Types of Solutions to Systems of Linear Equations

In this post, we will learn that any system of linear equations has one of three possible types of solution. Each type of solution has distinctive characteristics. These will be presented, so we can understand the type of system or situation we are dealing with when we observe a system of linear equations. The three types of solution are:

1. One unique solution
2. Infinitely many solutions
3. No solution

The type of solution can be determined by characteristics of lines used in analytic geometry. Those are the slope, the y-intercept, and the graph of the lines themselves. These are discussed in more detail below. However, it is important to understand that a system of equations has NO solution when the number of unknowns exceeds the number of equations. Therefore, a linear system that has 2 equations and 3 unknowns cannot be solved. Moreover, systems of 2 linear equations with 2 unknowns will be used in the examples below.

One Unique Solution

A system of linear equations has a unique solution when the graph of the lines intersect at a single point. This point of intersection is the solution of the system. Based on this, it can be inferred that the lines have distinct slope and y-intercept. A system of linear equations with an unique solution is shown below.

The graph generated in a TI-89 Titanium calculator shows the intersection of the lines at point (1, 3). This is the solution of the system of equations.

Infinitely Many Solutions

A system of linear equations has an infinite number of solutions when the graph of the lines are the same. In other words, the equations are the same. They are just written differently. Since the equations are the same, they obviously have the same slope and the same y-intercept. A system of linear equations with an infinite number of solutions is shown below.

The graph generated in a TI-89 Titanium calculator seems to show a single line. However, it shows the graph of both equations. This is because the equations are the same and their graphs touch each other at every point, hence infinitely many solutions. In fact, if the second equation is divided by -3 on both sides, the first equation is obtained.

No Solution

It was discussed previously that systems of linear equations with more unknowns than equations have no solution. Additionally, a linear system has no solution when the graph of the lines are parallel. In other words, they have no intersection point. These lines have the same slope, but different y-intercept. A system of linear equations with no solution is shown below.

The graph generated in a TI-89 Titanium calculator shows two parallel lines. This means the lines will never intersect and the system has no solution.

Conclusion

When dealing with system of linear equations, no matter the size, they will have one of these three types of solution.