The diagram of the circuit in the attached file is composed of two batteries (e1 = 9 V and e2 = 6 V) and four resistors (R1 = 110 ohm, R2 = 40 ohm, R3 = 50 ohm, and R4 = 50 ohm) as shown.
(a) What is the current I1 which flows through R1?
(b) What is the current I3 that flows through R3?
Because of the presence of batteries in more than one branch of the circuit, parts (a) and (b) of this problem can only be solved simultaneously. There is no way around this fact. Equivalent resistance tricks are of no help, except for resistances such as R1 and R4 in this circuit that are in the same "branch" and therefore must carry the same current. Begin by replacing R1 and R4 by an equivalent resistance; call it R14. Next express the current through R2 in terms of I1 and I3 using the Kirchhoff current rule.
Next write two independent voltage loop equations by going around the left-hand block of the circuit and, separately, the right-hand block. A loop around the entire periphery of the circuit is another possibility, but this does not give independent information because the resulting equation is the sum of the previous two loop equations. Solve the loop equations for I1 and I3.
This solution applies Kirchoff's laws to find the currents in a circuit.