Answer
$P = (12,-15,28)$
Work Step by Step
$l_1: (x.y.z) = (2,0,3)+n(2,-3,5)$
$l_2: (x.y.z) = (4,1,-4)+r(-1,2,-4)$
Let $P = (x,y,z)$.
The x-coordinate of $l_1$ and $l_2$ must both equal $x$:
$x = 2+n(2) = 4+r(-1)$
$2n = 2-r$
$n = \frac{2-r}{2}$
The y-coordinate of $l_1$ and $l_2$ must both equal $y$:
$y = 0+n(-3) = 1+r(2)$
$-3n = 1+2r$
$n = \frac{-1-2r}{3}$
We can equate the two expressions of $n$ to find $r$:
$\frac{2-r}{2} = \frac{-1-2r}{3}$
$6-3r = -2-4r$
$r = -8$
We can use $l_2$ to find $P$:
$P = (x,y,z) = (4,1,-4)+r(-1,2,-4)$
$P = (4,1,-4)+(-8)(-1,2,-4)$
$P = (4+8,1-16,-4+32)$
$P = (12,-15,28)$
