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 = 40 Ω, R2 = R6 = 156 Ω R3 = 69 Ω, and R4 = 113 Ω. The positive directions for the currents I1, I2 and I3 are indicated by the directions of the arrows.
The First question asks me to find the voltage at R4
V = IR
The Attempt at a Solution
I wasn't seeing any other way to find v4 than by making a loop for all three sections of the circuit
For the first loop I chose a clockwise orientation
-i3R3 - V1 +i1R1 = 0
For my second loop I chose another clockwise orientation
i2R2 - V2 - i6R6 + V1 + i3R3 = 0
and for the final loop I also choose a clockwise orientation
V2 -i5R5 - i4R4 = 0
Looking at the junction next to R2 I get one more equation
i2 - i3 = i1
Looking at junction B I get yet another, fifth, equation
i4 - i5 = i6
my problem is going to arise every time I have to label currents and come up with equations at junctions or loops. I understand how to create a system of equations using these but I will sometimes get solutions to the matrix with all my currents being zero. Also, i'm sometimes creating systems where I have Nx(N+2) rows and columns and I find that creates problems as well, I usually miss an equation. Any advice on how to fix these problems would be greatly appreciated, thank you.