The system of linear equations 3x - 2y - kz = 10 2x - 4y - 2z = 6 x+2y - z = 5m is inconsi…

Question

The system of linear equations
3x - 2y - kz = 10
2x - 4y - 2z = 6
x+2y - z = 5m
is inconsistent if :

Accepted Answer

$$\Delta  = \left| {\begin{pmatrix} 3 & { - 2} & { - k}  \
   1 & { - 4} & { - 2}  \
   1 & 2 & { - 1}  \

\end{pmatrix} } \right| = 0$$

$3(4 + 4) + 2( - 2 + 2) - k(4 + 4) = 0$

$$ \Rightarrow k = 3$$

$${\Delta _x} = \left| {\begin{pmatrix} {10} & { - 2} & { - 3}  \
   6 & { - 4} & { - 2}  \
   {5m} & 2 & { - 1}  \

\end{pmatrix} } \right| \ne 0$$

$$10(4 + 4) + 2( - 6 + 10m) - 3(12 + 20m) \ne 0$$

$$80 - 12 + 20m - 36 - 60m \ne 0$$

$$40m \ne 32 \Rightarrow m \ne {4 \over 5}$$

$${\Delta _y} = \left| {\begin{pmatrix} 3 & {10} & { - 3}  \
   2 & 6 & { - 2}  \
   1 & {5m} & { - 1}  \

\end{pmatrix} } \right| \ne 0$$

$$3( - 6 + 10m) - 10( - 2 + 2) - 3(10m - 6) \ne 0$$

$$ - 18 + 30m - 30m + 18 \ne 0 \Rightarrow 0$$

$${\Delta _z} = \left| {\begin{pmatrix} 3 & { - 2} & {10}  \
   2 & { - 4} & 6  \
   1 & 2 & {5m}  \

\end{pmatrix} } \right| \ne 0$$

$$3( - 20m - 12) + 2(10m - 6) + 10(4 + 4) - 40m + 32 \ne 0 \Rightarrow m \ne {4 \over 5}$$

Solutions · 1