Examine the consistency of the system of equations.

The given system of equations is:

2x − y = 5

x + y = 4

The given system of equations can be written in the form of AX = B, where

$A=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right], X=\left[\begin{array}{l}x \\ z\end{array}\right]$ and $B=\left[\begin{array}{l}5 \\ 4\end{array}\right]$

Now,

$|A|=2(1)-(-1)(1)=2+1=3 \neq 0$

$\therefore A$ is non-singular.

Therefore, $A^{-1}$ exists.

Hence, the given system of equations is consistent.