Simplifying $\beta_1$ estimate for a simple linear regression model

Question

For a simple linear regression model I am able to derive the normal equations and solve these to to obtain the following- $$\beta_0=\bar{Y}-\beta_1\bar{X}$$ $$\beta_1=\frac{\sum(X_i-\bar{X})(Y_i-\bar{Y})}{\sum(X_i-\bar{X})^2}$$

However in my text book and elsewhere online I see that $\beta_1$ is often (not always) simplified to $$\beta_1=\frac{\sum(X_i-\bar{X})Y_i}{\sum(X_i-\bar{X})^2}.$$

How does one arrive at this simplification? It is not at all obvious to me why?

Answer 1

\begin{align} \sum (X_i - \bar{X}) ( Y_i - \bar{Y}) &= \sum(X_i - \bar{X})Y_i - \sum(X_i - \bar{X})\bar{Y} \\ &= \sum(X_i - \bar{X})Y_i - \sum X_i \bar{Y} +\sum\bar{X}\bar{Y} \\ &= \sum(X_i - \bar{X})Y_i - n\bar{Y}\bar{X} +n\bar{Y}\bar{X}\\ &= \sum(X_i - \bar{X})Y_i. \end{align}