Solve eta=frac{{p}_{0}{V}_{0}}{5{p}_{0}{V}_{0}+3/2{p}_{0}{v}_{0}}

$$\eta = \frac{ { p }_{ 0 } { V }_{ 0 } }{ 5 { p }_{ 0 } { V }_{ 0 } + \frac{ 3 }{ 2 } { p }_{ 0 } { v }_{ 0 } }$$

Solve for V_0

$$\left\{\begin{matrix}V_{0}=\frac{3v_{0}\eta }{2\left(1-5\eta \right)}\text{, }&v_{0}\neq 0\text{ and }\eta \neq \frac{1}{5}\text{ and }p_{0}\neq 0\\V_{0}\neq 0\text{, }&v_{0}=0\text{ and }\eta =\frac{1}{5}\text{ and }p_{0}\neq 0\end{matrix}\right.$$

Solve for p_0

$$p_{0}\neq 0$$

$$\left(V_{0}=\frac{3v_{0}\eta }{2\left(1-5\eta \right)}\text{ and }v_{0}\neq 0\text{ and }\eta \neq \frac{1}{5}\right)\text{ or }\left(v_{0}=0\text{ and }\eta =\frac{1}{5}\text{ and }V_{0}\neq 0\right)$$