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Example №2. Solving of a System of Linear Equations by the Gauss Elimination (Many Solutions)

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.
Знак системы2x1+3x2-x3+x4 = 1
8x1+12x2- 9 x3+8x4 = 3
4x1+6x2+3x3- 2 x4 = 3
2x1+3x2+9x3- 7 x4 = 3
The equation 1 multiplied by -4 is added to the equation 2.   more info
( 8 x1 + 2 x1 * ( -4) )
+ ( 12 x2 + 3 x2 * ( -4) )
+ ( -9 x3 + ( - x3) * ( -4) )
+ ( 8 x4 + x4 * ( -4) )
= 3 + 1 * ( -4)
The "red" coefficient is zero.
Знак системы2x1+3x2-x3+x4 = 1
- 5 x3+4x4 = - 1
4x1+6x2+3x3- 2 x4 = 3
2x1+3x2+9x3- 7 x4 = 3
The equation 1 multiplied by -2 is added to the equation 3.   more info
( 4 x1 + 2 x1 * ( -2) )
+ ( 6 x2 + 3 x2 * ( -2) )
+ ( 3 x3 + ( - x3) * ( -2) )
+ ( -2 x4 + x4 * ( -2) )
= 3 + 1 * ( -2)
The "red" coefficient is zero.
Знак системы2x1+3x2-x3+x4 = 1
- 5 x3+4x4 = - 1
5x3- 4 x4 = 1
2x1+3x2+9x3- 7 x4 = 3
The equation 1 multiplied by -1 is added to the equation 4.   more info
( 2 x1 + 2 x1 * ( -1) )
+ ( 3 x2 + 3 x2 * ( -1) )
+ ( 9 x3 + ( - x3) * ( -1) )
+ ( -7 x4 + x4 * ( -1) )
= 3 + 1 * ( -1)
The "red" coefficient is zero.
Знак системы2x1+3x2-x3+x4 = 1
- 5 x3+4x4 = - 1
5x3- 4 x4 = 1
10x3- 8 x4 = 2
The equation 2 is added to the equation 3.   more info
( 5 x3 + ( -5 x3) )
+ ( -4 x4 + 4 x4 )
= 1 + ( -1)
The "red" coefficient is zero.
Знак системы2x1+3x2-x3+x4 = 1
- 5 x3+4x4 = - 1
0 = 0
10x3- 8 x4 = 2
The equation 2 multiplied by 2 is added to the equation 4.   more info
( 10 x3 + ( -5 x3) * 2 )
+ ( -8 x4 + 4 x4 * 2 )
= 2 + ( -1) * 2
The "red" coefficient is zero.
Знак системы2x1+3x2-x3+x4 = 1
- 5 x3+4x4 = - 1
0 = 0
0 = 0
Знак системы2x1+3x2-x3+x4 = 1
- 5 x3+4x4 = - 1
We will find the variable x3 from equation 2 of the system.
- 5 x3 + 4 x4 = - 1
x3 = 1/5 + 4/5 x4
We will find the variable x1 from equation 1 of the system.
2 x1 + 3 x2 - x3 + x4 = 1
2 x1 = 1 - 3 x2 + x3 - x4
2 x1 = 1 - 3 x2 + ( 1/5 + 4/5 x4 ) - x4
x1 = 3/5 - 3/2 x2 - 1/10 x4
Result:
x1 = 3/5 - 3/2 x2 - 1/10 x4
x3 = 1/5 + 4/5 x4








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