I think the answer in the back of the book is wrong. I have the following matrix
$$ \left[ \begin{array}{ccccc|c} \blacksquare&*&*&*&*&*\\ 0&\blacksquare&*&*&0&*\\ 0&0&0&0&\blacksquare&0\\ 0&0&0&0&*&\blacksquare \end{array} \right] $$
The black boxes are nonzero numbers, and the asterisks can be anything. The back of the book says there may be a solution, and if so, there will be infinite solutions.
If I make the bottom right black box equal to 1, then I don't see how we can get a solution.
Yes you are right, indeed from the third equation the $5^{th}$ unknown must be equal to zero and thus the black box in the last row should be equal to zero.