Let loop currents be I1 and I2 as shown in the fig.. We write the loop equation for the upper loop using Kirchhoff's loop rule :
E2 - R3I1 - R6(I1 - I2) - E3 - R5(I1 - I2) - R4I1 = 0
Substituting R for all resistances :
E2 - RI1 - R(I1 - I2) - E3 - R(I1 - I2) - RI1 = 0
Or E2 - 4RI1 + 2RI2 - E3 = 0
Or 4RI1 - 2RI2 = E2 - E3 ........(1)
Step by step solution provided.