  Let $\mathbb{F}_q$ denote the finite field of odd characteristic $p$ with $q$
elements ($q=p^{n},n\in \mathbb{N} $) and $\mathbb{F}_q^*$ represent the
nonzero elements of $\mathbb{F}_{q}$. In this paper, by using the Smith normal
form we give an explicit formula for the number of rational points of the
algebraic variety defined by the following system of equations over
$\mathbb{F}_{q}$: \begin{align*} {\left\{\begin{array}{rl}
&\sum_{i=1}^{r_1}a_{1i}x_1^{e^{(1)}_{i1}} ...x_{n_1}^{e^{(1)}_{i,n_1}}
+\sum_{i=r_1+1}^{r_2}a_{1i}x_1^{e^{(1)}_{i1}}
...x_{n_2}^{e^{(1)}_{i,n_2}}-b_1=0,\\ &\sum_{j=1}^{r_3}a_{2j}x_1^{e^{(2)}_{j1}}
...x_{n_3}^{e^{(2)}_{j,n_3}} +\sum_{j=r_3+1}^{r_4}a_{2j}x_1^{e^{(2)}_{j1}}
...x_{n_4}^{e^{(2)}_{j,n_4}}-b_2=0, \end{array}\right.} \end{align*} where the
integers $1\leq r_1<r_2$, $1\leq r_3<r_4$, $1\le n_1<n_2$, $1\le n_3<n_4$,
$n_1\leq n_3$, $b_1, b_2\in \mathbb{F}_{q}$, $a_{1i}\in \mathbb{F}_{q}^{*}$
$(1\leq i\leq r_2)$, $a_{2j}\in \mathbb{F}_{q}^{*}$$(1\leq j\leq r_4)$ and the
exponent of each variable is a positive integer. An example is also presented
to demonstrate the validity of the main result.
