I had laid my hand on an OMO (Online Math Open) problem which was
Convex pentagon $ABCDE$ is inscribed in circle $\gamma$. Suppose that $AB=14$, $BE=10$, $BC=CD=DE$, and $[ABCDE]=3[ACD]$. Then there are two possible values for the radius of $\gamma$. The sum of these two values is $\sqrt{n}$ for some positive integer $n$. Compute $n$
By using my Geometry Theorems (Ptolemy, Fact 5,etc) the following system of equations came up (after labelling some lengths as $a,b,c,d,x$)-
$$14a+bc=2ab$$ $$14x+bx=ad$$ $$xc+14d=10a$$ $$cd+14x=10b$$ $$xc+ax=d$$
There are five equations, and five variables, but are quite clumsily presented and do not look like how simple equations should look like, and we didn't find a solution to this till now, even after lot of algebraic manipulations. I suspect it cannot be solved only using these, perhaps we need more constraints. But can in general, equations like this (coefficients consist of variables) having # of variables$=$ # of equations have solutions which can be found out.
It would be great if this could be solved with Linear Algebra.
EDIT: I am interested in the methods to solve system of equations such as this, not the problem, the problem was mentioned just as an example to show such a system if equations.