If the system of simultaneous linear equations $3 x-4 y+k z+13=0, x+2 y-z-9=0$ and $k x-y+3 z+7=0$ has a unique solution $x=\alpha, y=\beta, z=\gamma$ for $k
eq m$ and $2 \beta-\gamma=8$, then $\alpha+m=$

Question

If the system of simultaneous linear equations $3 x-4 y+k z+13=0, x+2 y-z-9=0$ and $k x-y+3 z+7=0$ has a unique solution $x=\alpha, y=\beta, z=\gamma$ for $k 
eq m$ and $2 \beta-\gamma=8$, then $\alpha+m=$

Aicharya · Accepted Answer

$\begin{aligned} & 	ext { }\left|\begin{array}{ccc}3 & -4 & k \ 1 & 2 & -1 \ k & -1 & 3\end{array}ight| 
eq 0 \ & \Rightarrow \quad 3(6-1)+4(3+k)+k(-1-2 k) 
eq 0 \ & \Rightarrow 15+12+4 k-k-2 k^2 
eq 0 \ & \Rightarrow \quad 2 k^2-3 k-27 
eq 0 \Rightarrow 2 k^2-9 k+6 k-27 
eq 0 \ & \Rightarrow k 
eq \frac{9}{2},-3\end{aligned}$ On putting $x=\alpha, y=\beta, z=\gamma$ inequation $ \begin{gathered} x+2 y-z-9=0 \ \alpha+2 \beta-\gamma-9=0 \ \alpha+8-9=0 [It is given that 2 \beta-\gamma=8 ] \ \alpha=1 \end{gathered} $ As $k 
eq m \Rightarrow m=9 / 2-3$ $ \alpha+m=1+\frac{9}{2}=\frac{11}{2} 	ext { or } \alpha+m=1-3=-2 $

If the system of simultaneous linear equations $3 x-4 y+k z+13=0, x+2 y-z-9=0$ and $k x-y+3 z+7=0$ has a…

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