Examine the consistency of the system of equations.

The given system of equations is:

x + 3y = 5

2x + 6y = 8

The given system of equations can be written in the form of AX = 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.

Now,

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

$(\operatorname{adj} A) B=\left[\begin{array}{cc}6 & -3 \\ -2 & 1\end{array}\right]\left[\begin{array}{l}5 \\ 8\end{array}\right]=\left[\begin{array}{l}30-24 \\ -10+8\end{array}\right]=\left[\begin{array}{l}6 \\ -2\end{array}\right] \neq O$

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