find equation of plane through point and having the same xz trace as plane

Question

I need to find the equation of a plane through point $p(-4,1,6)$ and having the same trace in the $xz$-plane as the plane $x +4y -5z = 8$

I figured the $x$ and $z$ points remain the same, but I'm not sure how to find the correct $y$ value for this equation. The textbook lists the answer as $x + 42y - 5z = 8$, and I am unsure of where the $42$ is coming from.

Answer 1

Note that the trace of the given plane in $xz$-plane ($y=0$) is

$$x-5z=8$$

thus the plane we are looking for has equation

$$x+ty-5z=8$$

by the condition on P we have

$$-4+t(1)-5(6)=8 \implies t=8+4+30=42$$

thus

$$x+42y-5z=8$$

Answer 2

The trace in the $x$-$z$ plane is of course the solution set of the system of equations $$\begin{align} x+4y-5z-8 &= 0 \\ y &= 0, \end{align}$$ but there’s no need to find an explicit solution to this system. Every linear combination of these two equations also includes their intersection. Applying Plücker’s mu, the combination that also passes through the point $(-4,1,6)$ is $$(1)(x+4y-5z-8) - (-4+4\cdot1-5\cdot6-8)(y) = x+42y-5z-8 = 0.$$