Question

Examine the consistency of the system of equations

$x+3y=5,2x+6y=8$

Answer 1

The given system can be written as AX=B, where
$A=\left[\begin{array}{cc}1& 2\\ 2& 6\end{array}\right],X=\left[\begin{array}{c}x\\ y\end{array}\right]B=\left[\begin{array}{c}5\\ 8\end{array}\right]|A|=\left[\begin{array}{cc}1& 3\\ 2& 6\end{array}\right]=1\left(6\right)-3\left(2\right)=6-6=0$
Nothing can be said about consistency as yet. We compute.
$\left(adjA\right)B={\left[\begin{array}{cc}6& -2\\ -3& 1\end{array}\right]}^T\left[\begin{array}{c}5\\ 8\end{array}\right][\because adj(A)={\left[\begin{array}{cc}6& -2\\ -3& 1\end{array}\right]}^T]=\left[\begin{array}{cc}6& -2\\ -3& 1\end{array}\right]\left[\begin{array}{c}5\\ 8\end{array}\right]=\left[\begin{array}{cc}30-24& \\ -10+8& \end{array}\right]=\left[\begin{array}{c}6\\ -2\end{array}\right]\ne 0$
Thus, the solution of the given system of equations does not exist. Hence, the system of equations is inconsistent.