Question

For the system of equations:
$x+2y+3z=1$
$2x+y+3z=2$
$5x+5y+9z=4$

• A. there is only one solution
• B. there exists infinitely many soulution
• C. there is no soulution
• D. none of these

Answer 1

The correct option is A there is only one solution

$D=\left|\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\\ 5& 5& 9\end{array}\right|$
$\Rightarrow D=-7\ne 0$
Now, $D_1=\left|\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\\ 4& 5& 9\end{array}\right|$
$\Rightarrow D_1=0$
Now, $D_2=\left|\begin{array}{ccc}1& 1& 3\\ 2& 2& 3\\ 5& 4& 9\end{array}\right|$
$\Rightarrow D_2=-1\ne 0$
So, here $D\ne 0$ and at least one $D_1,D_2,D_3\ne 0$
Then there is a unique solution.