Infinitely many solutions vs one solution vs no solution in systems involving an unknown constant Just need a little clarification in case my assumptions are incorrect. If I were to have the matrix  
{{1, 3, 3}, {2, 7, 6}, {1, 4, k^2 - 18}} * {x1, x2, x3} = {1, 3, k+1}  
which has a row echelon form of: 
 {{1, 3, 3}, {0, 1, -12}, {0, 0, 1}} * {x1, x2, x3} = {1, 3, (k - 3)/(k^2 - 9)} , didn't want to reduce any further to avoid dealing with that ugly fraction. 
would it be the case where k = 3, -3 that would be no solution? I mean logically I suppose so but in terms of linear algebra I thought no solution required an inconsistency and not an undefined point. Or would an undefined k value make for an inconsistency? 
for one solution is it right to assume any unique result? suppose I chose k = 0, then would the unique solution be whatever I get from plugging in? And if so is it safe to assume that we have infinitely many unique solutions so long k does not equal 3, -3? 
and for infinitely many solutions is it again the case where any result so long k does not equal 3, -3?  
I would really appreciate a response with an explanation that can be understood. Thank you.  
oh and how would k = 3 play into the solutions? since if we were to simplify (k -3)/(k^2 - 9) we would end up with 1/(k+3) in which case k = 3 fails for the former but is valid for the latter.
 A: This is a case where complete row reduction is preferred, and leads to different values of $k$ to avoid: The complete row reduction leads to
$$x_1=\frac{3-3k}{k^2-21}-2,\\
x_2=1,\\
x_3=\frac{k-1}{k^2-21}.$$
So the only cases where the solution might not be unique are when $k = \pm \sqrt{21}.$ A good way to deal with these, which avoids the "zero divide" in the above equations, and issues of undefined answers, is just to substitute $k=\pm \sqrt{21}$ into the original matrix having the formulas in it, and row reduce each. Neither has any solution, i.e. both particular systems are inconsistent, in this case.
ADDED: I checked, and found your not-completely reduced matrix is off. It should have row 1 as $[1,3,3,1]$, row 2 as $[0,1,0,1]$, and the crucial row 3 as 
$$[0,0,1,(k-1)/(k^2-21)].$$
Thus you would be led to the same question about the only exceptions possible being $k=\pm \sqrt{21}$ which come from the completely reduced matrix. So the same course of action would be suggested, namely to substitute $k=\pm \sqrt{21}$ into the original system, obtaining matrices with only numbers, which would then be row reduced in the usual way to see if consistent. Note that before division the last row reads as 
$$[0,0,k^2-21,k-1].$$
So even here, without going back to the original system, you can see directly that if $k=\pm \sqrt{21}$ the last row reads $[0,0,0,(\pm \sqrt{21}-1]$ so the last entry is nonzero, making the system inconsistent. This avoids the plugging in of the radical expression for $k$ into the original system.
