That's a normal question. You can't get 1 number because these conditions define a map (a function from $R\rightarrow R$)- you need to reduce it to one parameter ${a_1}/{a_2}$ or ${b_1}/{b_2}$.
You have 2 bounds $c_1$, and $c_2$ plus two inequalities(exclusions) that $a_1\neq 0$ & $a_2\neq 0$
Your equation needs 3 bounds(it's just intuition, not a math) which means that you need 1 more bound.
To figure it out just derive further:
You don't have $c_1$ & $c_2$ in equation $\implies$ you need to substitute them:
$b_1 = a_1\times c_1; b_2 = a_2\times c_2$;
$c_3 = (a_1\times c_1 + a_2\times c_2)/(a_1+a_2) = c_1(a_1 + a_2)/(a_1+a_2) + (c_2-c_1)\times a_2/(a_1+a_2) =$
$c_1 + (c_2-c_1)/(1+ a_1/a_2)$ where the only unknown part is $a_1/a_2 = t; t\neq 0 (a_1\neq 0)$ So,
The answer:
$c_3 = c_1 + (c_2-c_1)/(1 + t)$, where t belongs $(-\infty; 0)\bigcup(0;+\infty)$
In other words if $c_1, c_2$ are concrete - it's just one graph.
If they are parameters ranging over $R$ you get $R^2$ space of of these hyporbolic functions.
This is a specific object in Math and has all the rights, just as a number from $R$ to exist and be used. You can call it a number in a different peculiar space.
What's important - there is no way to know what you'll get in the end until you solve it. Therefore you must always solve. Such questions mathematically always have an answer.