Rouché–Capelli theorem

Also known as Kronecker–Capelli theorem, Rouché–Fontené theorem, Rouché–Frobenius theorem

theorem in linear algebra that a system of linear equations with n variables has solution(s) iff the rk(A) = rk([A|b]), and that if there are solutions, they form an affine space of dimension n−rk(A)