We explicitly solve the following second-order system of difference equations:

$$x_{n+1}=\frac{a{x}_n{y}_{n-1}}{y_n-\beta y_{n-1}-\gamma }+b{x}_n+c,y_{n+1}=\frac{\alpha y_n{x}_{n-1}}{x_n-b{x}_{n-1}-c}+\beta y_n+\gamma ,$$

where $n\in {\mathbb{N}}_0$, $a$, $\alpha$, and the initial conditions $x_{-1}$, $y_{-1}$, $x_0$, and $y_0$ are nonzero real numbers, while the remaining parameters $b$, $c$, $\beta$, and $\gamma$ are real numbers. A detailed analysis of the solutions of our system when $\alpha =a$ with respect to the form, the periodicity and the limiting behavior is presented. To support and illustrate our theoretical results, numerical examples are provided. Our study, considerably generalizes some existing results in the literature.