I understand how to compute augmented matrices, but I am not understanding this specific answer.

Please look here:

Linear Algebra and Its Applications (9780321982384), Pg. 48, Ex. 4 :: Homework Help and Answers :: Slader
I do not understand the answer as to why that is like that. (I know trivial means where the only solution to Ax=0, is where x=0, and non-trivial exists where Ax=0, and x does not = 0.

how is x3 free?

Free implies where there are all 0's in the row does it not?

My buddy told me that the x3 is free because "the third row is missing." That does not make any sense to me, this problem was given 2 equations with 3 variables in each equation, denoting only 2 ROWS.