Examine the consistency of the system of equations$5x-y+4z=5,2x+3y+5z=2,5x-2y+6z=-1$

Given system of equations

$5x-y+4z=5$
$2x+3y+5z=2$
$5x-2y+6z=-1$

This can be written as

$AX=B$

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

Here, $|A|=5(18+10)+1(12-25)+4(-4-15)$

$\Rightarrow |A|=140-13-76=51$

Since, $|A|\ne 0$

Hence, the system of equations is consistent.