If the system of linear equations
$\begin{aligned} & 3 x+y+\beta z=3 \\ & 2 x+\alpha y-z=-3 \\ & x+2 y+z=4\end{aligned}$
has infinitely many solutions, then the value of $22 \beta-9 \alpha$ is :

Question

If the system of linear equations $\begin{aligned} & 3 x+y+\beta z=3 \ & 2 x+\alpha y-z=-3 \ & x+2 y+z=4\end{aligned}$ has infinitely many solutions, then the value of $22 \beta-9 \alpha$ is :

Aicharya · Accepted Answer

$\Delta=\left|\begin{array}{ccc}3 & 1 & \beta \\ 2 & \alpha & -1 \\ 1 & 2 & 1\end{array}\right|=0$
$3 \alpha+4 \beta-\alpha \beta+3=0$
$\Delta_3=\left|\begin{array}{ccc}3 & 1 & 3 \\ 2 & \alpha & -3 \\ 1 & 2 & 4\end{array}\right|=0$
$9 \alpha+19=0$
$\alpha=\frac{-19}{9}, \beta=\frac{6}{11}$
$\Rightarrow 22 \beta-9 \alpha=31$

If the system of linear equations $\begin{aligned} & 3 x+y+\beta z=3 \\ & 2 x+\alpha y-z=-3 \\ & x+2…

Solution