Answer
$P = (5,6,5)$
Work Step by Step
$l_1: (x.y.z) = (2,3,-1)+n(1,1,2)$
$l_2: (x.y.z) = (7,7,2)+r(-2,-1,3)$
Let $P = (x,y,z)$.
The x-coordinate of $l_1$ and $l_2$ must both equal $x$:
$x = 2+n(1) = 7+r(-2)$
$n = 5-2r$
The y-coordinate of $l_1$ and $l_2$ must both equal $y$:
$y = 3+n(1) = 7+r(-1)$
$n = 4-r$
We can equate the two expressions of $n$ to find $r$:
$5-2r = 4-r$
$r = 1$
We can use $l_2$ to find $P$:
$P = (x,y,z) = (7,7,2)+r(-2,-1,3)$
$P = (7,7,2)+(1)(-2,-1,3)$
$P = (7-2,7-1,2+3)$
$P = (5,6,5)$
