Have you noticed that solving your first nine equations for your first nine variables happens very quickly? And your tenth equation is just the sum of the previous nine variables? Then you can substitute the nine solutions into your tenth equation which should eliminate most of the variables and leave you with one large equation to study.
Most of those solutions appear as Roots. Using InputForm on those solutions might enable you to see the details. This might give you some insight into the form of the tenth equation as a sum of nine roots. Using Simplify on this may or may not help you.
You might be able to force some of those roots into algebraic form, but the results would likely be much larger and more complicated and make understanding the structure more difficult.
Hopefully this will give you another way of looking at your problem.
