Is there a way to solve this equation arising from a weighted sum? If I have a weighted finite sum that is equal to a known real value $x$, and I also know the real-valued and non-negative weights $a_i$, and I have unknown but also real and non-negative elements $b_i$ :
$x = \sum a_ib_i$
is it possible to extract the value $\sum b_i$? I don't need to know the value of each element $b_i$, only their sum.

No. Consider $x=3$, $a_1=1$, $a_2=2$.
$$3=1\cdot1+2\cdot1=1\cdot\frac12+2\cdot\frac54$$
$$1+1\neq\frac12+\frac54$$
So $\sum b_i$ is not determined by $\sum a_ib_i$.
