If the system of equations x + 2 y + 3 z = 3 4 x + 3 y - 4 z = 4 8 x + 4 y - z = 9 has infinitely many solutions, then the ordered pair is equal to [2023]

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Solution

Q.
If the system of equations

$x + 2 y + 3 z = 3$

$4 x + 3 y - 4 z = 4$

$8 x + 4 y - λ z = 9 + μ$

has infinitely many solutions, then the ordered pair $(λ, μ)$ is equal to [2023]

1 $(\frac{72}{5}, - \frac{21}{5})$

2 $(\frac{72}{5}, \frac{21}{5})$

3 $(- \frac{72}{5}, - \frac{21}{5})$

4 $(- \frac{72}{5}, \frac{21}{5})$

Ans.
(1)

$x + 2 y + 3 z = 3; 4 x + 3 y - 4 z = 4; 8 x + 4 y - λ z = 9 + μ$

$For infinite many solutions, Δ = 0 and Δ_{x} = 0$

$Δ =$ $| \begin{matrix} 1 & 2 & 3 \\ 4 & 3 & - 4 \\ 8 & 4 & - λ \end{matrix} |$ $= 0$

$\Rightarrow 1 (- 3 λ + 16) - 2 (- 4 λ + 32) + 3 (16 - 24) = 0$

$\Rightarrow 16 - 3 λ + 8 λ - 64 - 24 = 0 \Rightarrow 5 λ = 72$

$∴$ $λ = \frac{72}{5}$

$Δ_{x} =$ $| \begin{matrix} 3 & 2 & 3 \\ 4 & 3 & - 4 \\ 9 + μ & 4 & - λ \end{matrix} |$ $= 0$

$\Rightarrow 3 (- 3 λ + 16) - 2 (- 4 λ + 36 + 4 μ) + 3 (16 - 27 - 3 μ) = 0$

$\Rightarrow - 9 λ + 48 + 8 λ - 72 - 8 μ - 33 - 9 μ = 0$

$\Rightarrow - λ - 17 μ = 57 \Rightarrow - 17 μ = 57 + λ$

$∴$ $- μ = \frac{57 + \frac{72}{5}}{17} \Rightarrow μ = \frac{- 357}{85} = \frac{- 21}{5}$

$Thus, (λ, μ) \equiv (\frac{72}{5}, \frac{- 21}{5})$

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