\[ \begin{cases}
    n\hat\beta_0+\sum X_i
    \cdot\hat\beta_1 = \sum Y_i&\\
    \sum X_i \hat\beta_0+\sum X_i^2
    \cdot\hat\beta_1 = \sum X_iY_i&
\end{cases}\Rightarrow
\hat\beta_1=
\frac
{\begin{vmatrix}
    n & \sum Y_i\\
    \sum X_i & \sum X_iY_i
\end{vmatrix}}
{\begin{vmatrix}
    n & \sum X_i\\
    \sum X_i & \sum X_i^2
\end{vmatrix}}
= \frac{n\sum X_iY_i-\sum X_i\sum Y_i}
{n\sum X_i^2-(\sum X_i)^2}\]

\[\begin{align*}     &~\quad\frac{n\sum X_iY_i-\sum X_i\sum Y_i} {n\sum X_i^2-(\sum X_i)^2}\\     &= \frac{\displaystyle n \sum (X_i-\overline X+\overline X)     (Y_i - \overline Y+\overline Y) -\sum X_i\sum Y_i}     {\displaystyle n \sum (X_i-\overline X+\overline X)^2 -     (\sum X_i)^2}\\ &= \frac{\displaystyle     n\left(\sum (X_i-\overline X) (Y_i-\overline Y)-\overline Y     \sum (X_i-\overline X) -\overline X \sum (Y_i - \overline Y)     +\sum\overline X\overline Y\right)-\sum X_iY_i}     {\displaystyle n\left(\sum (X_i-\overline X)^2+2\overline X     \sum (X_i-\overline X)+\sum \overline X^2\right)-(\sum X_i)^2}\\     &= \frac{\displaystyle n\sum(X_i-\overline X)(Y_i-\overline Y)     +n\sum\overline X\overline Y-\sum X_i Y_i}     {\displaystyle n\sum (X_i-\overline X)^2+n\sum \overline X^2-     (\sum X_i)^2}\\     &= \frac{\displaystyle \sum (X_i-\overline X)(Y_i-\overline Y)}     {\displaystyle \sum(X_i-\overline X)^2} \end{align*}\]