﻿ Gauss Jordan Elimination Calculator Step by Step

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. - 4 x1 + 5 x2 - 3 x3 + 3 x4 = 20 4 x1 + 2 x2 + 3 x3 + 4 x4 = 10 5 x1 + 4 x2 + 4 x3 + 3 x4 = 20
( -4 x1 + 5 x1 )
+ ( 5 x2 + 4 x2 )
+ ( -3 x3 + 4 x3 )
+ ( 3 x4 + 3 x4 )
= 20 + 20
This transformation will allow us to count without fractions for some time. x1 + 9 x2 + x3 + 6 x4 = 40 4 x1 + 2 x2 + 3 x3 + 4 x4 = 10 5 x1 + 4 x2 + 4 x3 + 3 x4 = 20
( 4 x1 + x1 * ( -4) )
+ ( 2 x2 + 9 x2 * ( -4) )
+ ( 3 x3 + x3 * ( -4) )
+ ( 4 x4 + 6 x4 * ( -4) )
= 10 + 40 * ( -4)
The "red" coefficient is zero. x1 + 9 x2 + x3 + 6 x4 = 40 - 34 x2 - x3 - 20 x4 = - 150 5 x1 + 4 x2 + 4 x3 + 3 x4 = 20
( 5 x1 + x1 * ( -5) )
+ ( 4 x2 + 9 x2 * ( -5) )
+ ( 4 x3 + x3 * ( -5) )
+ ( 3 x4 + 6 x4 * ( -5) )
= 20 + 40 * ( -5)
The "red" coefficient is zero. x1 + 9 x2 + x3 + 6 x4 = 40 - 34 x2 - x3 - 20 x4 = - 150 - 41 x2 - x3 - 27 x4 = - 180
( -41 x2 + ( -34 x2) * ( -41/34) )
+ ( - x3 + ( - x3) * ( -41/34) )
+ ( -27 x4 + ( -20 x4) * ( -41/34) )
= -180 + ( -150) * ( -41/34)
The "red" coefficient is zero. x1 + 9 x2 + x3 + 6 x4 = 40 - 34 x2 - x3 - 20 x4 = - 150 7/34 x3 - 49/17 x4 = 15/17
The equation 3 is divided by 7/34. x1 + 9 x2 + x3 + 6 x4 = 40 - 34 x2 - x3 - 20 x4 = - 150 x3 - 14 x4 = 30/7
- 34 x2
+ ( - x3 + x3 )
+ ( -20 x4 + ( -14 x4) )
= -150 + 30/7
The "red" coefficient is zero. x1 + 9 x2 + x3 + 6 x4 = 40 - 34 x2 - 34 x4 = - 1020/7 x3 - 14 x4 = 30/7
x1
+ 9 x2
+ ( x3 + x3 * ( -1) )
+ ( 6 x4 + ( -14 x4) * ( -1) )
= 40 + 30/7 * ( -1)
The "red" coefficient is zero. x1 + 9 x2 + 20 x4 = 250/7 - 34 x2 - 34 x4 = - 1020/7 x3 - 14 x4 = 30/7
The equation 2 is divided by -34. x1 + 9 x2 + 20 x4 = 250/7 x2 + x4 = 30/7 x3 - 14 x4 = 30/7
x1
+ ( 9 x2 + x2 * ( -9) )
+ ( 20 x4 + x4 * ( -9) )
= 250/7 + 30/7 * ( -9)
The "red" coefficient is zero. x1 + 11 x4 = - 20/7 x2 + x4 = 30/7 x3 - 14 x4 = 30/7
Result:
x1 = - 20/7 - 11 x4
x2 = 30/7 - x4
x3 = 30/7 + 14 x4