System

First up, let's isolate x for $\frac{xy}{x+y}=2:\\ $

We get $x=\frac{2y}{y-2}$

Subbing this in, we get 

$\frac{\frac{2y}{y-2}z}{\frac{2y}{y-2}+z}=2\\ \frac{yz}{y+z}=2$

$\frac{2yz}{2y+z\left(y-2\right)}=2\\ \frac{yz}{y+z}=2$

Now, we isolate y. 

$\frac{2yz}{2y+z\left(y-2\right)}=2 \\ y=z$

Subsituting in y=z, we get

$\frac{zz}{z+z}=2$

$z=4$

We get z=4 as our answer. 

Thanks!:)