Weighted average of non-adjacent columns that excludes blanks

[HinesKetchup]

Hello, I need to calculate a weighted average that does not count blank cells as zeroes but instead excludes them from the average. I think I've figured out how to do this using sumproduct, however the added caveat is that the columns are non-adjacent (this is necessary due to other factors and cannot be changed). Is there any way to make this work?

Attaching a simple example worksheet.

 

[p45cal]

In row 2, something along the lines of:

=AVERAGE(FILTER($B$1:$H$1*$B2:$H2,($B$1:$H$1<>"")*($B2:$H2<>"")))

copied down.
Maybe AVERAGE isn't what you need wrapping the formula; try looking at the results without it.