In this paper, we first show the existence of solutions to the following system of nonlinear equations { a 11 x 1 + a 12 x 2 + ⋯ + a 1 n x n = b 11 1 x 1 + b 12 1 x 2 + ⋯ + b 1 n 1 x n , a 21 1 x 1 + a 22 x 2 x 1 + ⋯ + a 2 n x n x 1 = b 21 x 1 + b 22 x 1 x 2 + ⋯ + b 2 n x 1 x n , ⋯ ⋯ a k , k − 1 1 x k − 1 + ∑ 1 ≤ j ≤ n j ≠ k − 1 a k j x j x k − 1 = b k , k − 1 x k − 1 + ∑ 1 ≤ j ≤ n j ≠ k − 1 b k j x k − 1 x j , ⋯ ⋯ a n , n − 1 1 x n − 1 + ∑ 1 ≤ j ≤ n j ≠ n − 1 a n j x j x n − 1 = b n , n − 1 x n − 1 + ∑ 1 ≤ j ≤ n j ≠ n − 1 b n j x n − 1 x j , where n≥2 n ≥ 2 and a ij ,b ij ,1≤i,j≤n a i j , b i j , 1 ≤ i , j ≤ 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.