The given system of equations is:

$x+3 y=5$

$2 x+6 y=8$

The given system of equation can be written in the form of $A X=B$, where

$A=\left[\begin{array}{ll}1 & 3 \\ 2 & 6\end{array}\right], X=\left[\begin{array}{l}x \\ y\end{array}\right]$ and  $B=\left[\begin{array}{l}5 \\ 8\end{array}\right]$

Now, $|A|=1(6)-3(2)=6-6=0$

$\therefore A$ is a singular matrix.

$(a d j A)=\left[\begin{array}{cc}6 & -3 \\ -2 & 1\end{array}\right]$

$(\text {adjA}) B=\left[\begin{array}{cc}6 & -3 \\ -2 & 1\end{array}\right]\left[\begin{array}{l}5 \\ 8\end{array}\right]=\left[\begin{array}{c}30-24 \\ -10+8\end{array}\right]=\left[\begin{array}{c}6 \\ -2\end{array}\right] \neq 0$

Thus, the solution of the given system of equations does not exists, Hence, the given system of equations is inconsistent.