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A system of linear   Riemann-Liouville  fractional differential equations with a constant delay and impulses is studied.  Both cases are considered:   the case when the lower limit of the fractional derivative is fixed on the whole interval of consideration and the case when the lower limit of the fractional derivative is changed at any point of impulse. The initial conditions as well as impulsive conditions are defined in an appropriate way for both cases.  The explicit solutions are obtained and applied to study finite time stability.