1. What is the geometric sense of a linear system in two variables and a linear system in three variables?

A system with two variables has a linear equations that determines a line on the x and y plane. Since the equations are linear they describe lines. Because a solution to a linear system must satisfy all the equations, the solution is the intersection of these lines. This could be either a line, a single point, or the empty set. For three variables each linear equation determines a plane in a 3D space, and the solution set is the intersection of these planes. This means the solution may be a plane, a line, single point, or an empty set.