We propose a new technique for energizing coronal magnetic equilibria toward eruptions. We achieve this via a sequence of MHD relaxations of small line-tied pulses of magnetic helicity, each of which is simulated by a suitable rescaling of the current-carrying part of the field. The whole procedure is ‘magnetogram-matching’ because it involves no changes to the normal component of the field at the lower boundary. The technique is illustrated by application to bipolar force-free configurations whose magnetic flux ropes (MFRs) are modeled with our regularized Biot-Savart law method. We have found that, in spite of the bipolar character of the ambient potential field in these examples, the resulting MFR eruption is generally sustained by two reconnection processes. The first, which we refer to as breakthrough reconnection, is analogous to breakout reconnection in quadrupolar configurations. It occurs at a quasi-separator field line located inside the current layer that wraps around the erupting MFR, and results from taking into account the line-tying effect at the photosphere. The second process is the classical tether-cutting reconnection that develops at the second quasi-separator inside a vertical current layer formed below the erupting MFR. Both reconnection processes work in tandem to propel the MFR through the overlying ambient field. The considered examples suggest that our technique will be beneficial for both the modeling of particular eruptive events and theoretical studies of eruptions in idealized magnetic configurations. This research was supported by NASA programs HTMS (award no. 80NSSC20K1274) and HSR (80NSSC19K0858 and 80NSSC20K1317); NASA/ NSF program DRIVE (80NSSC20K0604); and NSF grants AGS-1923377 and ICER-1854790.