Find a perpendicular foot using the plane normal

A perpendicular line through A has direction equal to the plane’s normal:

$l_{AN}:\quad\mathbf r=(1,3,-1)+\lambda(0,-1,2),\quad\lambda\in\mathbb R.$

Since N is on this line, its position is $(1,3-\lambda,-1+2\lambda)$. Since N is also on the plane,

$\begin{aligned}-(3-\lambda)+2(-1+2\lambda)&=5,\\-3+\lambda-2+4\lambda&=5,\\\lambda&=2.\end{aligned}$

Substituting into the line gives $N=(1,3-2,-1+4)=(1,1,3)$, as in the Teacher solution. Check $-1+2(3)=5$ and $\overrightarrow{AN}=(0,-2,4)=2(0,-1,2)$: N lies on the plane and AN is normal to it.