Question 256455
<pre><font size = 4 color = "indigo"><b>
{{{system(x+3y-5z-w=9,
4x+y+3z+4w=11,
-3x-y+z-5w=-23,
x-y-3z-4w=-11)}}}

Put in all the 1's

{{{system(1x+3y-5z-1w=9,
4x+1y+3z+4w=11,
-3x-1y+1z-5w=-23,
1x-1y-3z-4w=-11)}}}

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, 3,-5,-1,"|",9,
                    4, 1, 3, 4,"|",11,
                   -3,-1, 1,-5,"|",-23,
                    1,-1,-3,-4,"|",-11))}}}

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 -4

{{{(matrix(4,6,    -4,-12,20,4,"|",-36,
                    4, 1, 3, 4,"|",11,
                   -3,-1, 1,-5,"|",-23,
                    1,-1,-3,-4,"|",-11))}}}

Add Row 1 to Row 2

{{{(matrix(4,6,    -4,-12,20,4,"|",-36,
                    0,-11,23,8,"|",-25,
                   -3,-1, 1,-5,"|",-23,
                    1,-1,-3,-4,"|",-11))}}}

Divide Row 1 by -4

{{{(matrix(4,6,     1,3,-5,-1,"|",9,
                    0,-11,23,8,"|",-25,
                   -3,-1, 1,-5,"|",-23,
                    1,-1,-3,-4,"|",-11))}}}

Multiply Row 1 by 3

{{{(matrix(4,6,     3,9,-15,-3,"|",27,
                    0,-11,23,8,"|",-25,
                   -3,-1, 1,-5,"|",-23,
                    1,-1,-3,-4,"|",-11))}}}

Add Row 1 to Row 3

{{{(matrix(4,6,     3,9,-15,-3,"|",27,
                    0,-11,23,8,"|",-25,
                    0,8,-14,-8,"|",4,
                    1,-1,-3,-4,"|",-11))}}}

Divide Row 1 by -3

{{{(matrix(4,6,    -1,-3, 5,1,"|",-9,
                    0,-11,23,8,"|",-25,
                    0,8,-14,-8,"|",4,
                    1,-1,-3,-4,"|",-11))}}}

Add Row 1 to row 4

{{{(matrix(4,6,    -1,-3, 5,1,"|",-9,
                    0,-11,23,8,"|",-25,
                    0,8,-14,-8,"|",4,
                    0,-4,2,-3,"|",-20))}}}

Multiply Row 4 by 2

{{{(matrix(4,6,    -1,-3, 5,1,"|",-9,
                    0,-11,23,8,"|",-25,
                    0,8,-14,-8,"|",4,
                    0,-8,4,-6,"|",-40))}}}

Add Row 3 to Row 4

{{{(matrix(4,6,    -1,-3, 5,1,"|",-9,
                    0,-11,23,8,"|",-25,
                    0,8,-14,-8,"|",4,
                    0,0,-10,-14,"|",-36))}}}

Divide Row 3 by 2
Divide Row 4 by 2

 
{{{(matrix(4,6,    -1,-3, 5,1,"|",-9,
                    0,-11,23,8,"|",-25,
                    0,4,-7,-4,"|",2,
                    0,0,-5,-7,"|",-18))}}}

Multiply Row 2 by 4
Multiply Row 3 by 11

{{{(matrix(4,6,    -1,-3, 5,1,"|",-9,
                    0,-44,92,32,"|",-100,
                    0,44,-77,-44,"|",22,
                    0,0,-5,-7,"|",-18))}}}

Add Row 2 to Row 3

{{{(matrix(4,6,    -1, -3, 5,1,"|",-9,
                    0,-44,92,32,"|",-100,
                    0,  0,15,-12,"|",-78,
                    0,  0,-5,-7,"|",-18))}}}

Multiply Row 4 by 3
Divide Row 2 by 4

{{{(matrix(4,6,    -1, -3, 5,1,"|",-9,
                    0,-11,23,8,"|",-25,
                    0,  0,15,-12,"|",-78,
                    0,  0,-15,-21,"|",-54))}}}

Add Row 3 to Row 4

{{{(matrix(4,6,    -1, -3, 5,1,"|",-9,
                    0,-11,23,8,"|",-25,
                    0,  0,15,-12,"|",-78,
                    0,  0,0,-33,"|",-132))}}}

Now put the letters and the equal signs back in-

{{{system(    -1x -3y+ 5z+1w=-9,
                    0x-11y+23z+8w=-25,
                    0x+  0y+15z-12w=-78,
                    0x+  0y+0z-33w=-132)}}}

Erase all the 0's and the 1's

{{{system(    -x -3y+ 5z+w=-9,
                    -11y+23z+8w=-25,
                      15z-12w=-78,
                    -33w=-132)}}}

Solve the 4th equation for w

{{{-33w=-132}}}
{{{w=4}}}

Substitute {{{w=4}}} into the 3rd equation and solve for z
{{{15z-12w=-78}}}
{{{15z-12(4)=-78}}}
{{{15z-48=-78}}}
{{{15z=-30}}}
{{{z=-2}}}

Substitute {{{w=4}}} and {{{z=-2}}} into the 2nd 
equation and solve for y:

{{{-11y+23z+8w=-25}}}
{{{-11y+23(-2)+8(4)=-25}}}
{{{-11y-46+32=-25}}}
{{{-11y-14=-25}}}
{{{-11y=-11}}}
{{{y=1}}}


Substitute {{{w=4}}}, {{{z=-2}}}, and {{{y=1))) into the 1st 
equation and solve for x:


{{{-x -3y+ 5z+w=-9}}}
{{{-x -3(1)+ 5(-2)+(4)=-9}}}
{{{-x -3-10+4=-9}}}
{{{-x -9=-9}}}
{{{-x=0}}}
{{{x=0}}}

So the solution is (x,y,z,w) = (0,1,-2,4)

Edwin</pre>