Show explicitly that the following identity holds under a Simple Linear Regression:

$\text{}\sum _{i=1}^{n}{r}_{i}\hat{{\mu}_{i}}=0$

with residuals ${r}_{i}={y}_{i}-\hat{{\mu}_{i}}$ and $\hat{{\mu}_{i}}=\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i}$.

my steps:

$\begin{array}{rl}& \sum _{i=1}^{n}{r}_{i}\hat{{\mu}_{i}}\\ =& \sum _{i=1}^{n}({y}_{i}-\hat{{\mu}_{i}})(\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i})\\ =& \text{}\sum _{i=1}^{n}({y}_{i}-\hat{{\beta}_{0}}-\hat{{\beta}_{1}}{x}_{i})(\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i})\\ =& \text{}\sum _{i=1}^{n}(\hat{{\beta}_{0}}{y}_{i}+\hat{{\beta}_{1}}{x}_{i}{y}_{i}-{\hat{{\beta}_{0}}}^{2}-\hat{{\beta}_{0}}\hat{{\beta}_{1}}{x}_{i}-\hat{{\beta}_{0}}\hat{{\beta}_{1}}{x}_{i}-{\hat{{\beta}_{1}}}^{2}{{x}_{i}}^{2})\\ =& \text{}\sum _{i=1}^{n}(\hat{{\beta}_{0}}{y}_{i}+\hat{{\beta}_{1}}{x}_{i}{y}_{i}-{(\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i})}^{2})\\ =& \text{}\sum _{i=1}^{n}({y}_{i}(\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i})-{(\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i})}^{2})\\ =& \text{}\sum _{i=1}^{n}(\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i})({y}_{i}-\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i})\\ =& \text{}\sum _{i=1}^{n}\hat{{\mu}_{i}}{r}_{i}\\ & \end{array}$

how to proceed?

$\text{}\sum _{i=1}^{n}{r}_{i}\hat{{\mu}_{i}}=0$

with residuals ${r}_{i}={y}_{i}-\hat{{\mu}_{i}}$ and $\hat{{\mu}_{i}}=\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i}$.

my steps:

$\begin{array}{rl}& \sum _{i=1}^{n}{r}_{i}\hat{{\mu}_{i}}\\ =& \sum _{i=1}^{n}({y}_{i}-\hat{{\mu}_{i}})(\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i})\\ =& \text{}\sum _{i=1}^{n}({y}_{i}-\hat{{\beta}_{0}}-\hat{{\beta}_{1}}{x}_{i})(\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i})\\ =& \text{}\sum _{i=1}^{n}(\hat{{\beta}_{0}}{y}_{i}+\hat{{\beta}_{1}}{x}_{i}{y}_{i}-{\hat{{\beta}_{0}}}^{2}-\hat{{\beta}_{0}}\hat{{\beta}_{1}}{x}_{i}-\hat{{\beta}_{0}}\hat{{\beta}_{1}}{x}_{i}-{\hat{{\beta}_{1}}}^{2}{{x}_{i}}^{2})\\ =& \text{}\sum _{i=1}^{n}(\hat{{\beta}_{0}}{y}_{i}+\hat{{\beta}_{1}}{x}_{i}{y}_{i}-{(\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i})}^{2})\\ =& \text{}\sum _{i=1}^{n}({y}_{i}(\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i})-{(\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i})}^{2})\\ =& \text{}\sum _{i=1}^{n}(\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i})({y}_{i}-\hat{{\beta}_{0}}+\hat{{\beta}_{1}}{x}_{i})\\ =& \text{}\sum _{i=1}^{n}\hat{{\mu}_{i}}{r}_{i}\\ & \end{array}$

how to proceed?