Let's define three loop currents, $$I_1$$, $$I_2$$, and $$I_3$$, for the three meshes in the circuit.
Mesh 1 (left loop): Contains the 12.0 V battery and resistor $$R_1$$.
Mesh 2 (middle loop): Contains the 3.00 V battery, resistor $$R_2$$, and resistor
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