The invention relates to a brake system for a motor vehicle, having a master brake cylinder (4, 5), which is driven by an electric drive, wherein a piston plunger (25) and/or a spindle (2a) is in operative connection with a piston (3) of the master brake cylinder (4, 5), having wheel brake valves (7) for selective opening and closing the pressure lines (BL) connecting the wheel brakes (RB) to the master brake cylinder (4, 5), having an actuation device (1), in particular in the form of a brake pedal, that acts on a travel simulator (W), wherein in the event of a malfunction, in particular in the event of failure of the drive (2), the actuation device (1) acts mechanically, in particular by means of the pedal shaft (1a), on the piston plunger (25) to build up pressure in at least one wheel brake, and in normal operation there is a particularly small free travel (ao); between the piston plunger (25) and the actuation device, in particular between the piston plunger (25) and the pedal shaft (1a), having a control device including a brake-by-wire function, which actuates the drive (2) with the piston (3) as well as the wheel brake valves (7), in particular for either simultaneous or consecutive pressure build-up and pressure reduction in the wheel brakes (RB) for the ABS, ESP and/or recuperation functions, wherein the storage tank (6) or a fluid reservoir (20, 20') is used to adjust the magnitude of the free travel (a).