System of linear equations

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If $2 x - 3 y + 4 z = 0, 5 x - 2 y - z = 0, 21 x - 8 y + a z = 0$ has infinitely many solutions, then a=

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$T h e c o n d i t i o n f o r a h o m o g e n e o u s s y s t e m t o h a v e i n f i n i t e l y m a n y s o l u t i o n s i s Δ = 0$
$\begin{array}{l} \Rightarrow |\begin{matrix} 2 & - 3 & 4 \\ 5 & - 2 & - 1 \\ 21 & - 8 & a \end{matrix}| = 0 \\ \Rightarrow 2 (- 2 a - 8) + 3 (5 a + 21) + 4 (- 40 + 42) = 0 \\ \Rightarrow - 4 a - 16 + 15 a + 63 + 8 = 0 \end{array}$
$\begin{array}{l} \Rightarrow 11 a + 55 = 0 \\ \Rightarrow 11 a = - 55 \\ \Rightarrow a = - 5 \end{array}$

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System of linear equations

Remember concepts with our Masterclasses.

If 2x−3y+4z=0,5x−2y−z=0,21x−8y+az=0has infinitely many solutions, then a=

The condition for a homogeneous system to have infinitely many solutions is Δ=0⇒2-345-2-121-8a=0⇒2-2a-8+35a+21+4-40+42=0⇒-4a-16+15a+63+8=0⇒11a+55=0⇒11a=-55⇒a=-5

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If $2 x - 3 y + 4 z = 0, 5 x - 2 y - z = 0, 21 x - 8 y + a z = 0$ has infinitely many solutions, then a=