How does
$$\frac{\overline{Y} \sum_{i=1}^{n} X_i^2 - \overline{X} \sum_{i=1}^{n} X_iY_i}{\sum_{i=1}^{n}(X_i - \overline{X})}= \overline{Y} - \frac{\sum_{i=1}^{n} (X_i - \overline{X})(Y_i - \overline{Y})}{\sum_{i=1}^{n} (X_i - \overline{X})} \cdot \overline{X}$$
All I know is
$\overline{X} = \frac{1}{n}\sum_{i=1}^{n} X_i$
$\sum_{i=1}^{n} (X_i - \overline{X})^2 = \sum_{i=1}^{n} X_i^2 - n \overline{X}^2$
Which other formulas are in use ?
There is an error in your denominator: It should be: $$ \sum_{i=1}^n(X_i-\overline X)^2\tag1 $$ With this correction, the only additional formula you need is: $$ \sum(X_i-\overline X)(Y_i-\overline Y)= \sum X_iY_i - n\overline X\overline Y.\tag2 $$ Plug (2) into the RHS of your first equation. Put everything on a common denominator, then substitute: $$ \sum_{i=1}^{n} (X_i - \overline{X})^2 = \sum_{i=1}^{n} X_i^2 - n \overline{X}^2\tag3 $$ After a bit of algebra you'll obtain the LHS of your equation.