The stochastic resonance (SR) is studied in an overdamped linear system when the bias-signal-modulated noise is introduced. The exact expressions of the first two moments and signal-to-noise ratio(SNR) are obtained when we introduce the asymmetric dichotomous noise into the system. We find that the system has three different forms of SR: the conventional SR, the bona fide SR and SR in the broad sense. Moreover, the additive noise weakens the SNR of the linear system while the cross-correlation enhances the SNR and improves the output signal.