For , R, suppose the system of linear equations gathered x-y+z=5 2x+2y+ z=8 3x-y+4z= gathe…

Question

For $\alpha ,\beta \in R$, suppose the system of linear equations

$\begin{gathered} x-y+z=5 \ 2x+2y+\alpha z=8 \ 3x-y+4z=\beta \end{gathered}$

has infinitely many solutions. Then $\alpha $ and $\beta $ are the roots of :

Accepted Answer

$\Delta=\begin{vmatrix} 1 \quad  -1 \quad  1 \ 2 \quad  2 \quad  \alpha \ 3 \quad  -1 \quad  4 \end{vmatrix}=0$

$\Rightarrow \alpha =4$

${\Delta}_{3}=0$

$=\begin{vmatrix} 1 \quad  -1 \quad  5 \ 2 \quad  2 \quad  8 \ 3 \quad  -1 \quad  \beta \end{vmatrix}=0$

$\Rightarrow \beta =14$

$	herefore $ ${x}^{2}-18x+56=0$

Solutions · 1