When a has two equations and two variables, the system can have zero, one or infinitely many solutions.
If a system has no solution, its graph might look similar to the graph shown.
Recall that the solution to a system is the point where the lines intersect. If a system has no solution, it must be then that the lines never intersect. In fact, the lines must be meaning that they have the same and different . An example of one such system is
The graph of a system that has one solution might look similar to the graph shown. Specifically, it will show that the lines intersect exactly once. The point of intersection is the solution to the system.
In contrast to parallel lines, lines that intersect once must have unequal slopes. For example, the system shown must have exactly one solutin as the two lines have different slopes.
Infinite Number of Solutions
For a system to have infinitely many solutions, it must mean that the lines intersect at infinitely many points. In fact, it means the lines lie on top of each other.
Such lines are said to be , and, as they have the same slope and y-intercept, they are different versions of the same line. One example of a system that has an infinite number of solutions is