Find-value-of-such-that-the-following-system-has-infinite-many-solutions-x-3z-3-2x-y-z-2-x-2y-z-1-

Question Number 42375 by Joel578 last updated on 24/Aug/18

Find value of α such that the following system

has infinite many solutions

x - 3 z = - 3

- 2 x - α y + z = 2

x + 2 y + α z = 1

Commented by Joel578 last updated on 24/Aug/18

What is the idea to solve this problem ?

Commented by maxmathsup by imad last updated on 24/Aug/18

f i r s t w e m u s t h a v e d e t A = 0 (i f d e t A \neq 0 w e g e t a u n i q u e s o l u t i o n)

w i t h A = (\begin{matrix} 1 0 - 3 \\ - 2 - α 1 \end{matrix})

(1 2 α)

d e t A = - α^{2} - 2 + 2 (6) - 3 α = - α^{2} - 3 α + 10

d e t A = 0 \Rightarrow α^{2} + 3 α - 10 = 0

Δ = 9 + 40 = 49 \Rightarrow α_{1} = \frac{- 3 + 7}{2} = 2 a n d α_{2} = \frac{- 3 - 7}{2} = - 5

f o r α = 2 w e g e t t h e s y s t e m {\begin{cases} x - 3 z = - 3 \\ - 2 x - 2 y + z = 2 \end{cases} \Rightarrow

{x + 2 y + 2 z = 1

{\begin{cases} x - 3 z = - 3 \\ - x + 3 z = 3 a n d x + 2 y + 2 z = 1 \Rightarrow {\begin{cases} x - 3 z = - 3 \\ 2 y = 1 - x - 2 z \end{cases} \end{cases}

\Rightarrow {\begin{cases} x = 3 z - 3 \\ y = \frac{1}{2} {1 - 3 z + 3 - 2 z} = \frac{1}{2} {2 - 5 z} \Rightarrow \end{cases}

(x, y, z) = (3 z - 3, 1 - \frac{5}{2} z, z) i n f i n i t e s o l u t i o n

f o r α = - 5 w e f o l l o w t h e s a m e m e t h o d \dots .

Commented by Joel578 last updated on 25/Aug/18

t h a n k y o u v e r y m u c h

Commented by math khazana by abdo last updated on 26/Aug/18

y o u a r e w e l c o m e .

Commented by Joel578 last updated on 27/Aug/18

Sir, if α = - 5, the system will have no solution

Answered by behi83417@gmail.com last updated on 24/Aug/18

1 x + 0 y - 3 z = - 3

- 2 x - α y + 1 z = 2

1 x + 2 y + α z = 1

▴ = 1 \times | \begin{matrix} - α 1 \\ 2 α \end{matrix} | - 0 \times | \begin{matrix} - 2 1 \\ 1 α \end{matrix} | - 3 \times | \begin{matrix} - 2 - α \\ 1 2 \end{matrix} | \neq 0

\Rightarrow - α^{2} - 2 - 3 (- 4 + α) \neq 0

\Rightarrow α^{2} + 3 α - 10 \neq 0

\Rightarrow α \neq \frac{- 3 \pm \sqrt{9 + 40}}{2} \neq \frac{- 3 \pm 7}{2} \neq 2, - 5 .

Commented by maxmathsup by imad last updated on 24/Aug/18

s i r b e h i i f d e t M \neq 0 t h e s y s t e m h a v e o n e s o l u t i o n \dots

Commented by behi83417@gmail.com last updated on 24/Aug/18

y e s s i r .

w h e n ▴ = 0 \Rightarrow i . e : α = 2 o r - 5, t h e

s y s t e m h a v e m a n y s o l u t i o n s .

Commented by Joel578 last updated on 25/Aug/18

t h a n k y o u v e r y m u c h

Commented by Joel578 last updated on 27/Aug/18

If α = - 5, the system will have no solution

Pls check

Commented by Joel578 last updated on 27/Aug/18

(\begin{matrix} 1 & 0 & - 3 & - 3 \\ - 2 & 5 & 1 & 2 \\ 1 & 2 & - 5 & 1 \end{matrix})

R_{2} + 2 R_{1} \to R_{2} (\begin{matrix} 1 & 0 & - 3 & - 3 \\ 0 & 5 & - 5 & - 4 \\ 1 & 2 & - 5 & 1 \end{matrix})

R_{3} - R_{1} \to R_{3} (\begin{matrix} 1 & 0 & - 3 & - 3 \\ 0 & 5 & - 5 & - 4 \\ 0 & 2 & - 2 & 4 \end{matrix})

R_{2} \leftrightarrow R_{3} (\begin{matrix} 1 & 0 & - 3 & - 3 \\ 0 & 2 & - 2 & 4 \\ 0 & 5 & - 5 & - 4 \end{matrix})

\frac{1}{2} R_{2} \leftrightarrow R_{2} (\begin{matrix} 1 & 0 & - 3 & - 3 \\ 0 & 1 & - 1 & 2 \\ 0 & 5 & - 5 & - 4 \end{matrix})

R_{3} - 5 R_{2} \to R_{3} (\begin{matrix} 1 & 0 & - 3 & - 3 \\ 0 & 1 & - 1 & 2 \\ 0 & 0 & 0 & - 14 \end{matrix})

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