Solving a Simple Circuit involving Current Source by the Loop Current Method

Homework Statement

For the resistor network in the picture, use loop currents to find the current through the 4Ω resistor and the change in voltage E across the current source.

Circuit Attached Below .

Homework Equations

• Kirchoffs Laws
• Loop Current Method
• V=IR

The Attempt at a Solution

Firstly I divided the circuit into 3 loops with 3 currents.
1. Top left Loop, Current A
2. Top right Loop, Current B
3. Bottom Loop, Current C

Looking For
• Voltage Change,E, across the current source
• Current A
• Current B
• Current C

Now I apply Kirchoffs laws for every loop. I go around in a counter clockwise manner and get the total drop in voltage for each.
• Loop A: -E+A+2(A-C)=0
• Loop B: E+(B-C)+2(B)=0
• Loop C: 12+(C-B)+2(C-A)+4C=0

Now i have 3 equations, but 4 unknowns,

So i need another equation coming from the current provided from the current source to get a system of equations and solve it.
But i know that the current source provides a current of 1amp for the loop, however this is where im not sure how to proceed,
does it mean that both A and B = 1 or A=1 only? How do i know which current the source affects?

I proceeded with the assumption that both A and B both equal one, but got an overdetermined system with no solution.

On second though maybe, both currents are related by lets say, the source gives out current = 1amp, so loop 1 and 2 share this current so, currents A+B=1 sort of thing? Proceed by substituting A=1-B into the system of equations and get 3 unknowns 3 equations?

Just need some checking on this fact and the rest of the problem is pretty straight forward.

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On second though maybe, both currents are related by lets say, the source gives out current = 1amp, so loop 1 and 2 share this current so, currents A+B=1 sort of thing? Proceed by substituting A=1-B into the system of equations and get 3 unknowns 3 equations?

Just need some checking on this fact and the rest of the problem is pretty straight forward.