Examine the consistency of the system of equations.

The given system of equations is:

5x − y + 4z = 5

2x + 3y + 5z = 2

5x − 2y + 6z = −1

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

$A=\left[\begin{array}{ccc}5 & -1 & 4 \\ 2 & 3 & 5 \\ 5 & -2 & 6\end{array}\right], X=\left[\begin{array}{c}x \\ y \\ z\end{array}\right]$ and $B=\left[\begin{array}{c}5 \\ 2 \\ -1\end{array}\right]$

Now,

$\begin{aligned}|A| &=5(18+10)+1(12-25)+4(-4-15) \\ &=5(28)+1(-13)+4(-19) \\ &=140-13-76 \\ &=51 \neq 0 \end{aligned}$

$\therefore A$ is non-singular.

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

Hence, the given system of equations is consistent.