In this paper, we considered some systems of rational difference equations of higher order as follows

$$\begin{array}{rcl}{u}_{n+1}& =& \frac{v_{n-6}}{1\pm v_n{u}_{n-1}{v}_{n-2}{u}_{n-3}{v}_{n-4}{u}_{n-5}{v}_{n-6}},\\ v_{n+1}& =& \frac{u_{n-6}}{1\pm u_n{v}_{n-1}{u}_{n-2}{v}_{n-3}{u}_{n-4}{v}_{n-5}{u}_{n-6}},\end{array}$$

where the initial conditions $u_{0,}$ $u_{-1},$ $u_{-2},$ $u_{-3},$ $u_{-4},$ $u_{-5},$ $u_{-6},$ $v_{0,}$ $v_{-1},$ $v_{-2},$ $v_{-3},$ $v_{-4},$ $v_{-5}$ and $v_{-6}$ were arbitrary real numbers. We obtained a closed form of the solutions for each considered system and also some periodic solutions of some systems were found. We presented some numerical examples to explain the obtained theoretical results.