A) no solution is possible
B) only one solution
C) infinite solutions
D) None of the above
Correct Answer: B
Solution :
Given, equations are \[x+2y+3z=1\] \[2x+y+3z=2\] and \[5x+5y+9z=4\] Write in the form \[AX=B\] \[\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 5 & 5 & 9 \\ \end{matrix} \right]\left[ \begin{matrix} x \\ y \\ z \\ \end{matrix} \right]=\left[ \begin{matrix} 1 \\ 2 \\ 4 \\ \end{matrix} \right]\] Now, \[|A|=\left| \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 5 & 5 & 9 \\ \end{matrix} \right|=\left| \begin{matrix} 1 & 2 & 3 \\ 0 & -3 & -3 \\ 0 & -5 & -6 \\ \end{matrix} \right|\] \[\left[ \begin{align} & {{R}_{2}}\to {{R}_{2}}-2{{R}_{1}} \\ & {{R}_{3}}\to {{R}_{3}}-5{{R}_{1}} \\ \end{align} \right]\] \[=1[(-3\times -6)-(-3\times -5)]\] \[=1[18-15]=3\ne 0\] Hence, equations have only one solution.
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