I found a plane given a point and two contained parametric lines and got a correct answer...
$14(x+2) + 7(z-1) = 0$
However the solution, in addition to my answer, listed
"... equivalently $2x+z+3=0$ "
Why are these planes equivalent? I see that you can expand this, simplify, and factor out a 3, but when I think about the component directions (e.g. $14x$ vs. $2x$) it doesn't make sense to me.
Why are the coefficients able to be factored?
EDIT:
I understand that $\left\langle14,7\right\rangle$ is the same direction as $\left\langle 2,1 \right\rangle$, but if you were to graph the planes in my example you would see the following:
link to image (not enough reputation points :/)
Please provide your reasoning, hopefully I'm not overthinking this :D