Circuit is constructed with six resistors and two batteries

[PhysicsNo]

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A circuit is constructed with six resistors and two batteries as shown. The battery voltages are V1 = 18 V and V2 = 12 V. The positive terminals are indicated with a + sign, The values for the resistors are: R1 = R5 = 70 Ω, R2 = R6 = 105 Ω R3 = 59 Ω, and R4 = 82 Ω. The positive directions for the currents I1, I2 and I3 are indicated by the directions of the arrows.I have gone through the algebra a lot of times, with a lot of different equations. I believe my problem is with my application of the loop rule. Please, point out my mistakes here.

Equations I came up with:
1. I2 = I1 + I3
2. V2 - I2R2 - I1R1 - I6R6 = 0
3. V2 - I2R2 - I3R3 - V1 - I6R6 = 0
4. V1 + I3R3 - I1R1 = 0

 

[Dale]

What is the current through R5?

 

[gneill]

What is the current through R5?

Did you mean R6?

@PhysicsNo, your loop equations look fine but you've introduced a current I6 that you should be able to write in terms of other, already defined currents. Consider what applying KCL at node a tells you.