Linear algebra Plane and line I have a plane with points: $ A = (-5, 1, 3), B = (-2, 6, -3), C = (-4, 3, 4)$
I also have a line with points: $ D = (-2, 7, 6), E = (-1, 8, -2)$
The vector equation form of the line:
$$
\left\{
   \begin{array}{c|}
      \begin{pmatrix}
        -2\\
         7\\
         6
      \end{pmatrix}
     + r         \begin{pmatrix}
         1\\
         1\\
         8
      \end{pmatrix}
\end{array}
r \in \Bbb R
\right.
$$
The vector equation form of the plane:
$$
\left\{
   \begin{array}{c|}
      \begin{pmatrix}
        -5\\
         1\\
         3
      \end{pmatrix}
     + s  
       \begin{pmatrix}
         3\\
         5\\
         -6
      \end{pmatrix}+t
    \begin{pmatrix}
         1\\
         2\\
         1
      \end{pmatrix}
\end{array}
s,t \in \Bbb R
\right.
$$
I am supposed to compute the intersection between these two.
What I know about intersection is that if I have a linear equation with no solution, then there is no intersection. If i have a Linear equation with one solution, then there is one intersection. If i have a linear equation with infinitely many solutions, then the line is contained within the plane.
I the answer is that the line is contained within the plane. I know the answer, but how is the solution squired is beyond me.
 A: Hint: We have that $\vec{x}$ lies in the intersection iff:
$\vec{x}=\begin{pmatrix}
        -2\\
         7\\
         6
      \end{pmatrix}
     + r         \begin{pmatrix}
         1\\
         1\\
         8
      \end{pmatrix}$ and $\vec{x}=\begin{pmatrix}
        -5\\
         1\\
         3
      \end{pmatrix}
     + s  
       \begin{pmatrix}
         3\\
         5\\
         -6
      \end{pmatrix}+t
    \begin{pmatrix}
         1\\
         2\\
         1
      \end{pmatrix}$. 
This leads to: $$\begin{pmatrix}
        -2\\
         7\\
         6
      \end{pmatrix}
     + r         \begin{pmatrix}
         1\\
         1\\
         8
      \end{pmatrix}=\begin{pmatrix}
        -5\\
         1\\
         3
      \end{pmatrix}
     + s  
       \begin{pmatrix}
         3\\
         5\\
         -6
      \end{pmatrix}+t
    \begin{pmatrix}
         1\\
         2\\
         1
      \end{pmatrix}$$ This is a system of 3 linear equations with 3 unknowns. Can you take it from here?
