Ordinary Least Squares - Derivation of Coefficient Value

[dreamingotter]

Assuming the model follows this equation: Y = Beta_0 + Beta_1*X, I understood everything until the first line of the following image:

But how come the Y_bar(sum(x_i-x_bar)) cancels out (or is equal to 0)? What assumptions are we making here?

 

[Dr.Peterson]

Are you aware that the sum of deviations from the mean is always zero? (That's something I do to check the correctness of my work when I am finding standard deviation by hand.) If not, it is easy to prove by breaking up [MATH]\sum_{i=1}^n(x_i-\bar{x})[/MATH] as [MATH]\sum_{i=1}^n x_i-\sum_{i=1}^n\bar{x}[/MATH].

As a result, [MATH]\overline{Y}[/MATH] is being multiplied by zero.