Give a geometric description of the intersection of three planes

Question

I had equations and expressed them as an augmented matrix. Shown below:

$$ \left[ \begin{array}{ccc|c} 1&2&-1&-3\\ 3&5&k&-4\\ 9&k+13&6&9\\ \end{array} \right] $$

I then row reduced the system to this.

$$ \left[ \begin{array}{ccc|c} 1&2&-1&-3\\ 0&1&-k-3&-5\\ 0&0&k^2-2k&5k+11\\ \end{array} \right] $$

I found out that it has no solutions when $k = 0$ and $k=2$, a unique solution for any real number other than $0$ and $2$ and that it isn't possible to have infinitely many solutions.

The very last part of the problem asks:

(i) Each of these equations represents a plane. In each case (no solutions, a unique solution and infinitely many solutions) give a geometric description of the intersection of the three planes.

I am stuck on this bit. Any help appreciated.

Answer 1

When $k=0$, none of the planes are parallel to each other. Any two of them intersect at a line but three of them do not intersect together.

Guide:

When $k=2$, the second and third equation have their left hand side being multiple of each other, try to interpret that.