My problem is the following. I want to determine if there is a solution $x\geq 0$ to equations of the following general type: $$W=\sum_{i=1}^n\frac{a_i}{b_i+c_ix},$$ where $W\geq 0$, $a_i\geq 0$, $b_i\geq 0$ and $c_i\geq 0$.
For example, the following equation $$3=\frac{2}{1+x} + \frac{5}{5+2x} + \frac{1}{7+x} + \frac{10}{3+3x}$$ has one solution for $x\geq 0$, which is around $1.38$. I know I could solve such an equation by writing the equation with a common denominator and then by using a numerical solver for the numerator. But is there, by any chance, a closed form solution to such equations?
I also noticed something special but I am not able to prove it. Is there a way to show that there is at most one solution to such equations with $x\geq 0$?
Thank you very much for your help!
$$f'(x):=\left(\sum_{i=1}^n\frac{a_i}{b_i+c_ix}\right)'=-\sum_{i=1}^n\frac{a_ic_i}{(b_i+c_ix)^2}<0$$ for all $x\ge0$.
As this function is strictly decreasing, the equation cannot have more than one solution. And it has one when
$$f(0)=\sum_{i=1}^n\frac{a_i}{b_i}\ge W,$$ because $f(\infty)=0$.
As said by others, the equation is an algebraic one so there is no closed formula for $n>4$.