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. $R -\{-11,11\}$

Answer 1

Answer: (D)

$\Delta=\begin{vmatrix}- k & 3 & -14 \\ -15 & 4 & - k \\ -4 & 1 & 3\end{vmatrix}=121- k ^{2}$
$\Delta \neq 0 $
$k \in R -\{11,-11\} $ (Unique sol.)
If $k =11$
$\Delta_{ z }=\begin{vmatrix}-11 & 3 & 25 \\ -15 & 4 & 3 \\ -4 & 1 & 4\end{vmatrix} \neq 0$
No solution
If $k =-11$
$\Delta_{ z }=\begin{vmatrix}11 & 3 & 25 \\ -15 & 4 & 3 \\ -4 & 1 & 4\end{vmatrix} \neq 0$