A System Is a Transformation Run Backwards
Solving systems of equations — at school you did it mechanically, elimination and substitution, two equations and you cancel out x and y. But you probably never got a feel for what it's actually doing. Here's the real question a system asks: "I applied transformation A to some input and got b. What was the input?" So you're rewinding the transformation. Lesson 5 said a matrix is a transformation; lesson 6 said that transformation can crush space flat. Those two ideas meet right here. When a system is solvable and when it isn't turns out to be exactly that picture.
Start with the equation Ax = b as a picture. A is the transformation, x is the input we don't know, b is the result. Apply A to the plane and every point goes somewhere. The one point that lands exactly on b — that's the x we're after. Drag a candidate point and push it through A. When the result of applying A sits right on b, that's your answer. Solving a system is, in the end, working backwards to find the input that arrives at b.
You can't just guess candidates every time, though. There's a smarter way. If A bent space, then there's a transformation that bends it exactly back. That's the inverse, A⁻¹. Watch: apply A and the grid warps, then apply A⁻¹ and the grid snaps right back to where it started. So the answer is x = A⁻¹b. Run b once through the reversing transformation and you land on the starting point x. That formula for solving a system was a "rewind button" all along.
But is there always exactly one answer? If the transformation didn't crush space — that is, if the determinant from lesson 6 isn't 0 — then yes. When the transformation spreads points out evenly, each result b has exactly one input landing on it. Move b around and watch the unique x follow it every time. A reversible transformation always has exactly one answer. That's the real meaning of "if det isn't 0, it solves cleanly."
Now drive the determinant to 0. As you saw in lesson 6, the plane gets crushed onto a single line. And here something strange happens. If the result b is not on that line? Nothing lands there at all — there's no solution. If b is on that line? A whole crowd of inputs got mashed onto that one spot, so there are infinitely many solutions. Drag b on and off the line. Either nothing, or far too much. That scary "no inverse" from lesson 6 shows up here, identically, as "no clean answer."
Finally, let's connect it to the picture you learned at school. Graph the two equations of a system and you get two straight lines, and the answer is where they cross. That "two lines crossing" picture is the very same thing as the "reverse the transformation" picture we just saw. And when the determinant is 0? The two lines are parallel and never meet (no solution), or they're the very same line and meet everywhere (infinitely many). A crushed transformation was parallel lines all along. Watch the two pictures turn out to be one and the same.