I have a basic background of calculus 1, and I'm self-teaching linear algebra using Strang's Linear Algebra and its Applications, 3rd ed. Early on, he describes singular cases, one of which is a 3-dimensional line of intersection (drawn myself using Python and PyPlot):
Strang explains that "the three planes have a whole line in common" and I can't get an intuition as to what he means by that. I only see the single point of intersection; how is there a line of solutions for these three vectors?
It looks like you have drawn three lines in the $x,y$ plane with equations \begin{align} y &= 2.5 \cr x+y &= 5 \cr x-y &= 0 \end{align} That intersect in the point $(2.5,2.5)$.
But you can interpret these equations as three planes in $x,y,z$ space, where it happens that $z$ does not appear in the equations. That means $z$ may take any value. Thus the planes intersect along the line where $(x,y,z) = (2.5,2.5,{\rm any value})$.
Generally in three dimensions an equation of a plane can be written in the form $ax+by+cz = d$. Strang is probably giving an examples of what can happen when you have 3 planes. Often they intersect at one point, sometimes in a line, sometimes they are all the same plane, and sometimes they have no point in common as when two of the planes are parallel. I didn't describe every case. The idea is to think of all the geometric possibilities and see how the algebra relates to each case.