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

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System of linear equations

Remember concepts with our Masterclasses.

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

The condition for a non-homogeneous system of equations to be inconsistent is rankA≠rankAD Augmented matrix AD=111612λ012310 R3→R3-R2, R2→R2-R1 ~111601λ-1-6003-λ10 Since rankA≠rankAD, we must have 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 $λ =$