Geometric description of linear equation 
Consider the system of equations. Give a geometric description of the
  intersection of the three planes when k=2 and k=0
eq.1) $x+2y-z=-3$
eq.2) $3x+5y+kz=-4$
eq.3) $9x+(k+13)y+6z=9$

My answer: 
The system will be inconsistent when k=2 or k=0. When k=2, eq.2 and eq.3 will be in parallel intersecting eq.1. When k=0...this is where I'm lost
 A: For $k = 0$ equation 2 says
$$
3x + 5 y = -4 \iff \\
\frac{3}{\sqrt{34}} x + \frac{5}{\sqrt{34}} y = -\frac{4}{\sqrt{34}} \iff \\
n \cdot(x - x_0, y - y_0, z - z_0) = 0 \Rightarrow \\
n \cdot (x,y,z) = n \cdot (x_0, y_0, z_0)
$$
This is an affine plane with normal vector
$$
n = \left( \frac{3}{\sqrt{34}}, \frac{5}{\sqrt{34}}, 0 \right)
$$
and position vector $p$
$$
p = (x_0, y_0, z_0) = \left(-\frac{4}{3}, 0, 0\right)
$$
We can do this with two other planes as well and end up with three normal vectors, where every two $n_i$ and $n_j$ of them are not parallel or antiparallel $n_i \cdot n_j \ne 1 \wedge n_i \cdot n_j \ne -1$, thus have a line $g_{ij}$ as intersection.
We need then to analyze the intersection of $g_{ij}$ and plane $k$.
A quick plot seems to indicate that the three pairwise intersections  $g_{ij}$ are parallel.

This would mean
$$
\emptyset 
= g_{ij} \cap g_{ik} 
= (E_i \cap E_j) \cap (E_i \cap E_k)
= E_i \cap E_j \cap E_k
$$
The direction vectors of the $g_{ij}$ are
$$
d_{ij} = \frac{n_i \times n_j}{\lVert n_i \times n_j \rVert}
$$
it is orthogonal to both $n_i$ and $n_j$.
The symbolical calculation is lengthy, but the numerical calculation gives
$$
d_{12} 
= d_{13} = d_{23} 
= (0.84515425472852, - 0.50709255283711, - 0.1690308509457)
$$
A vector $u$ common to plane $E_i$ and $E_j$ must fulfill
$$
0 = n_i \cdot (u - p_i) = n_j \cdot (u - p_j)
$$
so we need
$$
t_i d_{ij} = u - p_i \wedge t_j d_{ij} = u - p_j \Rightarrow \\
u = p_i + t_i d_{ij} = p_j + t_j d_{ij} \Rightarrow \\
(t_j - t_i) d_{ij} = p_i - p_j
$$
for some real numbers $t_i$ and $t_j$.
We have $p_1 = (-3,0,0)$, $p_2 = (-4/3, 0, 0)$ and $p_3 = (1,0,0)$ and note
that $p_i - p_j = (c, 0, 0)$ with non-vanishing $c$. So $p_i - p_j$ can not be a scalar multiple of $d_{ij}$, which has non-vanishing $y$- and $z$-components. So there is no common solution.
