\(\hept{\begin{cases}\frac{1}{x}=a\\\frac{1}{y}=b\end{cases}}\)
\(\hept{\begin{cases}a+b=5\\a-b=1\end{cases}\Rightarrow\hept{\begin{cases}a=3\\b=2\end{cases}\Rightarrow ab=6}}\)
\(\frac{1}{x}+\frac{1}{y}=\frac{x+y}{xy}=a.b.\left(x+y\right)=6\left(x+y\right)=5\Rightarrow\left(x+y\right)=\frac{5}{6}\)