Question

The system of equations
$x+4y-3z=3$
$x-y+7z=11$
$2x+8y-6z=7$
have

• A. unique solution
• B. infinitely many solutions
• C. no solutions
• D. only finite number of solutions

Answer 1

Answer: (C)

The given system of equations can be written in matrix form as,
$\left[\begin{array}{ccc}1& 4& -3\\ 1& -1& 7\\ 2& 8& -6\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}3\\ 11\\ 7\end{array}\right]$
i.e., $AX=B$
Now, $\left|A\right|=\left|\begin{array}{ccc}1& 4& -3\\ 1& -1& 7\\ 2& 8& -6\end{array}\right|=0$
$\therefore$ Solution is not unique.
Now, $adj\left(A\right)=\left[\begin{array}{ccc}-50& 0& 25\\ 20& 0& -10\\ 10& 0& -5\end{array}\right]$
Now, $\left(adjA\right)B=\left[\begin{array}{ccc}-50& 0& 25\\ 20& 0& -10\\ 10& 0& -5\end{array}\right]\left[\begin{array}{c}3\\ 11\\ 7\end{array}\right]\ne O$
$\therefore$ The given system of equations have no solution.