How to solve a set of equations where the unknowns are a function and some parameters? I'd like to know how to solve something like this:
$$\begin{eqnarray}
f(f(x_2)-f(x_1)) & = & 27.5\\
f(f(x_3)-f(x_1)) & = & 21.6\\
f(f(x_4)-f(x_1)) & = & 15.1\\
f(f(x_5)-f(x_1)) & = & 10.2\\
f(f(x_6)-f(x_1)) & = & 8.8\\
f(f(x_7)-f(x_1)) & = & 8.4\\
f(f(x_8)-f(x_1)) & = & 7.8\\
f(f(x_9)-f(x_1)) & = & 6.4\\
f(f(x_4)-f(x_3)) & = & 46.5\\
f(f(x_5)-f(x_3)) & = & 17.6\\
f(f(x_5)-f(x_4)) & = & 28\\
f(f(x_7)-f(x_4)) & = & 20.1\\
f(f(x_9)-f(x_7)) & = & 22
\end{eqnarray}$$
where I'm looking for $f(x)$ and $x_1,x_2,\ldots,x_9$.  It is given that $0<x_1<x_2<\ldots< x_9$ and that $f(x)$ is a decreasing function which we have reason to believe is linear.  (Also, there's a margin of error of about $\pm 0.2$.)

I see that it is impossible to obtain the exact solution for this question, since it needs a numerical solution. I am interested because I see it may be a real world calculation, based on the numbers given.
I will propose two methods.
(1) Since there are 13 equations and 9 $x_i$, we can propose a $f(x)$ with 4 parameters. For example, let: $$f(x) = ax^3+bx^2+cx+d$$ This system of equations would be very complex, and need a numerical way to find the solution.
(2) Assume that: $$f(x) = ax+b$$ and then use least square method to find the best fit of the line.
