Determine the piercing points of…

Determine the piercing points of $\tfrac{x-4}{1}=\tfrac{y-2}{2}=\tfrac{z-1}{-2}$ through the coordinate planes.

Explanation of Method

The points where a line pierces the three coordinate planes.

Parametric form: $x=4+t,\ y=2+2t,\ z=1-2t$.

Plane	Condition	$t$	Point
$xOy$	$z=0$	$\tfrac12$	$\left(\tfrac92,3,0\right)$
$xOz$	$y=0$	$-1$	$(3,0,3)$
$yOz$	$x=0$	$-4$	$(0,-6,9)$

Check, e.g. $(0,-6,9)$: $\tfrac{-4}{1}=\tfrac{-8}{2}=\tfrac{8}{-2}=-4$.

$(\tfrac92,3,0)$ $(0,-6,9)$ $(3,0,3)$

The line pierces $xOy$ at $\left(\tfrac92,3,0\right)$, $xOz$ at $(3,0,3)$ and $yOz$ at $(0,-6,9)$.

\[\begin{aligned} xOy&:\ \bigl(\tfrac92,3,0\bigr)\\ xOz&:\ (3,0,3)\\ yOz&:\ (0,-6,9) \end{aligned}\]