Remember that a 3 x 3 system is really three planes that are living in 3-D space...  They're actually locked into specific locations (not floating around) and we want to find where they intersect.

But, the three planes might not all intersect in a nice little point.

Two of the planes  might be parallel.

All three planes  might be parallel.

Or this could  happen!

In all of the cases, you won't get a single point where all three planes intersect...  So, there will be no solution.

When you are doing your elimination method, you'll be happily going along...

And you'll add two equations together and get a false statement like

When this happens, you'll know that there is no solution.  For your answer, write something like

"All three planes do not intersect in a single point.  
So, there is no solution."