Question

The equations $x+4y-2z=3$
$3x+y+5z=7$
$2x+3y+z=5$ 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, $D=|A|=\left|\begin{array}{ccc}1& 4& -2\\ 3& 1& 5\\ 2& 3& 1\end{array}\right|$
$D=-14+28-14=0$
Now, $D_1=\left|\begin{array}{ccc}3& 4& -2\\ 7& 1& 5\\ 5& 3& 1\end{array}\right|$
$\Rightarrow D_1=-42+72-32=2\ne 0$
Now, $D_2=\left|\begin{array}{ccc}1& 3& -2\\ 3& 7& 5\\ 2& 5& 1\end{array}\right|$
$\Rightarrow D_2=-18+21-2=1\ne 0$
Hence, $D=0$ and at least one of $D_1,D_2,D_3$ is non-zero.
Hence, the system of equation is inconsistent and has no solution