Determine (without doing row operations) that a system is consistent

In summary, the conversation revolves around the topic of linear algebra and the concept of consistency and infinite solutions in a system of linear equations. The first question discusses the consistency of a system without performing any row operations and the second question explores the idea of a system having infinite solutions when there are more variables than equations. The conversation ends with a resource for further learning on the topic.
  • #1
JJBladester
Gold Member
286
2
Hello,

I just started learning Elementary Linear Algebra. I've read through the first chapter in my book (Elementary Linear Algebra by Larson, Edwards, Falvo, 5th edition). There were two "Discovery" questions that try to see if you know how systems of equations work. I am unsure on both of them.

These are not homework questions, rather a test to see if you know how to determine if a system is consistent or if a system has an infinite # of solutions.

Consider the system of linear equations.
2x1 + 3x2 + 5x3 = 0
-5x1 + 6x2 -17x3 = 0
7x1 - 4x2 + 3x3 = 0

Without doing any row operations, explain why this system is consistent.

______________________________________________________________________

The following system has more variables than equations. Why does it have an infinite number of solutions?
2x1 + 3x2 + 5x3 + 2x4 = 0
-5x1 + 6x2 -17x3 - 3x4 = 0
7x1 - 4x2 + 3x3 + 13x4 = 0

Since I am just beginning to learn linear algebra, I am not sure exactly the answer for either question. I know a system is consistent if it has exactly one solution or if it has infinite solutions, so how could I tell if the system for question one is consistent without doing any row operations?

For the second problem, I know that a system has infinite solutions if after doing row operations you get a free variable, but how do you determine if a variable is considered free?
 
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  • #2
Ok... I read further along in the book and now I understand that for the first question, any homogeneous system will have at least one solution (where all variables = 0). Still wondering about question two.
 
  • #3
Ok... I found the solution to the second question. Every homogeneous system of linear equations is consistent. Moreover, if the system has fewer equations than variables, then it must have an infinite number of solutions. Bingo. Next time I'll read a bit further before posting.
 
  • #4
Well, glad you answered your own question. Keep these ideas in mind though, they'll come in handy once you reach vector spaces.
 
  • #5
Thanks for the encouragement and the heads-up, PowerIso. I'll make sure to bank this knowledge and look for it to come up again soon. I found some online Linear Algebra videos from MIT here:

http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/index.htm

Between my school's lectures and these, I am becoming more matrix-confident.
 

1. How can you determine if a system is consistent without doing row operations?

One way to determine if a system is consistent is by checking its rank. If the rank of the coefficient matrix is equal to the rank of the augmented matrix, then the system is consistent. This is because if the ranks are equal, there are no contradictory equations in the system, and a consistent solution exists.

2. Can I use a graph to determine if a system is consistent?

No, a graph cannot determine if a system is consistent. Graphs can only show the possible solutions of a system, but they cannot determine if a solution actually exists. To determine consistency, you need to analyze the equations and coefficients of the system.

3. Are there other methods to determine consistency besides checking the rank?

Yes, another method is to use the determinant of the coefficient matrix. If the determinant is not equal to zero, then the system is consistent. This is because a non-zero determinant means that the coefficient matrix is invertible, and a unique solution exists for the system.

4. What is the significance of a consistent system?

A consistent system means that there is at least one solution that satisfies all the equations. This is important in problem-solving as it allows us to find a solution that meets all the requirements and constraints of the system.

5. Can a system be consistent and have no solution?

No, a consistent system always has at least one solution. If a system has no solution, it is called inconsistent. This means that the equations in the system contradict each other, and there is no solution that satisfies all of them simultaneously.

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