If the system of linear equations:

x + y + z = 6

x + 2y + 5z = 10

2x + 3y + $λ$ z = $μ$

has infinitely many solutions, then the value of $λ + μ$ equals: [2026]

Question

If the system of linear equations:

x + y + z = 6

x + 2y + 5z = 10

2x + 3y + $λ$ z = $μ$

has infinitely many solutions, then the value of $λ + μ$ equals: [2026]

Smart Achievers · Accepted Answer

(3)

x + y + z = 6

x + 2y + 3z = 10

2x + 3y + $λ$ z = $μ$

For infinitely many solutions, $△ = 0$

$\Rightarrow | \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 5 \\ 2 & 3 & λ \end{matrix} | = 0$

$\Rightarrow (2 λ - 15) - (λ - 10) + (3 - 4) = 0$

$\Rightarrow λ = 6$

Also, $△_{1} = 0$

$\Rightarrow | \begin{matrix} 6 & 1 & 1 \\ 10 & 2 & 5 \\ μ & 3 & 6 \end{matrix} | = 0$

$\Rightarrow 6 (12 - 15) - (60 - 5 μ) + (30 - 2 μ) = 0$

$\Rightarrow - 18 - 60 + 5 μ + 30 - 2 μ = 0$

$\Rightarrow 3 μ = 48 \Rightarrow μ = 16$

$∴ λ + μ = 22$

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