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## Homework Statement

Look at the circuit given in figure 1. Solve for the three currents, I

_{1},I

_{2},I

_{3}.

## Homework Equations

KVL: Sum of voltages in a closed loop equals zero

KCL: Sum of currents entering a junction equals the sum of currents exiting a junction

Ohm's Law: V=IR

NOTE: Nodal analysis is not allowed for this solution

## The Attempt at a Solution

Through KCL, calling I

_{5}the current leaving the junction on the left-hand side of the circuit and I

_{4}the current leaving the junction on the right-hand side of the circuit: I

_{2}=I

_{5}+I

_{3}

I

_{4}=I

_{3}+I

_{1}

Then... the KVL equations for each loop:

5I

_{5}-10I

_{3}-8I

_{4}=0

5I

_{5}-12+I

_{2}=0

8I

_{4}-9+I

_{1}=0

-I

_{2}+12-10I

_{3}-9+I

_{1}=0

-I

_{2}+12-5I

_{5}+8I

_{4}-9+I

_{1}=0

-10I

_{3}-9+I

_{1}+5I

_{5}=0

-10I

_{3}-8I

_{4}-I

_{2}+12=0

Okay...so I can't come up with a methodical way of plugging these equations into one another and solving for any unknowns. It just seems...honestly impossible.

We haven't learned any clever tricks involving ignoring one battery (I've seen that in solutions elsewhere), and when the professor mentioned this question he presented it as pretty straightforward...is there something major that I'm missing?

I feel like I have a firm grasp of the mathematics and physics involved but I still can't produce any sort of solution. When I have I've found a mistake somewhere in my math.