Summary: In this paper, we first show the existence of solutions to the following system of nonlinear equations \[ \begin{cases} a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n=b_{11}\frac{1}{x_1}+b_{12}\frac{1}{x_2}+\cdots+b_{1n}\frac{1}{x_n},\\ a_{21}\frac{1}{x_1}+a_{22}\frac{x_2}{x_1}+\cdots+a_{2n}\frac{x_n}{x_1}=b_{21}x_1+b_{22}\frac{x_1}{x_2}+\cdots+b_{2n}\frac{x_1}{x_n},\\ \cdots\cdots\\ a_{k,k-1}\frac{1}{x_{k-1}}+\mathop{\sum}\limits_{\mathop{1\leq j\leq n}\limits_{j\neq k-1}}a_{kj}\frac{x_j}{x_{k-1}}=b_{k,k-1}x_{k-1}+\mathop{\sum}\limits_{\mathop{1\leq j\leq n}\limits_{j\neq k-1}}b_{kj}\frac{x_{k-1}}{x_j},\\ \cdots\cdots\\ a_{n,n-1}\frac{1}{x_{n-1}}+\mathop{\sum}\limits_{\mathop{1\leq j\leq n}\limits_{j\neq n-1}}a_{nj}\frac{x_j}{x_{n-1}}=b_{n,n-1}x_{n-1}+\mathop{\sum}\limits_{\mathop{1\leq j\leq n}\limits_{j\neq n-1}}b_{nj}\frac{x_{n-1}}{x_j},\end{cases} \] where \(n\geq 2\) and \(a_{ij}\), \(b_{ij}\), \(1\leq i\),\(j\leq n\), are positive constants. Then, we make use of this result to obtain the large deviation principle for the occupation time distributions of continuous-time finite state Markov chains with finite lifetime.