Step by Step Explanation of Sigmod odds function

Question

I am pursuing a Machine Learning Course but my poor maths basics are really bottlenecking the progress.

$$p = \frac1{1+e^{-(\beta_0 + \beta_1 x)}}$$

$$1-p = \frac{e^{-(\beta_0 + \beta_1 x)}}{1+e^{-(\beta_0 + \beta_1 x)}}$$

$$\frac{p}{1-p}= e^{\beta_0 + \beta_1 x}$$

I am trying to understand this derivation step by step. Can someone please me the transition from equation 1 to equation 3?

Thanks

Answer 1

If you let $z=e^{\beta_0 + \beta_1 x}$, your task is to solve for $z$ in

$$p=\frac{1}{1+e^{-(\beta_0+\beta_1x)}}=\frac1{1+z^{-1}}$$

Then we have

$$1+z^{-1}=\frac1p$$

Can you try to solve for $z$? Try to solve for $z^{-1}$ first.