Question

Consider the system? −3x + 5y + 3z = 0 6x + −7y + −4z = −2 9x + −6y + −3z = −6 Gaussian elimination of the augmented matrix for this system produces the matrix

Consider the system
−3x + 5y + 3z = 0
6x + −7y + −4z = −2
9x + −6y + −3z = −6
Gaussian elimination of the augmented matrix for this system produces the matrix

Answer 1

The solution is `((x),(y),(z))=((-10/9-1/9z),(-2/3-2/3z),(z))`

Explanation:

The augmented matrix is

`( (-3,5,3,|,0) , (6,-7,-4,|,-2),(9,-6,-3,|,-6) )`

Perform the following operations on the rows

`R3larr(R3)/3`

`( (-3,5,3,|,0) , (6,-7,-4,|,-2),(3,-2,-1,|,-2) )`

`R3larrR3+R1`

`( (-3,5,3,|,0) , (6,-7,-4,|,-2),(0,3,2,|,-2) )`

`R2larrR2+2R1`

`( (-3,5,3,|,0) , (0,3,2,|,-2),(0,3,2,|,-2) )`

`R3larrR3-R2`

`( (-3,5,3,|,0) , (0,3,2,|,-2),(0,0,0,|,0) )`

`R1larr(R1)/(-3)`

`( (1,-5/3,-1,|,0) , (0,3,2,|,-2),(0,0,0,|,0) )`

`R2larr(R2)/(3)`

`( (1,-5/3,-1,|,0) , (0,1,2/3,|,-2/3),(0,0,0,|,0) )`

`R1larr(R1+5/3R2)`

`( (1,0,1/9,|,-10/9) , (0,1,2/3,|,-2/3),(0,0,0,|,0) )`

The solution is

`((x),(y),(z))=((-10/9-1/9z),(-2/3-2/3z),(z))`