In post #9, you've shown an LTSpice simulation, but you didn't say whether you believe it's correct. Do you?

Looking at the original circuit, the left hand mesh shows that V1 = 20 + 7*I3. The right hand mesh shows that I3 = 5*V1. Making the obvious substitution, we get V1 = 20 + 5*7*V1; from this we get V1 = -20/34=-.588235

This is so simple that surely it is not mistaken, but this isn't what LTSpice gets in post #9. Is there something I'm not understanding?

I do not think it is correct, and I agree with you. The circuit does seem to be the same in the picture posted above though.

 

The current in B3 and V3 is going in the same direction. LTSpice's convention (conventional current, not electron current) is that the current that flows out of the + terminal of a voltage source carries a negative sign.

Then my next question would be, what do you get if you change the definition of B1 to be V=-7*I(V3) and of B2 to be V=-2*I(V3) in the simulation of post #9? I think that's what is needed so that the sign of the current out of V3 matches the direction of I3 in the original circuit.

 

TINA agrees with the hand calculations.

 

First, to clarify how LTSpice reports currents through a voltage source, here are the four cases of stimulus and response:



In light of the above, I added a zero volt source V2 just to monitor the current that flows clockwise in the right-hand mesh. Now I get the same answer as the Electrician.



In light of the previous discussion, here is the simplified circuit:



Interesting, If I simplify even further, it finds the other solution. I'm guessing that LTSpice has a problem because B1 is a function of G1 is a function of B1 is a function of G1 is a ....