# Kronecker-Capelli Theorem for system of congruences

Let $p$ be some prime. Given a system of linear congruences, \begin{align} m_1 x + n_1 y &\equiv c_1 \quad (mod\, p)\\ m_2 x + n_2 y &\equiv c_2 \quad (mod\,p)\\ \ldots \\ m_d x + n_d y &\equiv c_d \quad(mod\, p) \end{align} where $x$ and $y$ are the unknowns, can we say something about the solvability of the system based on the rank of the matrices of the coefficients, as in the Kronecker-Capelli theorem? I know that since $p$ is prime we can use Gaussian elimination to solve the system, but I wondered if it is straight forward that here also there is a solution if and only if $rank(A)=rank([A|b]),$ or something similar to the theorem for one linear congruence exists?

Thank you!

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Try this: math.stackexchange.com/questions/130517/… or math.stackexchange.com/questions/335986/…. See also related questions ("Related" column on the right). –  user35603 Jan 31 at 10:57