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@Jason Grout: 1) "..it is assumed that you can divide and things aren't zero...". Yes. Exactly that is what happens.
2) "..This happens in other math software systems too..". Yes again, eg. also in Mathematica:
In[4]:= RowReduce[{{1, 1, 2, b1}, {1, 0, 1, b2}, {2, 1, 3, b3}}]
Out[4]= {{1, 0, 1, 0}, {0, 1, 1, 0}, {0, 0, 0, 1}}
