Solve the system of equations?

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Homework Help Overview

The discussion revolves around solving a system of linear equations using Gaussian or Gauss-Jordan elimination. The equations provided are homogeneous and involve three variables.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts row reduction but finds discrepancies between their results and the book's answer. They question the correctness of their calculations and seek clarification on the presence of a parameter in the book's solution.

Discussion Status

Some participants suggest verifying the book's answer against the original equations and indicate that the original poster may not have fully completed the row reduction process. There is an exploration of the implications of having a free variable in the context of the solution.

Contextual Notes

The equations are homogeneous, and the discussion includes the concept of the determinant of the coefficient matrix being zero, which is relevant to the existence of non-trivial solutions.

Math9999

Homework Statement


Solve the system of equations
x1-3x2-2x3=0
-x1+2x2+x3=0
2x1+4x2+6x3=0
using either Gaussian or Gauss-Jordan elimination.

Homework Equations


None.

The Attempt at a Solution


R1+R2, I got
x1-3x2-2x3=0
-x2-x3=0
2x1+4x2+6x3=0
-----------------------------------------------------------------------
-2R1+R3, I got
x1-3x2-2x3=0
-x2-x3=0
10x2+10x3=0
----------------------------------------------------------------------
10R2+R3, I got
x1-3x2-2x3=0
-x2-x3=0
--------------------------------------------------------------------
Once I did the calculation, it doesn't match the answer from the book. Because I got x2=0 and x3=0 so therefore x1=0. But the answer in the book says t*[1, 1, -1]. Can anyone tell me what's wrong and how to get the right answer?
 
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Does the answer in the book check with the equations? I'd start there.

All your equations are equal to zero, so there exists a solution iff the determinant of the coefficients is zero.
 
Yes, when I plug in [1, 1, -1] into the given equations, it matches. But I don't know how they got the t variable in the answer.
 
You haven't completed the row reduction. When you do you should find a free variable that you can let be anything and get the others in terms of it.
 

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