﻿ Gauss Jordan Elimination Calculator

Example of Solving a System of Linear Equations by Gauss Jordan Elimination.

This solution has been made using the calculator presented on the site.

Please note that the coefficients will disappear which located in the "red" positions. 3 x1 + 2 x2 + x3 + x4 = - 2 x1 - x2 + 4 x3 - x4 = - 1 - 2 x1 - 2 x2 - 3 x3 + x4 = 9 x1 + 5 x2 - x3 + 2 x4 = 4
The equation 2 and equation 1 are reversed. x1 - x2 + 4 x3 - x4 = - 1 3 x1 + 2 x2 + x3 + x4 = - 2 - 2 x1 - 2 x2 - 3 x3 + x4 = 9 x1 + 5 x2 - x3 + 2 x4 = 4
( 3 x1 + x1 * ( -3) )
+ ( 2 x2 + ( - x2) * ( -3) )
+ ( x3 + 4 x3 * ( -3) )
+ ( x4 + ( - x4) * ( -3) )
= -2 + ( -1) * ( -3)
The "red" coefficient is zero. x1 - x2 + 4 x3 - x4 = - 1 5 x2 - 11 x3 + 4 x4 = 1 - 2 x1 - 2 x2 - 3 x3 + x4 = 9 x1 + 5 x2 - x3 + 2 x4 = 4
( -2 x1 + x1 * 2 )
+ ( -2 x2 + ( - x2) * 2 )
+ ( -3 x3 + 4 x3 * 2 )
+ ( x4 + ( - x4) * 2 )
= 9 + ( -1) * 2
The "red" coefficient is zero. x1 - x2 + 4 x3 - x4 = - 1 5 x2 - 11 x3 + 4 x4 = 1 - 4 x2 + 5 x3 - x4 = 7 x1 + 5 x2 - x3 + 2 x4 = 4
( x1 + x1 * ( -1) )
+ ( 5 x2 + ( - x2) * ( -1) )
+ ( - x3 + 4 x3 * ( -1) )
+ ( 2 x4 + ( - x4) * ( -1) )
= 4 + ( -1) * ( -1)
The "red" coefficient is zero. x1 - x2 + 4 x3 - x4 = - 1 5 x2 - 11 x3 + 4 x4 = 1 - 4 x2 + 5 x3 - x4 = 7 6 x2 - 5 x3 + 3 x4 = 5
( 5 x2 + ( -4 x2) )
+ ( -11 x3 + 5 x3 )
+ ( 4 x4 + ( - x4) )
= 1 + 7
This transformation will allow us to count without fractions for some time. x1 - x2 + 4 x3 - x4 = - 1 x2 - 6 x3 + 3 x4 = 8 - 4 x2 + 5 x3 - x4 = 7 6 x2 - 5 x3 + 3 x4 = 5
( -4 x2 + x2 * 4 )
+ ( 5 x3 + ( -6 x3) * 4 )
+ ( - x4 + 3 x4 * 4 )
= 7 + 8 * 4
The "red" coefficient is zero. x1 - x2 + 4 x3 - x4 = - 1 x2 - 6 x3 + 3 x4 = 8 - 19 x3 + 11 x4 = 39 6 x2 - 5 x3 + 3 x4 = 5
( 6 x2 + x2 * ( -6) )
+ ( -5 x3 + ( -6 x3) * ( -6) )
+ ( 3 x4 + 3 x4 * ( -6) )
= 5 + 8 * ( -6)
The "red" coefficient is zero. x1 - x2 + 4 x3 - x4 = - 1 x2 - 6 x3 + 3 x4 = 8 - 19 x3 + 11 x4 = 39 31 x3 - 15 x4 = - 43
( 31 x3 + ( -19 x3) * 31/19 )
+ ( -15 x4 + 11 x4 * 31/19 )
= -43 + 39 * 31/19
The "red" coefficient is zero. x1 - x2 + 4 x3 - x4 = - 1 x2 - 6 x3 + 3 x4 = 8 - 19 x3 + 11 x4 = 39 56/19 x4 = 392/19
The equation 4 is divided by 56/19. x1 - x2 + 4 x3 - x4 = - 1 x2 - 6 x3 + 3 x4 = 8 - 19 x3 + 11 x4 = 39 x4 = 7
- 19 x3
+ ( 11 x4 + x4 * ( -11) )
= 39 + 7 * ( -11)
The "red" coefficient is zero. x1 - x2 + 4 x3 - x4 = - 1 x2 - 6 x3 + 3 x4 = 8 - 19 x3 = - 38 x4 = 7
x2
- 6 x3
+ ( 3 x4 + x4 * ( -3) )
= 8 + 7 * ( -3)
The "red" coefficient is zero. x1 - x2 + 4 x3 - x4 = - 1 x2 - 6 x3 = - 13 - 19 x3 = - 38 x4 = 7
x1
+ - x2
+ 4 x3
+ ( - x4 + x4 )
= -1 + 7
The "red" coefficient is zero. x1 - x2 + 4 x3 = 6 x2 - 6 x3 = - 13 - 19 x3 = - 38 x4 = 7
The equation 3 is divided by -19. x1 - x2 + 4 x3 = 6 x2 - 6 x3 = - 13 x3 = 2 x4 = 7
x2
+ ( -6 x3 + x3 * 6 )
= -13 + 2 * 6
The "red" coefficient is zero. x1 - x2 + 4 x3 = 6 x2 = - 1 x3 = 2 x4 = 7
x1
+ - x2
+ ( 4 x3 + x3 * ( -4) )
= 6 + 2 * ( -4)
The "red" coefficient is zero. x1 - x2 = - 2 x2 = - 1 x3 = 2 x4 = 7
x1
+ ( - x2 + x2 )
= -2 + ( -1)
The "red" coefficient is zero. x1 = - 3 x2 = - 1 x3 = 2 x4 = 7
Result:
x1 = - 3
x2 = - 1
x3 = 2
x4 = 7