\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Consider the following system of linear equations, where \(k, m \in \mathbb{R}\):$$\left\{\begin{array}{rcrcrcr} x &+& y &+& z &=& 1 \\ x &+& 2y &+& 3z &=& 4 \\ 2x &+& 3y &+& kz &=& m\end{array}\right.$$
  1. Find the value of \(k\) for which the system does not have a unique solution.
  2. For this value of \(k\), find the value of \(m\) for which the system has infinitely many solutions.
  3. For the values of \(k\) and \(m\) found in parts (a) and (b), find the general solution to the system.

Capture an image of your work. AI teacher feedback takes approximately 10 seconds.