finding solution to simultaneous equations I have three equations
$$ 
\begin{align*}
      c_1 & = \frac{a}{1+a+b} + \frac{a}{a+b}+\frac{a}{1+a}\\
   c_2 & = \frac{b}{1+a+b} + \frac{b}{a+b}+\frac{b}{1+b}\\
c_3 & = \frac{1}{1+a+b} + \frac{1}{1+a}+\frac{1}{1+b}\\
\end{align*}$$
where $c_1,c_2,c_3$ are known constant. 
I want to know if I can find $a$ and $b$ or not.. 
Thanks,
 A: B. Goddard's great observation effectively answers your question (maybe not even he realises this). I had recognised the importance of his observation immediately, but only saw that it was the key on second thoughts, after casting about on the problem for some time.
Your question is whether the system has a solution for every choice of the (positive, according to you) constants $c_i$. Goddard has effectively shown that this is not the case, for a necessary condition that the constants must satisfy in order for the system to have a solution is that $\sum{c_i}=4$. It is easy to see that this restriction is also sufficient, so that we have that for every $$0<c_i<4$$ satisfying $$\sum{c_i}=4,$$ your system is soluble. An example is when the $c_i$ are equal, which gives the solution $a=b=1$.
A: Before undertaking calculations which are guaranteed to be tedious, sometimes it helps to graph the equations involved. In this case if we turn these into equations in $x$ and $y$ by letting $x=a$ and $y=b$. Using the graphing site desmos.com and letting $c_1,\,c_2,\,c_3$ be 'sliders' we see that the places where the three graphs contain the same point cannot actually be solutions since they would lead to division by $0$.
For example, $a=x=-1$ and $b=y=0$ involve division by zero, as does $a=x=0$ and $b=y=-1$. We can get a solution $a=x=b=y=0$ but only for the case $c_1=c_2=0$ and $c_3=3$.
There are cases where a common intersection can be obtained but they require specific values of $c_1,\,c_2,\,c_3$. For example, if we let $c_1=c_2=c_3=\frac{4}{3}$ then $a=b=1$ is a solution.
So in general, no, one cannot solve for $a$ and $b$.
Here is the link to the Desmos graphs.
