How to Solve a Linear System in Three Variables With No or Infinite Solutions

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  • 0:02 Linear System in 3 Variables
  • 1:51 No Solution
  • 3:42 Infinite Solutions
  • 5:12 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.

In this video lesson, you will learn to recognize when a linear system has no solution or an infinite number of solutions. You will also learn how to go about solving each of these types of systems.

A Linear System in Three Variables

You are probably familiar with linear equations, such as y = 3x + 4 or y - 3x = 4. These equations, when graphed, will give you a straight line. A linear system, then, is a collection of linear equations. Our usual linear equations have only two variables. Because we have two variables, a linear system using these kinds of equations will have two equations.

But, did you know that you can also have a linear system with three variables? Yes, you can. A linear system in three variables, then, will have three equations since it has three variables. These equations, just like the ones with only two variables, will not have any exponents. This is an example of a linear system in three variables:

Three variable linear system
three variable linear system

Looking at this system, we see our three variables, x, y, and z. These three letters are the most commonly used, although you can use any letter that you want; they don't have to be in order, either. If you have a friend named Sam, you could easily use those three letters, too. The important thing to remember is that we have three different variables, and none of them have exponents with them.

You will see these kinds of linear systems in higher math, where you will be asked to solve them. Just like other systems, our linear system in three variables can have one solution, no solution, or an infinite number of solutions. Just like the other equations that we are used to solving, if our system has only one solution, then we look for just one point. But, what about the other two? How do you find out if our system has no solution or an infinite number of solutions? Let's find out.

No Solution

Each equation in our system, when graphed, produces a plane - a flat surface that goes on forever. Because we are dealing with three variables, we are dealing with 3-dimensional space. So, picture the plane floating in space. We have three equations, so we have three of these flat surfaces floating around.

To simulate this on a smaller scale, simply take three pieces of paper, have a friend lend you a third hand, and hold onto them in random places in front of you. Imagine that these sheets of paper don't end; they keep going. As you keep playing, you will notice that many times, your three sheets of paper don't all meet together.

In this case, you will have no solution. Even if two of the planes meet, if the third one doesn't meet at the same point or points as the other two, then there is no solution. How does this look mathematically? Let's see. Let's try and solve this system:

three variable linear system

We can use any method that we are comfortable with to try and solve for our variables. Since we have a nicely laid out system, I'm going to use the elimination method to eliminate some of the variables to make it easier for me to solve.

I look at the first and third equations and see that they are ready for me to combine. If I add these two equations, I can make at least the x variable disappear, meaning I can make the coefficient of the x variable 0. I add these two equations together. I get 0 + 0 + 0 = 4.

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