The uncontrollability of the radio frequency (RF) environment is one of the main obstacles hindering the popularization of ambient backscatter communication devices, because the devices need strong signal to maintain the overhead of backscatter circuits and the reflection of the modulated signals. The intelligent reflector (IRS) can improve the radio frequency environment by adjusting the phase and amplitude of the incident signal, which provides the possibility for the widespread deployment of ambient backscatter communication devices. In this paper, we introduce a novel IRS-assisted ambient backscatter communications system (ABCS), in which the signal of ABCS rides on the signal of the primary system. The two systems share the same receiver, and the signals of the two systems can be demodulated separately based on continuous interference cancellation (SIC) technology. The purpose of this paper is to design the beamforming vector and IRS phase shift jointly to minimize the AP's transmit power while ensuring the quality of service of the ABCS and the primary communication system. Due to the non-convex nature of the problem, the time complexity of solving the problem through exhaustive search will be very high. Therefore, we propose an iterative-based beamforming vector and IRS phase shift joint design method to minimize the AP transmit power. This method can effectively reduce the transmission power of the access point, and the simulation results prove the effectiveness of the method.