System of linear equations

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The value of a for which the equations $3 x - y + a z = 1, 2 x + y + z = 2, x + 2 y - a z = - 1$ fail to have unique solution is __________

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Solution

$3 x - y + a z = 1______(1)$
$2 x + y + z = 2 - - - - - - - (2)$
$x + 2 y - a z = - 1 - - - - - - (3)$
Eq, $(1) + (2) \Rightarrow 5 x + (a + 1) z = 3 - - - - - (4)$
$2 \times (1) + (3) \Rightarrow 7 x + a z = 1 - - - - - - (5)$
Given system fail to have unique solution.
$\Rightarrow$ The system has no solution or infinitely many solutions
$\begin{array}{l} \Rightarrow \frac{5}{7} = \frac{a + 1}{a} \neq \frac{3}{1} \\ \Rightarrow 5 a = 7 a + 7 ∣ \\ \Rightarrow 2 a = - 7 \\ \Rightarrow a = - \frac{7}{2} \end{array}$

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System of linear equations

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The value of a for which the equations 3x−y+az=1, 2x+y+z=2, x+2y−az=−1 fail to have unique solution is __________

3x−y+az=1______12x+y+z=2−−−−−−−2x+2y−az=−1−−−−−−3Eq,1+2⇒5x+a+1z=3-----(4)2×(1)+(3)⇒7x+az=1------(5)Given system fail to have unique solution.⇒ The system has no solution or infinitely many solutions⇒57=a+1a≠31⇒5a=7a+7∣⇒2a=-7⇒a=-72

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The value of a for which the equations $3 x - y + a z = 1, 2 x + y + z = 2, x + 2 y - a z = - 1$ fail to have unique solution is __________