I think this discussion got way too involved for explaining anything at
the high school level. Here is my simple approach to answering the
question.
(1) You need as may equations as unknowns you seek. In this case it
sounds as if you are seeking 3 unknowns, so you need 3 equations.
(2) The equations need to be independent.
One obvious non-independent situation is when two equations differ
simply because one is a multiple of the other. For example, 2x + 4y -
5z = 0 is not independent with respect to 6x + 12y -15z = 0 because the
latter is simply 3 times the former. Explain this to the students by
saying that the second equation doesn't give any more information than
the first. Another way to see this is to realize we can get the second
equation from the first (or vice-versa) without the need for any new
information.
A less obvious situation is when you have three equations, two are
independent, but the third can be obtained from the first two by algebra
(without the need for any new information). 2x + 4y -5z = 0, 3x - 4y +
2a = 0, and 5x + 2a -5z = 0 are not independent because the last one is
simply the sum of the first two.
(3) The situation described immediately above is what happened to
Justin's student. How did this happen? When writing loop equations, a
"new" equation will not be independent of equations already obtained if
the new equation lacks a new circuit element that does not already
appear in the existing equations.
Therefore, when looking for new loops, you need to find loops that
contain at least one resistor or battery or some component that has not
already appeared in a loop. A "new" loop for which all components
already appear in existing loop equations will not contain any new
information that cannot be algebraically determined from the
previously-existing loop equations.
Another way to say this is the following... Once you have included every
circuit element in a loop, stop looking for more loop equations because
there won't be any more that are independent from the ones you already
have.
(4) A similar situation exists for junction equations. Once you have
included every current in a junction equation, stop looking for new
junction equations.
An additional pitfall with junctions is for students to think they have
incorporated a new current (not previously appearing in a junction
equation) but it only seems so because they used two different symbols
for the same current. It is common for students to look at one junction
and label the currents I1, I2, I3 and then look at another junction and
label the currents I4, I5, I6 when I5 is actually the same as I3. For
example, if the current into a junction at one end of a resistor is
labeled I3, then the current out of the junction at the other end of the
resistor is also I3 and should not be labeled I5 (or any number other
than 3).
In summary, explain to the students that a new equation must contain at
least one new piece of information (a new component or a new current)
that doesn't already appear in the existing equations.
Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Bluffton College
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu