiply: -y=x(a-1) making x=-y/(a-1) for all a except a=1.
For example, if a=2, x=-y. If a=0, x=y.
If the second expression had been ye^(xy)=1 then this is the same as ye^(xy)+1=2 so this makes a=2 and the solution would be x=-y.