SOLUTION: Working with matrices and thought I had a handle on them. But this one's got me puzzled. The last row does not appear to be valid...how can zeroes = 1? Here's the problem Con

Yes because: going left to right, the VERY FIRST non-zero element 
in each row which isn't all zeros has these properties:

A. It is a 1.

B. It has no elements directly above it or below it other than zeros. 

C. It is farther to the right than any first non-zero element which 
   may be above it.

D. If any row has only zeros then it is at the bottom.

There is no solution because, as you have discovered,
the bottom row translates to the following equation, 
where the 5th variable is  

     

which has no solution, since no matter what value you
substitute for  the left side will always be
0 and the right side is 1, and zero can never equal 1.
So the solution set is the empty set which is either
written  or 

But it is in rref, which means row reduced echelon form!

Edwin