Given a linear system of the form:
$$x_r = a$$ $$x_j = b$$ $$c_1x_1 + c_2x_2 ... c_nx_n = n$$ $$x_1 + x_2 + x_3 ... x_n = k $$ $$0 \leq a,b,x_1, x_2, x_3 ... x_n \leq 1$$ $$k \geq 0$$ How quickly can the feasibility of the system be checked? To clarify: $x_r, x_j$ are members of ${x_1, x_2 ... x_n}$. Would it be $O(n^{3.5})$ since I believe that is the general complexity for running a linear program or would it be less? Can one use gaussian elimination to quickly reduce the first 4 equations in $O(n^{3})$ and after that systematically move through the equations starting from the terms with largest coefficient and moving to terms with smallest coefficient assigning values that bring the equations as close to satisfactory as possible?