Ah, now that makes sense, especially when you frame the loop currents having their respective operators and the actual calculated loop currents with their own polarities (either positive or negative). But then, by that token, we must've set up the current loop equation for Loop z wrong, because while Loop z is drawn in the opposite direction of i3, the i3 would be set up as: *i3 = iy + (-iz) = (0.444 A) + (-1.33 A) = negative*, indicating that my assumption of iz direction was wrong, right?

Sort of...you would have known your assumption for Iz was wrong the second it popped out of the calculation as a negative number. if you need to display it as an answer then you can correct the direction at that point if required, but if you need to continue using the value in other calcuations in the mesh analysis then...

That must be the curveball you've been highlighting I think, which means I need to re-draw and re-calculate iz and iy in their correct directions to get their correct polarities and ultimately the right i3 value?

...you don't need to recalculate or re-draw if your assumption was wrong. Just continue working with all the values as is. This is probably best anyways since it will just cause confusion in your conventions if you make correct direction assumptions here and there. If you were consistent it should carry all the way through to the end. That's kind of why you treat it mechanically and blindly like a computer. The polarity of your calculated voltages and currents will work together to cancel out any wrong assumptions you made in their direction on your drawing. Whenever you combine the two together, then you know the actual direction but you only need to do this when you display the final result so the person reading it has an easier time.

You have to make a distinction between the polarity as drawn (direction of the currents that you assumed) from the polarity of the calculated value of the current (the actual polarity of the currents

**relative to your assumption)**. The calculated value and polarity plug into the variable and its polarity which represents the assumption of direction that you made. Only together can they tell you the actual direction (relative to what you have drawn).

If you drew I3 to point down on the drawing, Iy CW and Iz CW, then your equation would be

I3 = Iz - Iy

Drawing I3 like this and writing out this equation is the same as saying:

"I assume I3 is in the same direction as Iz and opposite direction of Iy,

as drawn (not in actuality)"

"I assume I3 is in the same direction as Iz, assuming that my direction assumption for Iz was correct."

"I hereby define down as the positive direction for I3 which just so happens to be the same definition that I have given Iz, both of which are the opposite definition given to Iy, which is defined as up being positive."

But then you might end up calculating that Iz = 3A, Iy = -1A.

That means that your assumption for Iz's direction was right, but your assumption for Iy was wrong. If you need to display Iy at this point you can draw a little schematic with the arrow in the proper direction for Iy and use +3A as the value. But if you need to use Iy for another calculation, then don't use the corrected value. Continue using what you originally defined and calculated with your conventions:

I3 = Iz - Iy = (3A) - (-1A) = 4A.

Since it is positive, that means your assumption of I3 (the down arrow) was correct

But if you got Iz = -1A and Iz = 3A

Then I3 = Iz - Iy = (-1A) - (3A) = -4A

Since I3 is negative, that means your down arrow for I3 was wrong and it actually flows in the opposite direction (up).

It's super mindless once you grasp it. You just blindly work through the math and stay consistent.