The left side of this equation can be reduced to: (4-i)/(2+i)=(4-i)(2-i)/((2+i)(2-i))=(8-6i-1)/5=(7/5)-(6i/5). This has a real and an imaginary part, but the right side has only a real part, assuming x and y are real. There cannot be a solution unless the right side has an imaginary component.
If the right side is y-i/x then y=7/5 and -i/x=6i/5 and x=-5/6.
If the right side is y/x-i/x, y/x=7/5 and, putting x=-5/6, y=-7/5*5/6=-7/6.
Assume x and y are complex: x=a+ib, y=c+id where a, b, c, d are real and rational.
7/5-6i/5=c+id+1/(a+ib)=c+id+a-ib/(a^2+b^2); a/(a^2+b^2)+c=7/5; d-b/(a^2+b^2)=-6/5; c+d+(a-b)/(a^2+b^2)=1/5.
If a=1 and b=2, 1/5+c=7/5, c=6/5; d-2/5=-6/5, d=-4/5.
x=1+2i; y=(1/5)(6-4i).
Alternatively: (a+c)/(a^2+b^2)=7/5; (d-b)/(a^2+b^2)=-6/5; (a+c)/(d-b)=-7/6.
If a=1 and b=2: (1+c)=7, c=6; d-2=-6, d=-4.q
x=1+2i; y=6-4i.