First slide

System of linear equations

Question

The values of $λ$ and $μ$ for which the system of equations $x + y + z = 6$ , $x + 2 y + 3 z = 10$ , $x + 2 y + λ z = μ$ has infinitely many solutions are

Easy

A

$λ = 3, μ = 10$
B

$λ = 3, μ \neq 10$
C

$λ \neq 3, μ = 10$
D

$λ \neq 3, μ \neq 10$

Solution

The given system is of the form $A X = B$ , where
$A = |\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & λ \end{matrix}|, X = [\begin{matrix} x \\ y \\ z \end{matrix}]$ and $B = [\begin{matrix} 6 \\ 10 \\ μ \end{matrix}]$
For infinitely many solutions
$| A | = 0, |\begin{matrix} 6 & 1 & 1 \\ 10 & 2 & 3 \\ μ & 2 & λ \end{matrix}| = 0$
$|\begin{matrix} 1 & 6 & 1 \\ 1 & 10 & 3 \\ 1 & μ & λ \end{matrix}| = 0$ and also $|\begin{matrix} 1 & 1 & 6 \\ 1 & 2 & 10 \\ 1 & 2 & μ \end{matrix}| = 0$
$\Rightarrow λ - 3 = 0$ , $2 λ + μ - 16 = 0$
$\Rightarrow 10 λ - 3 μ - 6 (λ - 3) + (μ - 10) = 0$ and $μ - 10 = 0$
$\Rightarrow λ - 3 = 0, 2 (λ - 3) + (μ - 10) = 0$
$\Rightarrow 4 (λ - 3) - 2 (μ - 10) = 0$
And $μ - 10 = 0$
$\Rightarrow λ = 3$ and $μ = 10$ .

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