Given two lines to find their intersection. I will fully disclose that this is a homework question. I would prefer not to be given an answer directly, and am looking for more of an indication as to whether I am on the right track. The problem with the courses I am working with is that they only show examples, and do not explain exactly how it "works".

Given $l_1 = (6,-1,0)+t(3,1,-4)$ and $l_2 = (4,0,5)+s(-1,1,5)$ find
  the intersection of $l_1$ and $l_2$.

First I took $d_1 = (3,1,-4)$ and $d_2 = (-1,1,5)$,
Then I made sure that they did not have the same ratio. (if they have the same ratio this would indicate that they are either coincident or parallel) they do not have the same ratio, so they either intersect at a point or are skew.

Then I made parametric equations:
$l_1:$
$$\begin{align}
x & = 6 + 3t\\
y & = -1 + t\\
z & = -4t\\
\end{align}$$
$l_2:$
$$\begin{align}
x & = 4 - s\\
y & = s\\
z & = 5 + 5s\\
\end{align}$$

Then I equated them to eachother:
$$\begin{align}
6 + 3t & = 4 - s\\
-1 + t & = s\\
-4t &= 5 + 5s\\
\end{align}$$

I moved the unknowns to one side:
$$\begin{align}
3t + s & = -2 \qquad & \text{(we'll call this equation $1$)}\\
t - s & = 1 \qquad & \text{(we'll call this equation $2$)}\\
-4t - 5s & = 5 \qquad & \text{(we'll call this equation $3$)}\\
\end{align}$$

This is where it gets tricky. If I take equation $(1)$ and $(2)$, I can cancel out the $s$ value, but the values both become strange, where $t$ is $\frac 34$ and $s$ is $-2 (\frac 34)$, obviously the left and right hand sides don't match.
But I I take equation $(2)$ and $(3)$, the left and right hand sides do match, and then if I go to find the point of intersection I get decimal values for coordinates (why would that be the case?)
Any help would be great. I just want to know what I'm doing wrong. Please don't just give me the answer.
Edit:
I am not sure why people are digging this up to down-vote it, and would appreciate a comment explaining your down-vote. 

Here is another way 
to find the distance
between two lines
in any number of dimensions.
If the lines intersect,
the distance will be zero.
Find shortest distance between lines in 3D
