Question

Examine the consistency of the system of equations :

$5x-y+4z=5$

$2x+3y+5z=2$

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

Answer 1

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 $AX=B$, 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, non-singular

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

$|A|=51\ne 0$

A is non-singular.

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

Hence, the given system of equations is consistent.