Answer
$\left\{\left(\dfrac{1}{5},\dfrac{1}{3}\right)\right\}$
Work Step by Step
We are given the system of equations:
$\begin{cases}
\dfrac{1}{x}+\dfrac{1}{y}=8\\
\dfrac{3}{x}-\dfrac{5}{y}=0
\end{cases}$
Note:
$u=\dfrac{1}{x}$
$v=\dfrac{1}{y}$
Rewrite the system in terms of $u$ and $v$:
$\begin{cases}
u+v=8\\
3u-5v=0
\end{cases}$
Use the elimination method. Multiply the first equation by 5 and add it to the second equation to eliminate $v$ and determine $u$:
$\begin{cases}
5u+5v=5(8)\\
3u-5v=0
\end{cases}$
$5u+5v+3u-5v=40+0$
$8u=40$
$u=5$
Determine $v$ using the first equation:
$u+v=8$
$5+v=8$
$v=3$
Determine $x$ and $y$:
$\dfrac{1}{x}=5\Rightarrow x=\dfrac{1}{5}$
$\dfrac{1}{y}=3\Rightarrow y=\dfrac{1}{3}$
The solution set of the system is:
$\left\{\left(\dfrac{1}{5},\dfrac{1}{3}\right)\right\}$
