Describing Solutions to Systems of Linear Equations
Number of Solutions for a System of Linear Equations
When a system of linear equations has two equations and two variables, the system can have zero, one or infinitely many solutions.
No Solution
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 parallel, meaning that they have the same slope and different -intercepts. An example of one such system is
One Solution
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 coincidental, and, as they have the same slope and -intercept, they are different versions of the same line. One example of a system that has an infinite number of solutions is
A system of equations has infinitely many solutions when the equations describe the same line. In other words, the lines must have the same slope and -intercept. The first line has a slope of Thus, the slope of the second line must also be This means, The system can now be written Notice that the -intercepts of the lines have already been given. They are and Since these values are different, and they cannot be changed, the lines are not coincidental. Therefore, the system will never have infinitely many solutions. In fact, since the lines are parallel, the system has no solution.
