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

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

$2 g + h + k = 0$

$g + 2 h + k = 0$

$g + h + 2 k = 0$

$g + h + k = 0$

answer is A.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

here, $D = |\begin{matrix} 1 & - 4 & 7 \\ 0 & 3 & - 5 \\ - 2 & 5 & - 9 \end{matrix}|$

$= 1 (- 27 + 25) + 4 (0 - 10) + 7 (0 + 6) [expanding along R_{1}]$

$= - 2 - 40 + 42 = 0$

The system of linear equations have infinite many solutions.
[ system is consistent and does not have unique solutions as D=0]
$\Rightarrow D_{1} = D_{2} = D_{3} = 0$

$Now, D_{1} = 0 \Rightarrow |\begin{array}{ccc} g & - 4 & 7 \\ h & 3 & - 5 \\ k & 5 & - 9 \end{array}| = 0$

$\begin{array}{l} \Rightarrow g (- 27 + 25) + 4 (- 9 h + 5 k) + 7 (5 h - 3 k) = 0 \\ \Rightarrow - 2 g - 36 h + 20 k + 35 h - 21 k = 0 \\ \Rightarrow - 2 g - h - k = 0 \Rightarrow 2 g + h + k = 0 \end{array}$