Consider the following system of equations :


Consider the following system of equations :

$x+2 y-3 z=a$

$2 x+6 y-11 z=b$

$x-2 y+7 z=c$

where $a, b$ and $c$ are real constants. Then the system of equations :

  1. has a unique solution when $5 a=2 b+c$

  2. has infinite number of solutions when $5 \mathrm{a}=2 \mathrm{~b}+\mathrm{c}$

  3. has no solution for all $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$

  4. has a unique solution for all a, b and c

Correct Option: , 2


$P_{1}: x+2 y-3 z=a$

$P_{2}: 2 x+6 y-11 z=b$

$P_{3}: x-2 y+7 z=c$


$5 P_{1}=2 P_{2}+P_{3} \quad$ if $5 a=2 b+c$

$\Rightarrow$ All the planes sharing a line of intersection

$\Rightarrow$ infinite solutions

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