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

This solution has been done by the calculator presented on the site.

Please note that the coefficients will disappear which located in the "red" positions.
Знак системы3x1+2x2+x3+x4 = - 2
x1 -x2+4x3-x4 = - 1
- 2 x1- 2 x2- 3 x3+x4 = 9
x1 +5x2-x3+2x4 = 4
The equation 2 and equation 1 are reversed.
Знак системыx1 -x2+4x3-x4 = - 1
3x1+2x2+x3+x4 = - 2
- 2 x1- 2 x2- 3 x3+x4 = 9
x1 +5x2-x3+2x4 = 4
The equation 1 multiplied by -3 is added to the equation 2.   more info
( 3 x1 + x1 * ( -3) )
+ ( 2 x2 + ( - x2) * ( -3) )
+ ( x3 + 4 x3 * ( -3) )
+ ( x4 + ( - x4) * ( -3) )
= -2 + ( -1) * ( -3)
The "red" coefficient is zero.
Знак системыx1 -x2+4x3-x4 = - 1
5x2- 11 x3+4x4 = 1
- 2 x1- 2 x2- 3 x3+x4 = 9
x1 +5x2-x3+2x4 = 4
The equation 1 multiplied by 2 is added to the equation 3.   more info
( -2 x1 + x1 * 2 )
+ ( -2 x2 + ( - x2) * 2 )
+ ( -3 x3 + 4 x3 * 2 )
+ ( x4 + ( - x4) * 2 )
= 9 + ( -1) * 2
The "red" coefficient is zero.
Знак системыx1 -x2+4x3-x4 = - 1
5x2- 11 x3+4x4 = 1
- 4 x2+5x3-x4 = 7
x1 +5x2-x3+2x4 = 4
The equation 1 multiplied by -1 is added to the equation 4.   more info
( x1 + x1 * ( -1) )
+ ( 5 x2 + ( - x2) * ( -1) )
+ ( - x3 + 4 x3 * ( -1) )
+ ( 2 x4 + ( - x4) * ( -1) )
= 4 + ( -1) * ( -1)
The "red" coefficient is zero.
Знак системыx1 -x2+4x3-x4 = - 1
5x2- 11 x3+4x4 = 1
- 4 x2+5x3-x4 = 7
6x2- 5 x3+3x4 = 5
The equation 3 is added to the equation 2.   more info
( 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.
Знак системыx1 -x2+4x3-x4 = - 1
x2 - 6 x3+3x4 = 8
- 4 x2+5x3-x4 = 7
6x2- 5 x3+3x4 = 5
The equation 2 multiplied by 4 is added to the equation 3.   more info
( -4 x2 + x2 * 4 )
+ ( 5 x3 + ( -6 x3) * 4 )
+ ( - x4 + 3 x4 * 4 )
= 7 + 8 * 4
The "red" coefficient is zero.
Знак системыx1 -x2+4x3-x4 = - 1
x2 - 6 x3+3x4 = 8
- 19 x3+11x4 = 39
6x2- 5 x3+3x4 = 5
The equation 2 multiplied by -6 is added to the equation 4.   more info
( 6 x2 + x2 * ( -6) )
+ ( -5 x3 + ( -6 x3) * ( -6) )
+ ( 3 x4 + 3 x4 * ( -6) )
= 5 + 8 * ( -6)
The "red" coefficient is zero.
Знак системыx1 -x2+4x3-x4 = - 1
x2 - 6 x3+3x4 = 8
- 19 x3+11x4 = 39
31x3- 15 x4 = - 43
The equation 3 multiplied by 31/19 is added to the equation 4.   more info
( 31 x3 + ( -19 x3) * 31/19 )
+ ( -15 x4 + 11 x4 * 31/19 )
= -43 + 39 * 31/19
The "red" coefficient is zero.
Знак системыx1 -x2+4x3-x4 = - 1
x2 - 6 x3+3x4 = 8
- 19 x3+11x4 = 39
56/19x4 = 392/19
The equation 4 is divided by 56/19.
Знак системыx1 -x2+4x3-x4 = - 1
x2 - 6 x3+3x4 = 8
- 19 x3+11x4 = 39
x4 = 7
The equation 4 multiplied by -11 is added to the equation 3.   more info
- 19 x3
+ ( 11 x4 + x4 * ( -11) )
= 39 + 7 * ( -11)
The "red" coefficient is zero.
Знак системыx1 -x2+4x3-x4 = - 1
x2 - 6 x3+3x4 = 8
- 19 x3 = - 38
x4 = 7
The equation 4 multiplied by -3 is added to the equation 2.   more info
x2
- 6 x3
+ ( 3 x4 + x4 * ( -3) )
= 8 + 7 * ( -3)
The "red" coefficient is zero.
Знак системыx1 -x2+4x3-x4 = - 1
x2 - 6 x3 = - 13
- 19 x3 = - 38
x4 = 7
The equation 4 is added to the equation 1.   more info
x1
+ - x2
+ 4 x3
+ ( - x4 + x4 )
= -1 + 7
The "red" coefficient is zero.
Знак системыx1 -x2+4x3 = 6
x2 - 6 x3 = - 13
- 19 x3 = - 38
x4 = 7
The equation 3 is divided by -19.
Знак системыx1 -x2+4x3 = 6
x2 - 6 x3 = - 13
x3 = 2
x4 = 7
The equation 3 multiplied by 6 is added to the equation 2.   more info
x2
+ ( -6 x3 + x3 * 6 )
= -13 + 2 * 6
The "red" coefficient is zero.
Знак системыx1 -x2+4x3 = 6
x2 = - 1
x3 = 2
x4 = 7
The equation 3 multiplied by -4 is added to the equation 1.   more info
x1
+ - x2
+ ( 4 x3 + x3 * ( -4) )
= 6 + 2 * ( -4)
The "red" coefficient is zero.
Знак системыx1 -x2 = - 2
x2 = - 1
x3 = 2
x4 = 7
The equation 2 is added to the equation 1.   more info
x1
+ ( - x2 + x2 )
= -2 + ( -1)
The "red" coefficient is zero.
Знак системыx1 = - 3
x2 = - 1
x3 = 2
x4 = 7
Result:
x1 = - 3
x2 = - 1
x3 = 2
x4 = 7
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