Q.Find the point where the line $\dfrac{x-2}{1} = \dfrac{y}{-1} = \dfrac{z-1}{2}$ meets the plane $2x + y + z = 2$.

Parametrizing the line and substituting into the plane equation gives the parameter value, hence the point of intersection.

Parametrize the line: $\dfrac{x-2}{1}=\dfrac{y}{-1}=\dfrac{z-1}{2}=t$

So $x=2+t$, $y=-t$, $z=1+2t$.

Substitute into the plane $2x+y+z=2$:

$2(2+t)+(-t)+(1+2t)=2$ …

Parametrize the line, substitute into the plane equation to find the parameter, then back-substitute for the point.

Line: $\dfrac{x-2}{1}=\dfrac{y}{-1}=\dfrac{z-1}{2} = t$ (say).

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

Plane: $2x+y+z=2$. Substitute:

$2(2+t) + (-t) + (1+2t) = 2$

$4+2t-t+1+2t = 2$

…