Example of Solving a System of Linear Equations by Gauss Jordan Elimination.
This solution was 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 |
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.
| | 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 |
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.
| | 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 |
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.
| | 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 |
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.
| | 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 |
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.
| | x1 | - | x2 | + | 4 | x3 | - | x4 | = | - 1 | ||||
| x2 | - | 6 | x3 | + | 3 | x4 | = | 8 | ||||||
| - | 19 | x3 | + | 11 | x4 | = | 39 | |||||||
| 6 | x2 | - | 5 | x3 | + | 3 | x4 | = | 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.
| | x1 | - | x2 | + | 4 | x3 | - | x4 | = | - 1 | ||||
| x2 | - | 6 | x3 | + | 3 | x4 | = | 8 | ||||||
| - | 19 | x3 | + | 11 | x4 | = | 39 | |||||||
| 31 | x3 | - | 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.
| | 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 |
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.
| | x1 | - | x2 | + | 4 | x3 | - | x4 | = | - 1 | ||||
| x2 | - | 6 | x3 | + | 3 | x4 | = | 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.
| | x1 | - | x2 | + | 4 | x3 | - | 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.
| | 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 |
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.
| | x1 | - | x2 | + | 4 | x3 | = | 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.
| | x1 | - | 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.
| | x1 | = | - 3 | |||||||||||
| x2 | = | - 1 | ||||||||||||
| x3 | = | 2 | ||||||||||||
| x4 | = | 7 |
Result:
x1 = - 3
x2 = - 1
x3 = 2
x4 = 7