jmc

First, we find $$\frac{1}{x}=\frac{1}{2}\left(\left(\frac{1}{x}+\frac{1}{y}\right)+\left(\frac{1}{x}-\frac{1}{y}\right)\right)=\frac{1}{2}(3+(-7))=-2.$$ Therefore, $x=-\frac{1}{2}$. Similarly, we find $$\frac{1}{y}=\frac{1}{2}\left(\left(\frac{1}{x}+\frac{1}{y}\right)-\left(\frac{1}{x}-\frac{1}{y}\right)\right)=\frac{1}{2}(3-(-7))=5.$$ Therefore, $y=\frac{1}{5}$. Our desired sum is $$x+y=-\frac{1}{2}+\frac{1}{5}=\boxed{-\frac{3}{10}}.$$