An error analysis of direct methods (i.e., Gaussian elimination or triangular factorization) of solving simultaneous linear algebraic equations is performed in the backward mode, in which the computational errors are expressed as perturbations on the data. Bounds are found for perturbations on the coefficients of the equations, leaving the right-hand sides unchanged. These bounds can be evaluated concurrently with the computation itself, with only a small increase in computing effort. Because they use information obtained during the solution process, these bounds avoid exaggerating the magnitude of the error, and so are also useful as error estimates.