Systems with two variables usually include “x” and “y” as the choices for variables.  These relationships compare two separate values in different ways.  Finding similarities in the equations, both geometrically and algebraically, can find one unique, one of several, or the presence of no common solutions.

In these three cases, one solution can be found when, in the y = mx + b format, the two equations have a different slope (m) and intersect at only one point.  No solutions are present when the equations have a different origin, but the same slope, resulting in parallel lines that never intersect.  In the third case, there are infinite solutions because the two equations are identical, thus resulting in the same solutions.

How to identify solutions, if any, using equations with two or more variables.

Systems of Equations