System of linear equations

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Let $(x, y, z)$ be a non-zero solution of the equation $x + λ y + 2 z = 0$ , $2 x + λ z = 0 a n d 2 λ x - 2 y + 3 z = 0$ where $λ \in R$ then the value of $\frac{x + y - z}{y} =$ ______

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Solution

$\begin{array}{l} |\begin{array}{l} 1 & λ & 2 \\ 2 & 0 & λ \\ 2 λ & - 2 & 3 \end{array}| = 0 \\ \Rightarrow 1 (0 + 2 λ) - λ (6 - 2 λ^{2}) + 2 (- 4) = 0 \\ \Rightarrow 2 λ - 6 λ + 2 λ^{3} - 8 = 0 \\ \Rightarrow 2 λ^{2} - 4 λ - 8 = 0 \\ \Rightarrow λ^{3} - 2 λ - 4 = 0 \end{array} \Rightarrow (λ - 2) (λ^{2} + 2 λ + 2) = 0 \Rightarrow λ = 2$
$∴ T h e g i v e n e q u a t i o n s b e c o m e$
$\begin{array}{l} x + 2 y + 2 z = 0, 2 x + 2 z = 0, 4 x - 2 y + 3 z = 0 \\ S o l v i n g, w e g e t \\ \frac{\begin{array}{l} x + 2 y + 2 z = 0 \\ 2 x + 2 z = 0 \end{array}}{\begin{array}{l} 2 y - x = 0 \\ \Rightarrow x = 2 y \end{array}} \\ A l s o, 4 x - 2 y + 3 z = 0 \Rightarrow z = - 2 y \end{array}$
$∴ \frac{x + y - z}{y} = \frac{2 y + y + 2 y}{y} = \frac{5 y}{y} = 5$

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System of linear equations

Remember concepts with our Masterclasses.

Let x,y,zbe a non-zero solution of the equation x+λy+2z=0, 2x+λz=0 and 2λx−2y+3z=0 where λ∈R then the value of x+y−zy= ______

1λ220λ2λ-23=0⇒1(0+2λ)-λ6-2λ2+2(-4)=0⇒2λ-6λ+2λ3-8=0⇒2λ2-4λ-8=0⇒λ3-2λ-4=0 ⇒λ-2λ2+2λ+2=0 ⇒λ=2∴The given equations become x+2y+2z=0, 2x+2z=0, 4x-2y+3z=0Solving, we getx+2y+2z=02x+2z=02y−x=0⇒x=2y Also, 4x-2y+3z=0⇒z=-2y∴x+y−zy=2y+y+2yy=5yy=5

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Let $(x, y, z)$ be a non-zero solution of the equation $x + λ y + 2 z = 0$ , $2 x + λ z = 0 a n d 2 λ x - 2 y + 3 z = 0$ where $λ \in R$ then the value of $\frac{x + y - z}{y} =$ ______