| \[x-4y+7z=g\] |
| \[3y-5z=h\] |
| \[-2x+5y-9z=k\] |
A) \[g+2h+k=0\]
B) \[g+h+2k=0\]
C) \[g+h+k=0\]
D) \[2g+h+k=0\]
Correct Answer: D
Solution :
\[x-4y+7z=g\] \[3y-5z=h\] \[-2x+5y-9z=k\] \[D=\left| \begin{align} & 1\,\,\,\,\,\,\,\,-4\,\,\,\,\,\,\,\,\,\,\,\,\,7 \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,3\,\,\,\,\,\,\,\,\,-5 \\ & -2\,\,\,\,\,\,\,\,\,5\,\,\,\,\,\,\,\,-9 \\ \end{align} \right|\] \[D=1(-27+25)-2\left( 20-21 \right)\] \[D=-2+2=0\] If system is consistent then \[{{D}_{1}}={{D}_{2}}^{~}={{D}_{3}}=0\] \[\left| \begin{align} & 1\,\,\,\,\,\,\,\,-4\,\,\,\,\,\,\,\,\,\,\,\,\,g \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,3\,\,\,\,\,\,\,\,\,\,\,\,\,h \\ & -2\,\,\,\,\,\,\,\,\,5\,\,\,\,\,\,\,\,\,\,\,\,\,k \\ \end{align} \right|=0\] \[1\left( 3k-5h \right)-2\left( -4h-3g \right)=0\] \[3k-5h+8h+6g=0\] \[6g+3h+3k=0\] \[2g+h+k=0\]
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