If the system of equations alpha x+y+z=5, x+2 | vedclass

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Question

For infinitely many solutions,

$\Delta=0=\Delta_{ x }=\Delta_{ y }=\Delta_{ z }$

$\Delta=\left|\begin{array}{lll}\alpha & 1 & 1 \ 1 &am

Vedclass · Accepted Answer

$(1,3)$

Vedclass · Answer

For infinitely many solutions,

$\Delta=0=\Delta_{ x }=\Delta_{ y }=\Delta_{ z }$

$\Delta=\left|\begin{array}{lll}\alpha & 1 & 1 \ 1 & 2 & 3 \ 1 & 3 & 5\end{array}ight|=0$

$\Rightarrow \alpha(10-9)-1(5-3)+1(3-2)=0$

$\Rightarrow \alpha-2+1=0$

$\Rightarrow \alpha=1$

$\Delta_{x}=\left|\begin{array}{lll}5 & 2 & 3 \ \beta & 3 & 5\end{array}ight|=0$

$\Rightarrow 5(10-9)-1(20-3 \beta)+1(12-2 \beta)$

$\Rightarrow 5-20+3 \beta+12-2 \beta$

$\Rightarrow-3+\beta=0$

$\Rightarrow \beta=3$

If the system of equations $\alpha x+y+z=5, x+2 y+$ $3 z=4, x+3 y+5 z=\beta$ has infinitely many solutions, then the ordered pair $(\alpha, \beta)$ is equal to:

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