w + 2x + 3y - 4z = -5
2w - x + y - 2z = -7
3w - x - 2y + z = 7
4w - 4x - 2y + 3z = 3
`
Get it in row echelon form. That is, this form:
`
w + 2x + 3y - 4z = -5
Ax + By + Cz = D
Ey + Fz = G
Hz = I
`
w + 2x + 3y - 4z = -5
2w - x + y - 2z = -7
3w - x - 2y + z = 7
4w - 4x - 2y + 3z = 3
`
Get rid of the 2w by adding -2 times the first equation
to it, and replacing it by what you get
`
w + 2x + 3y - 4z = -5
- 5x - 5y + 6z = 3
3w - x - 2y + z = 7
4w - 4x - 2y + 3z = 3
`
Get rid of the 3w by adding -3 times the first equation
to it, and replacing it by what you get
`
w + 2x + 3y - 4z = -5
- 5x - 5y + 6z = 3
- 7x - 11y + 13z = 22
4w - 4x - 2y + 3z = 3
`
Get rid of the 4w by adding -4 times the first equation
to it, and replacing it by what you get
`
w + 2x + 3y - 4z = -5
-5x - 5y + 6z = 3
-7x - 11y + 13z = 22
-12x - 14y + 19z = 23
`
Get rid of the -7x by adding -7 times the second equation, or
35x + 35y - 42z = -21, to 5 times the third equation, which is
-35x - 55y + 65z = 110, getting -20y + 23z = 89, and replacing
the third equation by it.
`
w + 2x + 3y - 4z = -5
-5x - 5y + 6z = 3
-20y + 23z = 89
-12x - 14y + 19z = 23
`
Get rid of the -12x by adding -12 times the second equation, or
60x + 60y - 72z = -36 to 5 times the fourthion, which is
-60x - 70y + 95z = 115, getting -10y + 23z = 79, and replacing the
third equation by it
`
w + 2x + 3y - 4z = -5
-5x - 5y + 6z = 3
-20y + 23z = 89
-10y + 23z = 79
`
Get rid of the -10y by adding -1 times the third equation, which is
20y - 23z = -89 to 2 times the fourth equation, which is
-20y + 46z = 158, getting 23z = 69, and replacing the third equation
by it.
`
w + 2x + 3y - 4z = -5
-5x - 5y + 6z = 3
-20y + 23z = 89
23z = 69
`
To finish,
(1) Solve the 4th equation for z,
(2) Substitute this value of z into the 3rd equation and solve for y
(3) Substitute the values for x and y into the 2nd equation and solve for x
(4) Substitute the values for z, y, and x in the 1st equation to find w.
Answer (w, x, y, z) = (2, 4, -1, 3)
`
Edwin McCravy
AnlytcPhil@aol.com
