### A very nonlinear system of three equations

Here's a fun little problem: determine the exponential curve f(x) = c + ba^x defined by three points, (2,10), (4,6), and (5,5).

The system of three equations and three unknowns is

10 = c + ba^2

6 = c + ba^4

5 = c + ba^5

It's not that hard to solve numerically. With a little algebraic substitution and iteration, the answer turns out to be

a = 0.640388203

b = 16.53456516

c = 3.219223594

But is there a more elegant way to use Sage to arrive at this result? I'm stumped.