Classifying Linear Systems in Math

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  • 0:01 A Linear System
  • 0:55 Inconsistent
  • 1:53 Independent
  • 3:14 Consistent
  • 3:30 Dependent
  • 4:40 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you'll understand how to classify linear systems and the reasons why a linear system's classification is dependent on the number of solutions it has.

A Linear System

As you work with more and more linear systems, you will see that some of them have no solutions, some have one solution, some have more than one solution, and others have an infinite number of solutions. In math, we have classifications for linear systems that tell us how many solutions our linear system has. Remember that a linear system is a system with more than one linear equation. Linear equations are those with variables that don't have exponents. We can have two dimensional linear equations, and we can have three-dimensional linear equations. A two dimensional linear system consists of two equations in two variables. A three-dimensional linear system consists of three equations in three variables.

linear systems

Let's take a look at the various classifications of linear systems. Knowing these classifications can help you in determining if you have found all the answers or not.


A linear system is inconsistent if the lines are parallel and therefore has no solution. If the equations are written in slope-intercept form, you will see that they have the same slope, but the intercept is different. When you solve this type of linear system, you will get an equation that doesn't make sense, such as 0 = 1. You see two numbers that you know aren't equal to each other. For example, solving this linear system gives you a nonsense equation in the end.

linear systems

Rewriting the equations in slope-intercept form, you get y = -x + 8 and y = -x + 6. You see that your slopes are the same, but your intercepts are different. You also get a nonsense answer when solving. 8 obviously does not equal 6.


If you rewrite the equations in slope intercept form, and you get two different equations with two different slopes, then you have an independent linear system. This type of linear system has one solution. The two lines cross at just one point. Your x will equal a number, and your y will equal a number. When written in slope-intercept form, the equations will have different slopes. This linear system is an example of an independent one.

linear systems

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