# gaussian elimination 2x1 +2x2 +3x3=1                              3x1- x2+ x3 = 3                             11x1 - 13x2 -7x3 = 2

sciencesolve | Certified Educator

You need to eliminate `x_1` from the first and the second equations, such that:

`3(2x_1 + 2x_2 + 3x_3) = 3 => 6x_1 + 6x_2 + 9x_3 = 3`

`-2(3x_1 - x_2 + x_3) = -6 => -6x_1 + 2x_2 - 2x_3 = -6`

`8x_2 + 7x_2 = -3`

You need to eliminate` x_1` from the first and the third equations, such that:

`11(2x_1 + 2x_2 + 3x_3) = 11 => 22x_1 + 22x_2 + 33x_3 = 11`

`-2(11x_1 - 13x_2 - 7x_3) = -4 => -22x_1 + 26x_2 + 14x_3 = -4`

`48x_2 + 47x_3 = 7`

You need to eliminate `x_2` from the equations `8x_2 + 7x_2 = -3 ` and `48x_2 + 47x_3 = 7` , hence, you need to multiply by -6 the equation `8x_2 + 7x_2 = -3` such that:

`{(-6(8x_2 + 7x_3) = 18),(48x_2 + 47x_3 = 7):}`

`{(-48x_2 - 42 x_3) = -6*18),(48x_2 + 47x_3 = 7):}`

`5x_3 = -101 => x_3 = -101/5`

`8x_2 - 707/5 = -3 => 8x_2 = 692/5 => x_2 = 173/10`

`3x_1 = 3 + x_2 - x_3 => 3x_1 = 3 + 173/10 + 101/5`

`3x_1 = 405/10 => 3x_1 = 81/2 => x_1 = 27/2`

Hence, evaluating the solution to the system, using gaussian elimination, yields `x_1 = 27/2 , x_2 = 173/10 , x_3 = -101/5.`