Question

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

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

Answer 1

The correct option is A there is only one solution

Given system of equation can be written as AX=B
where $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\\ 5& 5& 9\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}1\\ 2\\ 4\end{array}\right]$
Here, $D=\left|\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\\ 5& 5& 9\end{array}\right|$

$\Rightarrow D=-6-6+15=3$

Here, $D\ne 0$

Hence, the system of equations are consistent
Now,$D_1=\left|\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\\ 4& 5& 9\end{array}\right|$

$\Rightarrow D_1=-6-12+18=0$

$D_2=\left|\begin{array}{ccc}1& 1& 3\\ 2& 2& 3\\ 5& 4& 9\end{array}\right|$

$\Rightarrow D_2=6-3-6=-3\ne 0$

So, $D\ne 0$ and at least one of $D_1,D_2,D_3\ne 0$

Hence, the system has a unique non-trivial solution.