Determining if lines are skewed, parallel or intersecting using matrices. Determine whether the following pair of lines is​ parallel, intersecting, or skew. If the lines​ intersect, determine the​ point(s) of intersection.
$$ r(t) = \langle 4+4t,1-8t,5-3t \rangle $$
$$ R(s) =\langle 14+5s, 5+2s, 13 +4s \rangle $$
I am trying to find an alternative way of resolving this problem using linear algebra. I don't know if it is possible to prove that the lines are parallel or skewed with matrices, if that is possible it would be nice to see how the process looks like. my question is: how i can determine whether these two lines are intersecting. 
 A: Guide:
Clearly the problem lines are not parallel since $(4, -8, -3)$ is not a multiple of $(5, 2, 4)$. What remains to check if they intersect.
Hence it suffices to check if 
$$r(t)=R(s)$$ has a solution, which is just $3$ linear equations with $2$ variables, we can of course form a linear system and reduced it to RREF to check if it has a solution.
A: you have
$$
\eqalign{
  & {\bf x} = \left( {\matrix{
   x  \cr 
   y  \cr 
   z  \cr 
 } } \right) = \left( {\matrix{
   4  \cr 
   1  \cr 
   5  \cr 
 } } \right) + t\left( {\matrix{
   4  \cr 
   { - 8}  \cr 
   { - 3}  \cr 
 } } \right)  \cr 
  & {\bf x} = \left( {\matrix{
   x  \cr 
   y  \cr 
   z  \cr 
 } } \right) = \left( {\matrix{
   {14}  \cr 
   5  \cr 
   {13}  \cr 
 } } \right) + s\left( {\matrix{
   5  \cr 
   2  \cr 
   4  \cr 
 } } \right) \cr} 
$$
So the two lines have common points given by the following linear system
$$
\left( {\matrix{
   4  \cr 
   1  \cr 
   5  \cr 
 } } \right) + t\left( {\matrix{
   4  \cr 
   { - 8}  \cr 
   { - 3}  \cr 
 } } \right) = \left( {\matrix{
   {14}  \cr 
   5  \cr 
   {13}  \cr 
 } } \right) + s\left( {\matrix{
   5  \cr 
   2  \cr 
   4  \cr 
 } } \right)\quad  \Rightarrow \quad \left\{ \matrix{
  4t - 5s = 10 \hfill \cr 
   - 8t - 2s = 4 \hfill \cr 
   - 3t - 4s = 8 \hfill \cr}  \right.
$$
and you know that this type of systems in general can have $0$, $1$ or infinite solutions according to whether ....
