#### Answer

$z=-15$

#### Work Step by Step

$\dfrac{10}{z}=\dfrac{5}{z}-\dfrac{1}{3} \\
\dfrac{10}{z}=\dfrac{15-z}{3z} \\
30z=z(15-z) \\
30z=15z-z^2 \\
z^2+30z-15z=0 \\
z^2+15z=0 \\
z(z+15)=0 \\
$
But the solution is only $z=-15$, because if $z=0$ the original equation would be undefined.