Answer
$x = \frac 15$ and $y = \frac 13$
Work Step by Step
If $u = \frac{1}{x}$ and $v = \frac{1}{y}$, the system can be written as:
$u + v = 8$
$3u - 5v = 0$
1. Solve for $u$ in the first equation:
$u + v = 8$
Subtract v from both sides.
$u = 8 - v$
2. Substitute this value into the second equation, and solve for v:
$3u - 5v = 0$
$3(8-v) - 5v = 0$
$24 - 3v - 5v = 0$
$24 - 8v = 0$
Add 8v to both sides.
$24 = 8v$
Divide both sides by 3:
$3 = v$
3. Find the value of $u$:
$u = 8 -v = 8 -3 = 5$
4. Calculate x and y
$\frac 1x = u $
$\frac 1u = x$
$\frac 1 {5} = x$
$\frac 1y = v$
$\frac 1v = y$
$\frac 13 = y$
