Two Variables · Superchallenge problem

A least-squares line $\hat y = a + bx$ is fitted to $(1, 3)$, $(2, 7)$, $(3, 4)$, $(4, 9)$, $(5, 8)$, $(6, 12)$, giving residuals $e_i = y_i - \hat y_i$. Without computing the line, find $\displaystyle\sum_{i} e_i\,\hat y_i$.