Answer
Pivots are
$
\begin{bmatrix}
(2) && -3 && 0 && 3\\
0 && (1) && 1 && 1\\
0 && 0 && -5 && 0\\
\end{bmatrix}$
Solutions are
$x=3,y=1,z=0$
Work Step by Step
$
2x − 3y = 3\\
4x − 5y + z = 7\\
2x − y − 3z = 5\\
$
Converting them into matrix from, we get
$\begin{bmatrix}
2 && -3 && 0 && 3\\
4 && -5 && 1 && 7\\
2 && -1 && -3 && 5\\
\end{bmatrix}$
Subtract 2 times row 1 from row 2 &
Subtract 1 times row 1 from row 3
$\sim
\begin{bmatrix}
2 && -3 && 0 && 3\\
0 && 1 && 1 && 1\\
0 && 2 && -3 && 2\\
\end{bmatrix}$
Subtract 2 times row 2 from row 3
$\sim
\begin{bmatrix}
(2) && -3 && 0 && 3\\
0 && (1) && 1 && 1\\
0 && 0 && -5 && 0\\
\end{bmatrix}$
Where the numbers in brackets are pivot.
Now, back substituting, we get
$-5z = 0, y+z=1,2x-3y=3$
Which on solving gives,
$z=0,y=1,x=3$
