The invention relates to a brake system (1) of a motor vehicle. The brake system for the motor vehicle has at least a first wheel brake (5, 7) and at least a second wheel brake (6, 8). A first inlet valve (20, 22) is fluidically connected, on the one hand, to the first wheel brake (5, 7), and, on the other hand, to a brake pressure source (4), and a second inlet valve (21, 23) is fluidically connected, on the one hand, to the second wheel brake (6, 8) and on the other hand, to the brake pressure source (4). In this case, a switching valve (34, 35) is provided, which fluidically connects the one fluid outlet (36, 37) of the second inlet valve (21, 23) in a first switching position to the second wheel brake (6, 8) and in a second switching position to the first wheel brake (5, 7). Additionally, the invention also relates to a method for operating the brake system for the motor vehicle.