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

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

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Answer 1 · 2017-09-26T14:52:24

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

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

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