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Answer to one of the questions : try the 3D plotting.

I don't clearly understand the other questions : "geometric interpretation of linear algebra ?" ... it looks like a math question (not related sagemath). Geometry and linear algebra are both parts of "mathematics. They are linked sometimes but not always. For example, if you use polar coordinates instead of cartesian coordinates, you have no more "linear" algebra.

Your words are inaccurate shortcuts : there is no "difference" between to planes. There are geometric objects which doesn't exist in real life : a line has no width, a Moebius strip has only one "face"... But OK, you mean "intersection" of two sets, the intersection of two sets of points elements in the 3D space (the "geometric interpretation", in fact a bijective map, of R^3 the set of real triples).

If you want to work in cartesian coordinates (R^3 for the real world space) then yes, you have to code using symbolics functions. For example : var('x,y,z"); f1(x,y,z) = x + y + z ; f2(x,y,z) = x + y -2*z ; res = solve([f1 == 0,f2 == 0],x,y,z). The solve() function returns values for x,y,z and here, because intersection of two planes is usually a line (could be a plane or an empty set of points too), you will get a parametrized answer.

"why are ... have ..." : two verbs in one sentence. In geometry, there are "axioms" which are sentences you cannot prove. So no answer for that "why" question or the philosophical : "faith" is not "truth". One of geometry axioms is Euclid's : two parallel lines have no intersection.

"Invertible matrixes have solutions" : not at all, it's a crazy verbal mix (read again what you type). Mix of: "solutions set of a system of equations" (for example the empty set of (x,y,z) triples if your two planes are parallel) and "a matrix is invertible or not".