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 $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ are real constants. Then the system of equations :
Correct Option: , 2
$D=\left|\begin{array}{ccc}1 & 2 & -3 \\ 2 & 6 & -11 \\ 1 & -2 & 7\end{array}\right|$
$=20-2(25)-3(-10)$
$=20-50+30=0$
$D_{1}=\left|\begin{array}{ccc}a & 2 & -3 \\ b & 6 & -11 \\ c & -2 & 7\end{array}\right|$
$=20 a-2(7 b+11 c)-3(-2 b-6 c)$
$=20 a-14 b-22 c+6 b+18 c$
$=20 a-8 b-4 c$
$=4(5 a-2 b-c)$
$D_{2}=\left|\begin{array}{ccc}1 & a & -3 \\ 2 & b & -11 \\ 1 & c & 7\end{array}\right|$
$=7 b+11 c-a(25)-3(2 c-b)$
$=7 b+11 c-25 a-6 c+3 b$
$=-25 a+10 b+5 c$
$=-5(5 a-2 b-c)$
$D_{3}=\left|\begin{array}{ccc}1 & 2 & a \\ 2 & 6 & b \\ 1 & -2 & c\end{array}\right|$
$=6 c+2 b-2(2 c-b)-10 a$
$=-10 a+4 b+2 c$
$=-2(5 a-2 b-c)$
for infinite solution
$\mathrm{D}=\mathrm{D}_{1}=\mathrm{D}_{2}=\mathrm{D}_{3}=0$
$\Rightarrow 5 a=2 b+c$