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Question

If the system of equations $$x+y+z=5$$, $$x+2y+3z=9$$, $$x+3y+\alpha z=\beta$$ has infinitely many solutions, then $$\beta -\alpha$$ equals?


A
5
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B
18
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C
21
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D
8
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Solution

The correct option is D $$8$$
$$D=\begin{vmatrix} 1 & 1 & 1\\ 1 & 2 & 3\\ 1 & 3 & \alpha\end{vmatrix}=\begin{vmatrix} 1 & 1 & 1\\ 0 & 1 & 2\\ 0 & 2 & \alpha -1\end{vmatrix}=(\alpha -1)-4=(\alpha -5)$$
for infinite solutions $$D=0\Rightarrow \alpha =5$$
$$D_x=0\Rightarrow \begin{vmatrix} 5 & 1 & 1\\ 9 & 2 & 3\\ \beta & 3 & 5\end{vmatrix}=0$$
$$\Rightarrow \begin{vmatrix} 0 & 0 & 1\\ -1 & -1 & 3\\ \beta -15 & -2 & 5\end{vmatrix}=0$$
$$\Rightarrow 2+\beta -15=0$$
$$\Rightarrow \beta -13=0$$
on $$\beta =13$$ we get $$D_y=D_z=0$$
$$\alpha =5, \beta =13$$.

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