The next example illustrates the possibilities for the solution set of a system of linear equations. We will not be too formal here, and the necessary theorems to back up our claims will come in subsequent sections. So read for feeling and come back later to revisit this example.
This example exhibits all of the typical behaviors of a system of equations. A subsequent theorem will tell us that every system of linear equations has a solution set that is empty, contains a single solution or contains infinitely many solutions (Theorem PSSLS). Example STNE yielded exactly two solutions, but this does not contradict the forthcoming theorem. The equations in Example STNE are not linear because they do not match the form of Definition SLE, and so we cannot apply Theorem PSSLS in this case.