Subtracting a constant from the OLS summation

Question

On a proof for the OLS of $\beta$, I have seen this step:

$\sum x_i (y_i - \alpha - \beta x_i) = \sum (x_i - k) (y_i - \alpha - \beta x_i) $ for any constant $k$.

Why is this true?

Answer 1

That's because $y_i - \alpha -\beta x_i$ is the residual $r_i$, and the sum of residuals ($\sum\limits_i r_i$) is zero (see e.g. this answer).

In this step you are just adding $k\sum (y_i - \alpha -\beta x_i)$ on the right-hand-side, that is equal to zero.