Question

For which of the following ordered pairs $(\mu ,\delta ),$ the system of linear equations
$x+2y+3z=1$
$3x+4y+5z=\mu$
$4x+4y+4z=\delta$
is inconsistent?

• A. $(4,6)$
• B. $(3,4)$
• C. $(1,0)$
• D. $(4,3)$

Answer 1

The correct option is D $(4,3)$

$D=\left|\begin{array}{ccc}1& 2& 3\\ 3& 4& 5\\ 4& 4& 4\end{array}\right|$

$R_3\to R_3+2R_1-2R_2$
$D=\left|\begin{array}{ccc}1& 2& 3\\ 3& 4& 5\\ 0& 0& 0\end{array}\right|=0$

For inconsistent system, one of $D_x,D_y,D_z$ should not be equal to $0.$
$D_x=\left|\begin{array}{ccc}1& 2& 3\\ \mu & 4& 5\\ \delta & 4& 4\end{array}\right|=2(2\mu -\delta -2)\cdots \left(i\right)$

$D_y=\left|\begin{array}{ccc}1& 1& 3\\ 3& \mu & 5\\ 4& \delta & 4\end{array}\right|=-4(-\delta +2\mu -2)\cdots \left(ii\right)$

$D_z=\left|\begin{array}{ccc}1& 2& 1\\ 3& 4& \mu \\ 4& 4& \delta \end{array}\right|=-2(\delta -2\mu +2)\cdots \left(iii\right)$
For inconsistent system, we get from $\left(i\right),\left(ii\right)$ and $\left(iii\right)$
$2\mu \ne \delta +2$
$\therefore$ From the provided options, the system will be inconsistent for $\mu =4,\delta =3.$