In a problem I am working on I can show that solutions can be found as roots of a polynomial. Additionally I can show that this polynomial can be factored into purely linear terms, like

(x - x_1)(x - x_2)(x - x_3)...(x - x_n) = 0

note that all of the linear terms have unity slope so the only unknowns are the intercepts. Does this case have a general solution for n terms, n >= 1