In deriving the OLS estimators for 1 variable, I am trying to convert
$$ \begin{align} \hat{\beta_1} = \frac{\sum_{i=1}^N X_iY_i- N\bar{X}\bar{Y} }{\sum_{i=1}^NX^2_i - N(\bar{X})^2} \end{align} $$
to the final form
$$ \hat{\beta_1} = \frac{\sum^N_{i=1}\begin{bmatrix} \begin{pmatrix} X_i-\bar{X} \end{pmatrix} \begin{pmatrix} Y_i-\bar{Y} \end{pmatrix} \end{bmatrix}} {\sum^N_{i=1}\begin{pmatrix}X_i - \bar{X}\end{pmatrix}^2} $$
In particular, I'm trying to show the following 2 sums are valid: $\sum_{i=1}^N\begin{bmatrix}(X_i -\bar{X})(Y_i -\bar{Y})\end{bmatrix}\sum_{i=1}^N X_iY_i- N\bar{X}\bar{Y} $
and $\sum_{i=1}^N(X_i -\bar{X})^2 = \sum_{i=1}^NX^2_i - N(\bar{X})^2 $
What's the best way to show the following sums?
\begin{align} \sum_{i=1}^N[(X_i -\bar{X})(Y_i -\bar{Y})]&=\sum_{i=1}^N[X_i Y_i-\bar{X}Y_i-X_i\bar{Y}+\bar{X}\bar{Y}] \\&=\sum_{i=1}^N X_i Y_i- \sum_{i=1}^N\bar{X}Y_i- \sum_{i=1}^NX_ i\bar{Y}+\sum_{i=1}^N\bar{X}\bar{Y} \\&=\sum_{i=1}^N X_i Y_i- \bar{X}\sum_{i=1}^NY_i- \bar{Y}\sum_{i=1}^NX_ i+\bar{X}\bar{Y}\sum_{i=1}^N 1 \\&=\sum_{i=1}^N X_i Y_i- \bar{X} (N \bar{Y})- \bar{Y}(N \bar{X})+N\bar{X}\bar{Y} \\&=\sum_{i=1}^N X_iY_i- N\bar{X}\bar{Y} \end{align}
The other one follows similarly.