I'm assuming that you mean
[tex]\dfrac{uv}{u+v}[/tex]
because if you meant
[tex]\dfrac{uv}{u}+v[/tex]
then u would simplify and you couldn't make it the subject.
Under this assumption, we start with
[tex]F=\dfrac{uv}{u+v}[/tex]
We multiply both sides by [tex]u+v[/tex]
[tex]F(u+v)=uv[/tex]
We expand the left hand side:
[tex]Fu+Fv=uv[/tex]
We move all terms involving u to the left and all terms not involving u to the right:
[tex]Fu-uv=-Fv[/tex]
We factor u on the left hand side:
[tex]u(F-v)=-Fv[/tex]
We divide both sides by [tex]F-v[/tex]
[tex]u=\dfrac{-Fv}{F-v}[/tex]
