If the system of equations: x + y + z = 5; x + 2y + 3z = 9; x + 2y + z = has infinitely many solutions, then the value of is: [2026]

Question

If the system of equations: x + y + z = 5; x + 2y + 3z = 9; x + 2y + $λ$ z = $μ$ has infinitely many solutions, then the value of $λ + μ$ is: [2026]

Smart Achievers · Accepted Answer

(2)

Since, given system equations have infinitely many solutions.

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

$\Rightarrow 1 (2 λ - 9) - 1 (λ - 3) + 1 (3 - 2) = 0$

$\Rightarrow 2 λ - 9 - (λ - 3) + 1 = 0 \Rightarrow λ - 5 = 0$

$\Rightarrow λ = 5$

$∴ △_{3} = | \begin{matrix} 1 & 1 & 5 \\ 1 & 2 & 9 \\ 1 & 3 & μ \end{matrix} | = 0$

$\Rightarrow 1 (2 μ - 27) - 1 (μ - 9) + 5 (3 - 2) = 0$

$\Rightarrow (2 μ - 27) - (μ - 9) + 5 (1) = 0$

$\Rightarrow μ - 13 = 0 \Rightarrow μ = 13$

So, $λ + μ = 5 + 13 = 18$

Solution

Q.
If the system of equations: x + y + z = 5; x + 2y + 3z = 9; x + 2y + $λ$ z = $μ$ has infinitely many solutions, then the value of $λ + μ$ is: [2026]

1 16

2 18

3 19

4 21

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