The system of equations -k x+3 y-14 z=25 -15 x+4 y-k z=3 -4 x+y+3 z=4 is consistent for all k in the set

Question

The system of equations -k x+3 y-14 z=25 -15 x+4 y-k z=3 -4 x+y+3 z=4 is consistent for all k in the set (A) R (B) R - -11,13  (C) R - 13  (D) R - -11,11

Tardigrade Admin · Accepted Answer

Δ=|- k 3 -14 -15 4 - k -4 1 3|=121- k 2 Δ ≠ 0 k ∈ R - 11,-11 (Unique sol.) If k =11 Δ z =|-11 3 25 -15 4 3 -4 1 4| ≠ 0 No solution If k =-11 Δ z =|11 3 25 -15 4 3 -4 1 4| ≠ 0

Q. The system of equations
$- k x + 3 y - 14 z = 25$
$- 15 x + 4 y - k z = 3$
$- 4 x + y + 3 z = 4$
is consistent for all $k$ in the set

$R$

$R - {- 11, 13}$

$R - {13}$

$R - {- 11, 11}$

$Δ = ∣ ∣ - k - 15 - 4 341 - 14 - k 3 ∣ ∣ = 121 - k^{2}$
$Δ \neq = 0$
$k \in R - {11, - 11}$ (Unique sol.)
If $k = 11$
$Δ_{z} = ∣ ∣ - 11 - 15 - 4 3412534 ∣ ∣ \neq = 0$
No solution
If $k = - 11$
$Δ_{z} = ∣ ∣ 11 - 15 - 4 3412534 ∣ ∣ \neq = 0$

Q. The system of equations −kx+3y−14z=25 −15x+4y−kz=3 −4x+y+3z=4 is consistent for all k in the set

R

R−{−11,13}

R−{13}

R−{−11,11}

Δ=∣∣​−k−15−4​341​−14−k3​∣∣​=121−k2 Δ=0 k∈R−{11,−11} (Unique sol.) If k=11 Δz​=∣∣​−11−15−4​341​2534​∣∣​=0 No solution If k=−11 Δz​=∣∣​11−15−4​341​2534​∣∣​=0

Q. The system of equations
$- k x + 3 y - 14 z = 25$
$- 15 x + 4 y - k z = 3$
$- 4 x + y + 3 z = 4$
is consistent for all $k$ in the set

$R$

$R - {- 11, 13}$

$R - {13}$

$R - {- 11, 11}$

$Δ = ∣ ∣ - k - 15 - 4 341 - 14 - k 3 ∣ ∣ = 121 - k^{2}$
$Δ \neq = 0$
$k \in R - {11, - 11}$ (Unique sol.)
If $k = 11$
$Δ_{z} = ∣ ∣ - 11 - 15 - 4 3412534 ∣ ∣ \neq = 0$
No solution
If $k = - 11$
$Δ_{z} = ∣ ∣ 11 - 15 - 4 3412534 ∣ ∣ \neq = 0$