I am working through the derivation of an adiabatic process of an ideal gas $pV^{\gamma}$ and I can't see how to go from one step to the next. Here is my derivation so far which I understand:
$$dE=dQ+dW$$ $$dW=-pdV$$ $$dQ=0$$ $$dE=C_VdT$$
therefore
$$C_VdT=-pdV$$
differentiate the ideal gas equation $pV=Nk_BT$
$$pdV+Vdp=Nk_BdT$$
rearrange for $dT$ and substitute into the 1st law:
$$\frac{C_V}{Nk_B}(pdV+Vdp)=-pdV$$.
The next part is what I am stuck with I can't see how the next line works specifically how to go from $\frac{C_V}{C_p-C_V}=\frac{1}{\gamma -1}$
using the fact that $C_p-C_V=Nk_B$ and $\gamma = \frac{C_p}{C_V}$ it can be written
$$\frac{C_v}{Nk_B}=\frac{C_V}{C_p-C_V}=\frac{1}{\gamma -1}$$.
If this could be explained to me, I suspect it is some form of algebraic rearrangement that I am not comfortable with that is hindering me.