During linear algebra class, I was explaining that given 2 equations + 2 unknowns where we expect there to be a unique solution but sometimes there can be 0 solutions or a line's worth.

At some point I started counting configurations of 3 lines in the plane. We expect three different configurations with three intersection point for the three pairs of lines. I wonder which configurations are the next order down in degeneracy.

My gut is that setting a line parallel to another and having all three intersect at a point should count the same, since they can be both deformed by a single line.

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Theory of simultaneous equations is intersection theory in the plane. Can this hierarchy of degenerate configurations be interpreted geometrically with Schubert calculus?