If the system of equations has infinitely many solutions, then is equal to: [2025]

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Solution

Q.
If the system of Linear equations

$3 x + y + β z = 3$

$2 x + α y - z = - 3$

$x + 2 y + z = 4$

has infinitely many solutions, then the value of $22 β - 9 α$ is : [2025]

1 31

2 49

3 37

4 43

Ans.
(1)

Since, the given system of equations has infinitely many solutions, $△ = △_{1} = △_{2} = △_{3} = 0$

$△ =$ $| \begin{matrix} 3 & 1 & β \\ 2 & α & - 1 \\ 1 & 2 & 1 \end{matrix} |$ $= 0$

$\Rightarrow 3 α + 4 β - α β + 3 = 0$ ...(i)

$△_{3} =$ $| \begin{matrix} 3 & 1 & 3 \\ 2 & α & - 3 \\ 1 & 2 & 4 \end{matrix} |$ $= 0$

$\Rightarrow 9 α + 19 = 0 \Rightarrow α = \frac{- 19}{9}$

From (i), we get

$β = \frac{6}{11}$

$∴ 22 β - 9 α = 22 \times \frac{6}{11} - 9 \times \frac{- 19}{9} = 12 + 19 = 31$ .

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