I am trying to understand how to determine the difference between no solutions for a system and infinitely many solutions.

I have the set of equations:

x + 2y - z = 1

2x + y + z = 1

x - y + 2z = 1

To start, I make sure the determinants do not equal zero.

They do indeed equal zero so there are no solutions.

My next system is :

u + 2v -w = 2

2u + v + w = 1

u - v + 2w = - 1

Again, I check and the determinants equal zero. So, I assume that there are no solutions.

However, my textbook states that for this second system there are infinitely many solutions where u = - w, v = w + 1 and w can have any value. Please explain the difference to me!!!! Thanks Frostking