This is going to be a different approach.
Please investigate what happens to your system if .
After doing so, try to convince yourself that is the answer.
Find b such that the homogeneous system with augmented matrix below has non-trivial solutions:
So I know a system of equations is homogeneous if all the constant terms are zero, and I know to make a nontrivial solution I need at least one variable to have a nonzero value.
My question is, what is b supposed to be? Is there a single number that it should end up being? Or can it be 1 or 2 or 3, etc.?
Yes, you're getting very close. If b = -12 then the bottom row of that matrix becomes all zeros. The corresponding equation is 0 = 0 (which is automatically satisfied, of course), and we're left with only two genuine equations for three unknowns. So you should be able to find a nontrivial solution for them.
Yes. A homogeneous system of linear equations always has at least one solution, namely when all the variables are zero. That is called the "trivial" solution. Any other solution is "nontrivial" (even if some of the variables are zero, just so long as they are not all zero).