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In this paper, by using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over $\\mathbb{F}_{q}$: $$ \\sum\\limits_{j=0}^{t-1}\\sum\\limits_{i=1}^{r_{j+1}-r_j} a_{k,r_j+i}x_1^{e^{(k)}_{r_j+i,1}}...x_{n_{j+1}}^{e^{(k)}_{r_j+i,n_{j+1}}}=b_k, \\ k=1,...,m. 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Let $m$ and $t$ be positive integers. In this paper, by using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over $\\mathbb{F}_{q}$: $$ \\sum\\limits_{j=0}^{t-1}\\sum\\limits_{i=1}^{r_{j+1}-r_j} a_{k,r_j+i}x_1^{e^{(k)}_{r_j+i,1}}...x_{n_{j+1}}^{e^{(k)}_{r_j+i,n_{j+1}}}=b_k, \\ k=1,...,m. 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