Summary
In this chapter, we covered a lot of ground! We began by taking the familiar idea of a linear equation in two variables and demonstrated that the set of points that satisfy the equation are exactly those that form a straight line. We then extended this to a system of two linear equations of two variables, which represent, geometrically speaking, two lines. A solution to the system is a point that satisfies not one, but both equations. Geometrically, this means a solution can only be a point of intersection of the two lines. As we know from elementary geometry, two lines must either be parallel, intersect, or coincide entirely. This characterizes three possible conclusions about solutions: a system must have no solutions (if they are parallel), one unique solution (if they intersect), or infinitely many solutions (if they coincide).
Then, the real fun started as we introduced systems of many linear equations and many unknowns, which are not so easily interpretable from a...