Solve the system of equations?

[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
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-2R₁+R₃, I got
x₁-3x₂-2x₃=0
-x₂-x₃=0
10x₂+10x₃=0
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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?

 

[Dr.D]

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.

 

[Math9999]

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.

 

[LCKurtz]

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.