Passage: A line L passing through the point (-1,2,-3) is...

The line $L$ is perpendicular to the plane $2x+3y+z+5=0$, so its direction ratios are proportional to the normal vector of the plane, which is $(2, 3, 1)$.

The equation of the line passing through $(-1, 2, -3)$ with direction ratios $(2, 3, 1)$ is given by:

$\dfrac{x+1}{2} = \dfrac{y-2}{3} = \dfrac{z+3}{1}$

Multiplying the entire equation by $6$, we get:

$3(x+1) = 2(y-2) = 6(z+3)$

$3x+3 = 2y-4 = 6z+18$

Answer: $3x+3=2y-4=6z+18$