a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

A system of linear equations is pretty much exactly what it sounds like: two linear equations that should be solved at the same time. If we have two linear equations, that also means graphing those lines on the coordinate plane.

Students should know that if two lines intersect on the coordinate plane, the point of intersection corresponds to the values of x and y that work for both equations. They share the same solution, just like a bland sandwich and a smelly person share the same solution: bacon.

Of course, sometimes that doesn't happen—a shared solution just doesn't work out because there are no coordinates that work for both equations at the same time. For example, x + y = 4 and x + y = 5 don't share any solutions. Whatever x and y are, they can either add up to 4 or 5, but not both. The technical name for these kinds of systems is "inconsistent."

Students should understand that the intersection marks the solution to both equations because really, these linear graphs are visual representations of all x and y values that make their respective equations true. If the lines intersect on the graph, the intersection is really the intersection of the solutions to these equations. They share a point because they share a solution.