Find the value of \(k\) for which the set of linear equations
$$\begin{aligned}
x + 3 y + k z & = 4
4 x - 2 y - 10 z & = - 5
x + y + 2 z & = 1
\end{aligned}$$
has no unique solution.
For this value of \(k\), find the set of possible solutions, giving your answer in the form
$$\left( \begin{array} { c }
x
y
z
\end{array} \right) = \mathbf { a } + t \mathbf { b } ,$$
where \(\mathbf { a }\) and \(\mathbf { b }\) are vectors and \(t\) is a scalar.