If the system of equations has infinitely many solutions, then equals:

Question

If the system of equations $x + y + z = 5, x + 2 y + 3 z = 9, x + 3 y + α z = β$ has infinitely many solutions, then $β - 2 α$ equals:

Smart Achievers · Accepted Answer

(2)

$x + y + z = 5,$ $x + 2 y + 3 z = 9,$ $x + 3 y + α z = β$

$D =$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & α \end{matrix} |$ $= 0$

$\Rightarrow (2 α - 9) + (3 - α) + (3 - 2) = 0$

$\Rightarrow α = 5$

$Now,$ $D_{3} =$ $| \begin{matrix} 1 & 1 & 5 \\ 1 & 2 & 9 \\ 1 & 3 & β \end{matrix} |$ $= 0$

$\Rightarrow 2 β - 27 + 9 - β + 5 (3 - 2) = 0$

$\Rightarrow β = 13$

$\Rightarrow at α = 5, β = 13$

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