Starting with Eq. (7.9), derive Eqs. (7.10) and (7.11). [begin{align*} & e_{2}-e_{1}=frac{p_{1}+p_{2}}{2}left(frac{1}{ho_{1}}-frac{1}{ho_{2}} ight) & e_{2}-e_{1}=frac{p_{1}+p_{2}}{2}left(v_{1}-v_{2} ight)

Starting with Eq. (7.9), derive Eqs. (7.10) and (7.11).

\[\begin{align*}
& e_{2}-e_{1}=\frac{p_{1}+p_{2}}{2}\left(\frac{1}{ho_{1}}-\frac{1}{ho_{2}}\right) \\
& e_{2}-e_{1}=\frac{p_{1}+p_{2}}{2}\left(v_{1}-v_{2}\right) \tag{7.9}
\end{align*}\]

\[\begin{equation*}
\frac{T_{2}}{T_{1}}=\frac{p_{2}}{p_{1}}\left(\frac{\frac{\gamma+1}{\gamma-1}+\frac{p_{2}}{p_{1}}}{1+\frac{\gamma+1}{\gamma-1} \frac{p_{2}}{p_{1}}}\right) \tag{7.10}
\end{equation*}\]

\[\begin{equation*}
\frac{ho_{2}}{ho_{1}}=\frac{1+\frac{\gamma+1}{\gamma-1}\left(\frac{p_{2}}{p_{1}}\right)}{\frac{\gamma+1}{\gamma-1}+\frac{p_{2}}{p_{1}}} \tag{7.11}
\end{equation*}\]