If the system of linear equations
x - 4y + 7z = g
3y - 5z = h
-2x + 5y - 9z = k
is consistent, then

Options

A. 2g + h +k = 0
B. g + 2h + k =0
C. g + h + k =0
D. g + h + 2k =0

Explanation

(a) Here, `D = |{:(1, -4, 7), (0, 3, -5), (-2, 5, -9):}|`
` = 1(-27+25) +4(0-10) +7(0+6) " " ["expanding along "R_(1)]`
` =-2-40+42=0`
`therefore ` The system of linear equations have infinite many solutions.
[`because`system of consistent and does not have unique solution as D = 0]
`rArr D_(1) = D_(2) = D_(3) =0`
Now, `D_(1) = 0 rArr |{:(g, -4, 7), (h, 3, -5), (k, 5, -9):}| =0`
`rArr g(-27 +25) +4(-9h +5k) +7 (5h -3k) =0`
`rArr -2g-36h + 20k + 35h-21k =0`
`rArr -2g-h-k =0 rArr 2g + h + k =0`