The exceptions are for dependent systems of equations, where eventually one of your equations becomes always true or always false. If it becomes always true, then you throw it out, and from then on, you have fewer equations. If it becomes always false, then you throw the whole system out; it is inconsistent and has no solutions.
Otherwise, you keep the same number of equations and the same number of variables ―that is, even if some individual equations have fewer variables, the system as a whole should keep the same number of variables― until the system is solved.
Here's an example that I originally prepared for my College Algebra courses,
where you are sometimes expected to solve
systems of equations with 3 variables instead of only 2.
(But the basic techniques are the same.)
So this is more complicated
than anything that you'll have to do in this course,
but you can see how I kept track of everything without losing my place.
Or download it: WebM format, Ogg Vorbis format, MPEG-4 format.
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