For the simple linear regression model, show that the values for \(\widehat{\beta_{1}}\) and \(\widehat{\beta_{0}}\) that solve the equations (5.9) are:

\[ \begin{gather*} \widehat{\beta_{1}}=\frac{\sum_{i=1}^{n}\left(x_{i}-x\right)\left(y_{i}-y\right)}{\sum_{i=1}^{n}\left(x_{i}-x\right)^{2}} \tag{5.40}\\ \widehat{\beta_{0}}=y-\widehat{\beta_{1}} x \tag{5.41} \end{gather*} \]

provided that not all \(x_{i}\) are the same.