If the system of equations 3x – y + z = 4 has infinitely many solutions, then is equal to [2025]

Question

If the system of equations

2x – y + z = 4

$5 x + λ y + 3 z = 12$

$100 x - 47 y + μ z = 212$

has infinitely many solutions, then $μ - 2 λ$ is equal to [2025]

Smart Achievers · Accepted Answer

(4)

As the given system of equations has infinitely many solutions, then

$△ = △_{1} = △_{2} = △_{3} = 0$

$△_{2} =$ $| \begin{matrix} 2 & 4 & 1 \\ 5 & 12 & 3 \\ 100 & 212 & μ \end{matrix} |$ $= 0$

$\Rightarrow 2 (12 μ - 636) - 4 (5 μ - 300) + 1 (1060 - 1200) = 0$

$\Rightarrow 4 μ - 212 = 0$

$\Rightarrow μ = 53$

$△_{3} =$ $| \begin{matrix} 2 & - 1 & 4 \\ 5 & λ & 12 \\ 100 & - 47 & 212 \end{matrix} |$ $= 0$

$\Rightarrow 2 (212 λ + 564) + 1 (1060 - 1200) + 4 (- 235 - 100 λ) = 0$

$\Rightarrow 24 λ + 48 = 0 \Rightarrow λ = - 2$

$∴ μ - 2 λ = 53 - 2 (- 2) = 57$ .

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