Question

When graphing the following equation, where does the solution plane cross the z-axis?

Answer 1

Your plane is 3x-y+4z=32 and you want to know when it crosses the z-axis or in other words intersects it. Along the z axis only z changes, additionally it goes through the origin (0,0,0).
So we can insert x=y=0 into the plane to calculate the point along the z-axis where it intersects the plane:
3x-y+4z=32
insert 0s:
4z=32
z=8

so the solution plane intersects the z-axis at (0,0,8)