Question

The system of equations $x+4y-2z=3,3x+y+5z=7,2x+3y+z=5$ will have

• A. a unique solution
• B. infinite number of solutions
• C. no solution
• D. two solutions

Answer 1

The correct option is C no solution

The given system of equations can be written as $AX=B$
Where $A=\left[\begin{array}{ccc}1& 4& -2\\ 3& 1& 5\\ 2& 3& 1\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}3\\ 7\\ 5\end{array}\right]$

Here, $D=\left|A\right|=\left|\begin{array}{ccc}1& 4& -2\\ 3& 1& 5\\ 2& 3& 1\end{array}\right|$
$\Rightarrow D=-14+28-14=0$

Now, $D_x=\left|\begin{array}{ccc}3& 4& -2\\ 7& 1& 5\\ 5& 3& 1\end{array}\right|$
$\Rightarrow D_x=-42+72-32=-2\ne 0$
If $D=0$ and at least one of $D_x,D_y,D_z$ is non-zero then the system of equations has no solution.
Hence, by Cramer's rule the given system of equations is inconsistent .