In the present paper, we mainly investigate the qualitative behavior of the solutions of a discrete system of difference equations Sigma mi=1 xn-i xn+1 = alpha + yn Pmi=1 yn-i , yn+1 = beta + xn ,nENwhere alpha, beta E (0, infinity), m E Z+, x-i and y-i are non-negative real numbers for i E {0, 1, ... , m}. Namely, we discuss the boundedness character and the asymptotic stability properties of steady states of the mentioned system. Finally, for this system, we give a rate of convergence result which has an important place in the discrete dynamical systems. Besides, some numerical simulations with graphs are given to emphasize the efficiency of our theoretical results in the article.