Inconsistent System of Equations: Definition & Example

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  • 0:03 System of Equations
  • 1:21 Inconsistent System of…
  • 2:25 Identify Inconsistent Systems
  • 5:25 Example
  • 6:37 Lesson Summary
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Lesson Transcript
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.

In this lesson, we'll review systems of equations, commonly used in mathematical problem solving, while concentrating on one particular type: inconsistent systems of equations.

System of Equations

A system of equations is a group of equations with the same variables. For example, consider the following system of equations:

x - y = -1

3x + y = 9

This is a system of equations in two variables, namely x and y. As we know, an equation represents a set of points on a graph. Therefore, a system of equations can be represented graphically as well as algebraically.

inconsistent system1

The solution set of a system of equations consists of all the points of intersection of the equations in the system. These points represent values we can plug in for the variables to make all of the equations true. For example, we see that the solution of the system in our graphic is (2,3), because this is where the two equations intersect. There are three possibilities for solutions of a system of equations:

1.) A finite number of solutions: the equations intersect at a finite number of points. Our example above is an example of this - the equations intersect at exactly one point.

2.) Infinitely many solutions: the equations intersect everywhere (they are the same graph).

3.) No solution: the equations do not intersect at all.

Inconsistent System of Equations

Of the three possibilities for the solutions of a system of equations, one possibility is that the system has no solution. When this is the case, we call the system an inconsistent system of equations. Look at the following system:

3x + 2y = 12

3x + 2y = 6

Notice that the first equation and the second equation of this system have the same left-hand side. However, their right-hand sides differ. So? This means it's not possible to find numbers, x and y, to plug into 3x + 2y to equal two different numbers, namely 6 and 12. Therefore, there is no solution to this system, and it's an inconsistent system of equations.

These types of systems are extremely useful when analyzing a problem. If we represent a problem using a system and the system is inconsistent, then we know there is no solution to that problem. This tells us we have to make adjustments or start over altogether. While that may not be ideal in a real-world setting, it can be incredibly time-saving.

Identify Inconsistent Systems

There are a number of ways to identify an inconsistent system. One approach is to logically deduce that there is no solution, as we did before. If you can manipulate the equations to get a common left- or right-hand side, and the other side differs between equations, you can logically deduce that there is no solution to the system.

Another way to identify an inconsistent system is by trying to solve the system algebraically. As you're going through the process of solving the system, you will run into a contradiction. That is, you will run into something that doesn't make sense, such as 1 = 2 or 0 = 5. Let's look at our example of an inconsistent system of equations. There are multiple ways to solve a system of equations algebraically, but let's just use substitution for simplicity here.

To use substitution to solve this system, use the following steps:

1.) Solve one of the equations for one of the variables, say x.

2.) Plug the x-value you found in step 1 into the other equation to solve for the remaining variable, y. You now have a numerical value for y.

3.) Plug the numerical y-value found in step 2 into either of the original equations. This will result in an equation with just one variable, x. Solve for x.

At that point, you have numerical values for both variables, and you have your solution.

Let's try applying this to our example system. First, solve one of the equations for one of the variables: y. (If you wanted to solve for x instead, that's fine. It doesn't matter which variable you solve for in the first step.)

3x + 2y = 12 Subtract 3x from both sides...
2y = 12 - 3x Divide both sides by 2...
y = 6 - 1.5x We've solved for y!

The next step is to plug our value for y into the second equation, and then solve for x.

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