Question 86464
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solve the following system of equtions and give the general
solution in parametric vector form. 

  x + y - 2z     = 6
     3y +  z - w = 3
-2x + y + 2z - w = 9
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Associated with this linear system is the augmented matrix

{{{(matrix(3,5,1,1,-2,0,6,0,3,1,-1,3,-2,1,2,-1,9))}}}

Reducing it to row reduced echelon form:

{{{(matrix(3,5,1,0,0,1/3,-9,0,1,0,-1/3,3,0,0,1,0,-6))}}}

So the general solution is 

{{{(matrix(4,1,x,y,z,w))}}} = {{{(matrix(4,1,-9-(1/3)w,3+(1/3)w,-6,w))}}}

Write the -6 as -6+0w, and the w as 0 + 1w

{{{(matrix(4,1,x,y,z,w))}}} = {{{(matrix(4,1,-9-(1/3)w,3+(1/3)w,-6+0w,0+1w))}}}

Write the matrix on the right as the sum of two matrices:

{{{(matrix(4,1,x,y,z,w))}}} = {{{(matrix(4,1,-9,3,-6,0))}}} + {{{(matrix(4,1,-(1/3)w,(1/3)w,0w,1w))}}}

Now factor out a scalar w from the right-hand matrix:

{{{(matrix(4,1,x,y,z,w))}}} = {{{(matrix(4,1,-9,3,-6,0))}}} + w{{{(matrix(4,1,-1/3,1/3,0,1))}}}

That's fine in that form, however, it looks neater when you factor out
1/3 from the matrix on the right:

{{{(matrix(4,1,x,y,z,w))}}} = {{{(matrix(4,1,-9,3,-6,0))}}} + {{{(1/3)w}}}{{{(matrix(4,1,-1,1,0,3))}}}

Edwin</pre>