﻿ Gaussian Elimination Method Calculator

Example of Solving a System of Linear Equations by Gaussian 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
We will find the variable x3 from equation 3 of the system.
7/34 x3 - 49/17 x4 = 15/17
x3 = 30/7 + 14 x4
We will find the variable x2 from equation 2 of the system.
- 34 x2 - x3 - 20 x4 = - 150
- 34 x2 = - 150 + x3 + 20 x4
- 34 x2 = - 150 + ( 30/7 + 14 x4 ) + 20 x4
x2 = 30/7 - x4
We will find the variable x1 from equation 1 of the system.
x1 + 9 x2 + x3 + 6 x4 = 40
x1 = 40 - 9 x2 - x3 - 6 x4
x1 = 40 - 9 * ( 30/7 - x4 ) - ( 30/7 + 14 x4 ) - 6 x4
x1 = - 20/7 - 11 x4
Result:
x1 = - 20/7 - 11 x4
x2 = 30/7 - x4
x3 = 30/7 + 14 x4