MHB Solving Linear Systems with a Variable: How to Handle Unknown Constants?

[delc1]

Hello,

I am stuck on a question for some time now and I am unsure how to solve this. I have tried the substitution and gaussian elimination methods but have had no luck at all.

Identify the value(s) of k for which the following linear system
x + 2y + z = 2
2x − 2y + 3z = 1
x + 3y + (k^2 − 3)z = k
has:
(i) no solutions
(ii) one solution
(iii) infinitely many solutions

I am particularly having trouble with the variable 'k'. I am not sure how to handle it?

 

[mathmari]

We have the following:

$$\begin{pmatrix}
1 & 2 & 1\\
2 & -2 & 3\\
1 & 3 & k^2-3
\end{pmatrix}\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}=\begin{pmatrix}
2\\
1\\
k
\end{pmatrix}
$$

Applying gaussian elimination we have the following:

(Let $(a)$ be the first line, $(b)$ the second and $(c)$ the third one.)

$$\begin{pmatrix}
1 & 2 & 1 & | & 2\\
2 & -2 & 3 & | & 1\\
1 & 3 & k^2-3 & | & k
\end{pmatrix} \overset{(b)'=(b)-2(a), (c)'=(c)-(a)}{ \longrightarrow } \begin{pmatrix}
1 & 2 & 1 & | & 2\\
0 & -6 & 1 & | & -3\\
0 & 1 & k^2-4 & | & k-2
\end{pmatrix} \overset{(c)''=6(c)'+(b)}{ \longrightarrow } \begin{pmatrix}
1 & 2 & 1 & | & 2\\
0 & -6 & 1 & | & -3\\
0 & 0 & 6k^2-23 & | & 6k-15
\end{pmatrix} $$

When we get from the last line $0=0$ we have infinitely many solutions, so when $6k^2-23=0$ AND $6k-15=0$.

When only one of the two is zero ($6k^2-23=0$ OR $6k-15=0$ but not both) we have no solutions.

In any other case we have one solution.

 

[Uberjew]

We have the following:
When only one of the two is zero ($6k^2-23=0$ OR $6k-15=0$ but not both) we have no solutions.

Wouldn't there only be no solutions when the second last column in the bottom row of this matrix equeled 0?

If we found k such that $6k-15=0$, then one of the equations would be:

0x + 0y + 6k²-23z = 0

and since z can be 0, this would result in a solution?

Every text I've read says it must be when the last non-zero row of [U|c] ends with c, where c != 0, that there is no solutions and nothing about when there is another row which also has an unknown constant, for instance k, which is in both the final column in the last non-zero row and another column in the same row.

Please correct me if I am wrong