Hi ctlawson, it doesn't matter which loops you choose. Your goal with 'choosing loops' is to create a collection of equations on which you can then do algebra to figure out the unknowns. You need as many equations as you have unknowns. Each loop is an example of Conservation of Energy, in that the total gain and loss in potential around a closed loop must equal zero. You begin the loop at some point, travel along, and then return to where you started, but if in doing so you arrived with a different amount of energy than when you started, then some energy must have been created or destroyed along the way, which is a violation of Conservation of Energy. Anyhow, I say all this to try and be convincing that the choice of loop doesn't matter. The discrepancy between our answers could be due to intermediate rounding error, or check that you included the $1.0 \Omega$ resistors, or something like that.

All the best,
Mr. Dychko

Hi bmuniz8219, that's a good question. Each current "starts" at a junction. While the diagram draws $I_1$ to the left along the very top, it's understood (even though not explicitly drawn) that the current originates at the junction below the $12\Omega$ resistor where the $I_2$ and $I_3$ currents also meet. So $I_1$ initially goes up through the $12\Omega$ resistor, and THEN, left along the top.

All the best with your studies,
Mr. Dychko

Additionally when solving for I1 should the addition of the two I1 coefficients not yield 2.5032?

Ah nevermind I realized that I was missing the one I1 on the left side. Thank you