Summation Reduction

Question

Does $\sum\frac{y_i}{n}-\sum(\frac{x_i-\bar x}{(x_i-\bar x)^2}\cdot y_i)\sum\frac{x_i}{n}$ reduce to:
$\sum[y_i(\frac{1}{n}-\frac{x_i-\bar x}{(x_i-\bar x)^2}\sum\frac{x_i}{n})]$
Basically, what does $\sum a-\sum b\sum c$ reduce to?
edit: it should be noted I'm trying to rewrite the first expression in terms of $\sum y_ik_i$ for some constant $k$. See comment for original equation.

Answer 1

Yes it does.

Let $a=y_{i}, b=\frac{1}{n}, c=\frac{x_i-\bar x}{(x_i-\bar x)^2}, d=\frac{_{}}{n}$.

Then you have:

$\sum\frac{y_i}{n}-\sum(\frac{x_i-\bar x}{(x_i-\bar x)^2}\cdot y_i)\sum\frac{x_i}{n} =$

$\sum ab - \sum ca \sum d$ =

$\sum ab - \sum ac \sum d$ =

$\sum ab - \sum a(c \sum d)$ =

$\sum a(b - c \sum d)$ =

$\sum[y_i(\frac{1}{n}-\frac{x_i-\bar x}{(x_i-\bar x)^2}\sum\frac{x_i}{n})]$