Q.Find the coordinates of the point where the line $\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z-2}{4}$ crosses the $XY$-plane.

Let $\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z-2}{4}=t$. The line meets the $XY$-plane where $z=0$:

$\frac{0-2}{4}=t \quad\Rightarrow\quad t=-\frac12.$ …

Recall that the $XY$-plane is exactly the set of points with $z=0$ (Section 2); set the $z$-expression in the symmetric form equal to zero to …

Setting $x=0$ or $y=0$ instead of $z=0$ (confusing which coordinate vanishes on the $XY$-plane); sign …