This lesson serves two main goals. The first goal is to revisit the idea (first learned in middle school) that not all systems of linear equations have a single solution. Some systems have no solutions and others have infinitely many solutions. The second goal is to investigate different ways to determine the number of solutions to a system of linear equations.

Earlier in the unit, students learned that the solution to a system of equations is a pair of values that meet both constraints in a situation, and that this condition is represented by a point of intersection of two graphs. Here, students make sense of a system with no solutions in a similar fashion. They interpret it to mean that there is no pair of values that meet both constraints in a situation, and that there is no point at which the graph of each equation would intersect.

Next, students use what they learned about the structure of equations and about equivalent equations to reason about the number of solutions. For instance, students recognize that equivalent equations have the same solution set. This means that if the two equations in a system are equivalent, we can tell—without graphing—that the system has infinitely many solutions. These exercises are opportunities to look for and make use of structure (MP7).

Likewise, students are aware that the graphs of linear equations with the same slope but different vertical intercepts are parallel lines. If the equations in a system can be rearranged into slope-intercept form (where the slope and vertical intercept become "visible"), it is possible to determine how many solutions a system has without graphing.