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Let p be a prime number. We propose a new method for solving sparse linear systems of equations modulo p when the coefficient matrix has column degree at most 2. This algorithm is based on a well-known decoding algorithm for Low-Density Parity-Check (LDPC) codes called bit-flipping (BF) algorithm. We modify and extend this hard-decision decoding algorithm. The complexity of this algorithm is linear in terms of the number of columns n and the number of nonzero coefficients u of the matrix. We give a detailed small example, and report on computational results for larger systems.