System of linear equations

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If the system of equations $x + y + z = 6$ , $x + 2 y + λ z = 0, x + 2 y + 3 z = 10$ has no solution, Then $λ =$

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Solution

$T h e a u g m e n t e d m a t r i x [A D] = [\begin{matrix} 1 & 1 & 1 & 6 \\ 1 & 2 & λ & 0 \\ 1 & 2 & 3 & 10 \end{matrix}] R_{3} \to R_{3} - R_{2} ~ [\begin{matrix} 1 & 1 & 1 & 6 \\ 1 & 2 & λ & 0 \\ 0 & 0 & 3 - λ & 10 \end{matrix}] H e r e r a n k [A D] = 4 I f t h e g i v e n s y s t e m h a s n o s o l u t i o n, t h e n r a n k A \neq r a n k [A D] \Rightarrow 3 - λ = 0 \Rightarrow λ = 3$

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System of linear equations

Remember concepts with our Masterclasses.

If the system of equationsx+y+z=6,x+2y+λz=0, x+2y+3z=10 has no solution, Thenλ=

The augmented matrix AD=111612λ012310 R3→R3-R2 ~111612λ0003-λ10 Here rank AD=4 If the given system has no solution, then rankA≠rankAD ⇒3-λ=0 ⇒λ=3

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If the system of equations $x + y + z = 6$ , $x + 2 y + λ z = 0, x + 2 y + 3 z = 10$ has no solution, Then $λ =$