Question 107919
Start with the given system

{{{system(-x+y+3z=6,x+y+2z=7,2x+3y+z=4)}}}


Write all of the coefficients and the right side constants in an augmented matrix


{{{(matrix(3,4,-1,1,3,6,1,1,2,7,2,3,1,4))}}}




Multiply Row 1 by {{{-1/1}}} to make the pivot 1:


{{{(matrix(3,4,1,-1,-3,-6,1,1,2,7,2,3,1,4))}}}


 Add  -1*Row 1 to Row 2 to get the new Row 2


{{{(matrix(3,4,1,-1,-3,-6,0,2,5,13,2,3,1,4))}}}


 Add  -2*Row 1 to Row 3 to get the new Row 3


{{{(matrix(3,4,1,-1,-3,-6,0,2,5,13,0,5,7,16))}}}


Multiply Row 2 by {{{1/2}}} to make the pivot 1:


{{{(matrix(3,4,1,-1,-3,-6,0,1,5/2,13/2,0,5,7,16))}}}


 Add  -5*Row 2 to Row 3 to get the new Row 3


{{{(matrix(3,4,1,-1,-3,-6,0,1,5/2,13/2,0,0,-11/2,-33/2))}}}


Multiply Row 3 by {{{-2/11}}} to make the pivot 1:


{{{(matrix(3,4,1,-1,-3,-6,0,1,5/2,13/2,0,0,1,3))}}}



 Add  -5/2*Row 3 to Row 2 to get the new Row 2


{{{(matrix(3,4,1,-1,-3,-6,0,1,0,-1,0,0,1,3))}}}




 Add  3*Row 3 to Row 1 to get the new Row 1


{{{(matrix(3,4,1,-1,0,3,0,1,0,-1,0,0,1,3))}}}



 Add  1*Row 2 to Row 1 to get the new Row 1


{{{(matrix(3,4,1,0,0,2,0,1,0,-1,0,0,1,3))}}}



The matrix is now in <font size=4><b>reduced row echelon form</b></font>


If you need more help with row reduction, check out the <a href="http://www.math.odu.edu/~bogacki/lat/">Linear Algebra Toolkit</a>