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If $\small x = a, y = b, z = c$ is a solution of the system of linear equations

$\small x+8y+7z=0$

$\small 9x+2y+3z=0$

$\small x+y+z=0$

such that the point (a, b, c) lies on the plane $\small x+2y+z=6,$ then $\small 2a+b+c$ equals :

Option 1)
$?1$ - 1

Option 2)
0

Option 3)
1

Option 4)
2

Answers (2)

best_answer

As we learnt in

Cramer's rule for solving system of linear equations -

When $\Delta =0$ and $\Delta _{1}=\Delta _{2}=\Delta _{3}=0$ ,

then the system of equations has infinite solutions.

- wherein

$a_{1}x+b_{1}y+c_{1}z=d_{1}$

$a_{2}x+b_{2}y+c_{2}z=d_{2}$

$a_{3}x+b_{3}y+c_{3}z=d_{3}$

and

$\Delta =\begin{vmatrix} a_{1} &b_{1} &c_{1} \\ a_{2} & b_{2} &c_{2} \\ a_{3}&b _{3} & c_{3} \end{vmatrix}$

$\Delta _{1},\Delta _{2},\Delta _{3}$ are obtained by replacing column 1,2,3 of $\Delta$ by $\left ( d_{1},d_{2},d_{3} \right )$ column

$\\ a+8b+7c=0 \\ 9a+2b+3c=0 \\ a+b+c=0$

and $a+2b+c=6$ given

solve for a,b, and c

so that

$\\ a = 1 \\ b=6 \\ c= -7$

$\\ \therefore 2a + b+c = 2\times 1 + 6 -7 \\ = 2 +6 -7 \\= 8-7 =1$

Option 1)

$?1$ - 1

This option is incorrect.

Option 2)

0

This option is incorrect.

Option 3)

1

This option is correct.

Option 4)

2

This option is incorrect.

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prateek

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