@WBahn Thank you for this intricate detail. Wow. And thank you for the privilege of your time. And I apologize on leaving units off my left side equation, they were the ohms*Amps. Note taken.

One last clarifying question. I understand the right side in full now (and how you broke it down with letters - that really helps). However, the left side, with your equations, it seems as though you went counter-clockwise from a to b. I thought we were doing the b-to-a walking approach to get the same result (+20V)

But going from B to A we have to sum the gains. I thought we were going clockwise since we had to pick a direction when doing the "big loop."

Where did I go wrong? Or does the "b-to-a" clockwise approach not work anymore?

I feel like an idiot asking these tedious questions, but I want to be solid. Thank you for your patience and breaking it down. I hope someday I can pay it forward on these forums.

 

So there's three things that got a bit intermingled.

First, the only place where clockwise or counter clockwise was particularly relevant was in writing the loop equation (i.e., applying KVL around a closed loop). When doing that, it is extremely valuable to pick a direction and then sum either the voltage gains or the voltage drops in that direction. This is just to make it a lot harder to mess up the signs of one of the voltages along the way.

Second, when finding Vxy, the voltage at Node 'x' relative to the voltage at Node 'y', this has nothing to do with directions around a loop. Vxy is the voltage drop as we go from Node 'x' to Node 'y' or, equivalently, the voltage gain as we go from Node 'y' to Node 'x'. Those are straight from the definition of the voltage difference between two nodes. The path we take is irrelevant (as long as we are dealing with conservative electric fields, which you are right now).

Finally, when talking about using the double-subscript notation, that is completely independent of traversing a circuit along any particular path in any particular direction. It's merely a convention that says that a double-subscripted voltage means the voltage at the node identified by the first subscript minus the voltage at the node identified by the second node. We can then leverage this to manipulate equations very easily.

If I have

(something1) + Vxz + (something2) + Vzy + (something3) = (something4)

Then I can replace Vxz + Vzy with Vxy.

Essentially, if two voltages being added have a common first subscript in one and second subscript in the other, I can combine them removing the common subscript.

Other things that are useful are the fact that Vxy = -Vyx

For instance, let's say that someone built this circuit and used a voltmeter to determine that Vha = -35 V. I can use this to find Vab very easily.

Vab = Vah + Vhb = -Vha + Vhb

I'm given that Vha = -35 V and, by inspection, Vhb = -15 V. So I have

Vab = -(-35 V) + (-15 V) = 20 V

 

Thanks @WBahn for the very clear explanation. I will keep working at more problems in the hope of getting better. But I thank you for your patience!