Physics Teacher's Basic Circuit Analysis Recipe Needs Tweaking

I teach intro, calc-based physics. Below is a terse summary of my recipe for analyzing simple battery-resistor multi-loop circuits. It was working well, until we tried the circuit seen in the attachment. Depending on our selection of simultaneous equations, sometimes our analysis produced correct results, but other times (again depending on our choice of simultaneous equations) didn't work because we ended-up with an inverted matrix that is undefined.



How does the following need to be augmented of fixed?



RECIPE

1. Identify all junctions and arbitrarily label all currents entering/leaving each junction. This sets the number of unknowns.


2. Apply Junction Rule to every junction that produces a different equation.


3. Apply Loop Rule to as many loops as it takes to include each branch (each circuit element) at least once.


4. The number of simultaneous equations you generate needs to equal the number of unknown variables you need to investigate. Use all your Loop Rule equations plus as many Junction Rule equations needed to meet this number. You must use at least one Junction Rule equation.

I'm pretty sure we're correctly handling loop voltage rises/drops across batteries and resistors (with and against the loop direction).



HELP, and thank you, and HELP

Attached Images

circuit.JPG (23.0 KB)