Find the intersection of these two planes. 
Find the intersection of $8x + 8y +z = 35$ and
$x  =  \left(\begin{array}{cc} 6\\
 -2\\ 3\\ \end{array}\right)
 +$ $ \lambda_1 \left(\begin{array}{cc}
 -2\\ 1\\ 3\\ \end{array}\right)
 +$ $ \lambda_2 \left(\begin{array}{cc} 1\\ 1\\
 -1\\ \end{array}\right) $

So, I have been trying this two different ways. One is to convert the vector form to Cartesian (the method I have shown below) and the other was to convert the provided Cartesian equation into a vector equation and try to find the equation of the line that way, but I was having some trouble with both methods.
Converting to Cartesian method:
normal = $
\left(\begin{array}{cc}
-4\\
1\\
-3\\
\end{array}\right)
$
Cartesian of x $=-4x + y -3z = 35$
Solving simultaneously with $8x + 8y + z = 35$, I get the point $(7, 0, -21)$ to be on both planes, i.e., on the line of intersection.
Then taking the cross of both normals, I get a parallel vector for the line of intersection to be $(25, -20, -40)$.
So, I would have the vector equation of the line to be:
$
\left(\begin{array}{cc}
7\\
0\\
-21\\
\end{array}\right)
+$
$\lambda
\left(\begin{array}{cc}
25\\
-20\\
-40\\
\end{array}\right)
$
But my provided answer is:
$
\left(\begin{array}{cc}
6\\
-2\\
3\\
\end{array}\right)+
$
$ \lambda
\left(\begin{array}{cc}
-5\\
4\\
8\\
\end{array}\right)
$
I can see that the directional vector is the same, but why doesn't the provided answer's point satisfy the Cartesian equation I found?
Also, how would I do this if I converted the original Cartesian equation into a vector equation? Would I just equate the two vector equations and solve using an augmented matrix? I tried it a few times but couldn't get a reasonable answer, perhaps I am just making simple errors, or is this not the correct method for vector form?

Note: In this case it might be quicker to "plug in"
$$
8(6-2\lambda_1+\lambda_2)+8(-2+\lambda_1+\lambda_2)+3(3+3\lambda_1-\lambda_2)=0
$$
which means $\lambda_1=-41-13\lambda_2$ and then you plug that into the original line equation.
