For what values of $k$ is the set linearly independent? For what values of $k$ is the following set linearly independent?
$$
        \begin{bmatrix}
        1 & 2 & 1 \\
        0 & -1 & 3 \\
        2 & 3 & k \\
        \end{bmatrix}
$$
I reduced it down to
$$
        \begin{bmatrix}
        1 & 0 & 7 \\
        0 & 1 & -3 \\
        0 & 0 & -5+k\\
        \end{bmatrix}
$$

If $k=5$ then your last matrix is
$$
\begin{bmatrix}
1 & 0 & 7 \\
0 & 1 & -3 \\
0 & 0 & 0
\end{bmatrix}$$
which means that its rank is $2$ and the rows of this matrix (and also of the original matrix) are linearly dependent.
If $k\ne5$, you can divide the third row by $(k-5)$ (since $k-5\ne0$ in this case) to get
$$ 
\begin{bmatrix}
1 & 0 & 7 \\
0 & 1 & -3 \\
0 & 0 & 1
\end{bmatrix}$$
This matrix is in row echelon form. It has three non-zero row. So the row is equal to $3$ and the rows are linearly independent. (And the same is true for the original matrix.) 
