Example:
-4*w + 3*x - 4*y - z = -37
-2*w - 5*y + 3*z = -20
-w - x - 3*y - 4*z = -27
-3*w + 2*x + 4*y - z = 7
Let w be the first pivot variable, and the first equation the first
pivot equation. w appears in the second equation, so we subtract
(-2)/(-4) = 1/2 times the first equation from the second, getting
-2*w - 5*y + 3*z = -20,
-2*w + (3/2)*x - 2*y - (1/2)*z = -37/2
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-(3/2)*x - 3*y + (7/2)*z = -3/2
w appears in the third equation so we subtract (-1)/(-4) times the
first equation from the third. w appears in the fourth equation, so we
subtract (-3)/(-4) times the first equation from the fourth. This
leaves
-4*w + 3*x - 4*y - z = -37
-(3/2)*x - 3*y + (7/2)*z = -3/2
-(7/4)*x - 2*y - (15/4)*z = -71/4
-(1/4)*x + 7*y - (1/4)*z = 139/4
Now the last three equations are a system of three equations in three
unknowns, and we treat that similarly. Let x be the next pivot
variable, and the second equation the pivot equation. Since x appears
in the third equation, we subtract (-7/4)/(-3/2) = 7/6 times the
second equation from the third. Since x appears in the fourth equation
we subtract (-1/4)/(-3/2) = 1/6 times the second equation from the
fourth. This leaves
-4*w + 3*x - 4*y - z = -37
-(3/2)*x - 3*y + (7/2)*z = -3/2
(3/2)*y - (47/6)*z = -16
(15/2)*y - 16*z = 35
Now the last two equations are a system of two equations in two
unknowns, and we treat that similarly. Let y be the next pivot
variable, and the third equation the next pivot equation. Since y
appears in the fourth equation, we subtract (15/2)/(3/2) = 5 times the
third equation from the fourth. This leaves
-4*w + 3*x - 4*y - z = -37
-(3/2)*x - 3*y + (7/2)*z = -3/2
(3/2)*y - (47/6)*z = -16
(115/3)*z = 115
This completes the first phase.
Now we start with the last equation, and solve for z. This gives
z = 3, and we substitute that into the preceding equations:
-4*w + 3*x - 4*y = -34
-(3/2)*x - 3*y = -12
(3/2)*y = 15/2
The last remaining equation tells us that y = 5, and we substitute
that into the preceding equations:
-4*w + 3*x = -14
-(3/2)*x = 3
The last remaining equation tells us that x = -2, and we substitute
that into the preceding equation:
-4*w = -8
That gives us the value w = 2, and so the solution is
(w,x,y,z) = (2,-2,5,3)
Understood?