Types of Systems of Equations

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson will help you become familiar with the different types of systems of equations. After a quick review of the definition of a system of equations, we will look at each of the different types of systems and how to identify them.

System of Equations

Before we get to the different types of systems of equations, let's quickly review the definition of a system of equations and their solutions. A system of equations is a group of two or more equations with the same variables. A solution to a system of equations consists of the values of the variables that make all of the equations in the system true. A solution to a system of equations is also a point of intersection on the graphs of the equations in the system. A solution set of a system of equations is the set of all solutions to the system.

Let's consider an example to illustrate this. Suppose you are having a party as a fundraiser for your favorite charity. You charge $10 for an adult ticket and $5 for a child's ticket. About 258 people show up, and you make $2,150. You can use this information to figure out how many adults and how many children were at the party.

If you let a equal the number of adults at the party and c equal the number of children at the party, then it must be the case that a + c = 258, since there were a total of 258 people. Also, you brought in 10a dollars from the adult tickets and 5c dollars from children's tickets. Since you brought in $2,150 all together, 10a + 5c = 2,150, which gives us the following two equations:

a + c = 258

10a + 5c = 2,150

This is a system of equations using the variables a and c. If we plug in 172 for a and 86 for c, we see that both of the equations are true.

172 + 86 = 258

10(172) + 5(86) = 2,150

Therefore, a = 172, and c = 86 is a solution to this system. We also see this solution in the image below, since the graphs of these two equations intersect at the point (172, 86).

System of Equations

As a result of applying a system of equations to a real-world situation, we know that there were 172 adults and 86 children at the party. Now that we've reviewed what a system of equations is, let's look at the different types of systems.

Consistent and Inconsistent Systems

A solution set of a system of equations consists of all the solutions to that system. There are three possibilities for the solution set to a system.

1.) The system has a finite number of solutions.

2.) The system has an infinite number of solutions.

3.) The system has no solutions.

Systems of equations fall into two categories: consistent systems and inconsistent systems. A consistent system of equations has at least one solution. An inconsistent system has no solutions. Of the three possibilities for the solution set of a system, the first two represent a consistent system because in both cases the system has at least one solution. The third possibility represents an inconsistent system because there is no solution to the system.

For example, consider our opening party example. We saw that the system had a solution. Therefore, the system is a consistent system. We can also see this in the graph of that system. If a system is consistent, it has at least one solution, so it follows that the graphs of the equations of the system will intersect at least once. Since the graph of our system shows an intersection point, we know the system is consistent.

To further our understanding, consider the following example of an inconsistent system:

x^2 + y^2 = 1

y = x - 2

Observe that the graph of this system has no intersection points. Thus, it follows that the system has no solution, and it is an inconsistent system.

Inconsistent System

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