find the value of k such that the line x-1 = y-2 = z-k and the plane
3 1 5
2x-y-z+10 = 0 intersect only at one pin .
we have that
$\displaystyle \frac{x-1}{3}=\frac{y-2}{1}=\frac{z-k}{5}$
setting each equal to t we can get the parametric form.
$\displaystyle \frac{x-1}{3}=t \iff x=3t+1$
doing the same for the others gives
$\displaystyle y=t+2 \mbox{ and } z=5t+k$
plugging these into the equation of the plane gives
$\displaystyle 2(3t+1)-(t+2)-(5t+k)+10=0 \iff -k+10 =0 \iff k=10$