Question

I need to solve the equation for z

Answer 1

I think this is how you do it...

Answer 2

to the risk of sounding redundant.

well, let's bear in mind that the LCD for all three fractions will just be "xyz", so then let's multiply both sides by that LCD in order to do away with the denominators and then solve for "z".

$\bf \cfrac{2}{x}=\cfrac{3}{y}+\cfrac{1}{z}\implies \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{xyz}}{xyz\left( \cfrac{2}{x} \right)=xyz\left( \cfrac{3}{y}+\cfrac{1}{z} \right)}\implies 2yz=3xz+xy \\\\\\ 2yz-3xz=xy\implies \stackrel{\textit{common factor}}{z(2y-3x)}=xy\implies z=\cfrac{xy}{2y-3x}$