Question 686473
x+2y-w+z=9 2x-y+2w+3z=-3 3x+y+w-z=-4 x-y-3w+z=4

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{{{system(x+2y-w+z=9,
2x-y+2w+3z=-3,
3x+y+w-z=-23,
x-y-3w+z=4)}}}

Put in all the 1's

{{{system(1x+2y-1w+1z=9,
2x-1y+2w+3z=-3,
3x+1y+1z-1w=-4,
1x-1y-3z+1w=4)}}}

Erase the letters and the plus signs,
put a bar where the equal signs are, 
and put the whole thing in parentheses:

{{{(matrix(4,6,     1, 2,-1, 1,"|",9,
                          2,-1, 2, 3,"|",-3,
                          3, 1, 1,-1,"|",-4,
                          1,-1,-3, 1,"|",4))}}}

The idea is to get all 0's below the diagonal.

Caution: 

1. After you've finished getting zeros under the diagonal
in column 1, NEVER USE ROW 1 AGAIN!
2. After you've finished getting zeros under the diagonal
in column 2, NEVER USE ROW 2 AGAIN!

Multiply the Row 1 by -2

{{{(matrix(4,6,    -2, -4, 2, -2,"|",-18,
                          2,-1, 2, 3,"|",-3,
                          3, 1, 1,-1,"|",-4,
                          1,-1,-3, 1,"|",4))}}}

Add Row 1 to Row 2

{{{(matrix(4,6,   -2, -4, 2, -2,"|",-18,
                         0,-5, 4, 1,"|",-21,
                          3, 1, 1,-1,"|",-4,
                          1,-1,-3, 1,"|",4))}}}

Divide Row 1 by -2

{{{(matrix(4,6,        1, 2,-1, 1,"|",9,
                         0,-5, 4, 1,"|",-21,
                          3, 1, 1,-1,"|",-4,
                          1,-1,-3, 1,"|",4))}}}


Multiply Row 1 by -3

{{{(matrix(4,6,    -3, -6, 3, -3,"|",-27,
                         0,-5, 4, 1,"|",-21,
                          3, 1, 1,-1,"|",-4,
                          1,-1,-3, 1,"|",4))}}}


Add Row 1 to Row 3

{{{(matrix(4,6,     -3, -6, 3, -3,"|",-27, 
                         0,-5, 4, 1,"|",-21,
                          0, -5, 4,-4,"|",-31,
                          1,-1,-3, 1,"|",4))}}}

Divide Row 1 by -3

{{{(matrix(4,6,    1, 2,-1, 1,"|",9,
                         0,-5, 4, 1,"|",-21,
                          0, -5, 4,-4,"|",-31,
                          1,-1,-3, 1,"|",4))}}}

Multiply Row 1 by -1

{{{(matrix(4,6,    -1, -2,1, -1,"|",-9,
                         0,-5, 4, 1,"|",-21,
                          0, -5, 4,-4,"|",-31,
                          1,-1,-3, 1,"|",4))}}}

Add Row 1 to row 4

{{{(matrix(4,6,     -1, -2,1, -1,"|",-9,
                         0,-5, 4, 1,"|",-21,
                          0, -5, 4,-4,"|",-31,
                          0,-3,-2, 0,"|",-5))}}}



Divide Row 1 fo -1

{{{(matrix(4,6,     1,  2,-1, 1,"|",9, 
                         0,-5, 4, 1,"|",-21,
                          0, -5, 4,-4,"|",-31,
                          0,-3,-2, 0,"|",-5))}}}

Multiply row 2 by -1

{{{(matrix(4,6,     1,  2,-1, 1,"|",9, 
                         0, 5, -4, -1,"|",21,
                          0, -5, 4,-4,"|",-31,
                          0,-3,-2, 0,"|",-5))}}}


Add row 2 to row 3

{{{(matrix(4,6,     1,  2,-1, 1,"|",9, 
                         0, 5, -4, -1,"|",21,
                          0, 0, 0, -5,"|",-10,
                          0,-3,-2, 0,"|",-5))}}}

Multiply Row 2 by 3 and Row 4 by 5

{{{(matrix(4,6,     1,  2,-1, 1,"|",9, 
                         0, 15, -12, -3,"|",63,
                          0, 0, 0, -5,"|",-10,
                          0,-15,-10, 0,"|",-25))}}}

Add Row 2 to Row 4

{{{(matrix(4,6,     1,  2,-1, 1,"|",9, 
                         0, 15, -12, -3,"|",63,
                          0, 0, 0, -5,"|",-10,
                          0,  0,-22, -3,"|",38))}}}

Swap rows 3 and 4


{{{(matrix(4,6,     1,  2,-1, 1,"|",9, 
                         0, 15, -12, -3,"|",63,
                           0,  0,-22, -3,"|",38,
                             0, 0, 0, -5,"|",-10


))}}}

Now we have only 0's under the diagonal,
so we put the letters and the equal signs back in-

{{{system(    1x +2y-1w+1z=9,
                    0x+15y-12w-3z=63,
                    0x+  0y-22w-3z=38,
                    0x+  0y+0w-5z=-10)}}}

Erase all the 0's and the 1's

{{{system(    x +2y-w+z=9,
                    15y-12w-3z=63,
                    -22w-3z=38,
                    -5z=-10)}}}


Solve the 4th equation for z

{{{-5z=-10}}}
{{{z=2}}}

Substitute {{{z=2}}} into the 3rd equation and solve for w{{{-22w-3z=38}}}
{{{-22w-3(2)=38}}}
{{{-22w-6=38}}}
{{{-22w=44}}}
{{{w=-2}}}

Substitute {{{z=2}}} and {{{w=-2}}} into the 2nd 
equation and solve for y:

{{{15y-12w-3z=63}}}
{{{15y-12(-2)-3(2)=63}}}
{{{15y+24-6=63}}}
{{{15y+18=63}}}
{{{15y=45}}}
{{{y=3}}}


Substitute {{{z=2}}}, {{{w=-2}}}, and {{{y=3))) into the 1st 
equation and solve for x:

 
{{{  x +2y-w+z=9,}}}
{{{x+2(3)-(-2)+(2)=9}}}
{{{x+6+2+2=9}}}
{{{x+10=9}}}
{{{x=-1}}}

So the solution is (x,y,w,z) = (-1,3,-2,2)

Edwin</pre>