I assume the first equation is supposed to be
[tex]5x-3y+2z=13[/tex]
and not
[tex]5x-3x+2x=4x=13[/tex]
As an augmented matrix, this system is given by
[tex]\left[\begin{array}{ccc|c}5&-3&2&13\\2&-1&-3&1\\4&-2&4&12\end{array}\right][/tex]
Multiply through row 3 by 1/2:
[tex]\left[\begin{array}{ccc|c}5&-3&2&13\\2&-1&-3&1\\2&-1&2&6\end{array}\right][/tex]
Add -1(row 2) to row 3:
[tex]\left[\begin{array}{ccc|c}5&-3&2&13\\2&-1&-3&1\\0&0&5&5\end{array}\right][/tex]
Multiply through row 3 by 1/5:
[tex]\left[\begin{array}{ccc|c}5&-3&2&13\\2&-1&-3&1\\0&0&1&1\end{array}\right][/tex]
Add -2(row 3) to row 1, and add 3(row 3) to row 2:
[tex]\left[\begin{array}{ccc|c}5&-3&0&11\\2&-1&0&4\\0&0&1&1\end{array}\right][/tex]
Add -3(row 2) to row 1:
[tex]\left[\begin{array}{ccc|c}-1&0&0&-1\\2&-1&0&4\\0&0&1&1\end{array}\right][/tex]
Multiply through row 1 by -1:
[tex]\left[\begin{array}{ccc|c}1&0&0&1\\2&-1&0&4\\0&0&1&1\end{array}\right][/tex]
Add -2(row 1) to row 2:
[tex]\left[\begin{array}{ccc|c}1&0&0&1\\0&-1&0&2\\0&0&1&1\end{array}\right][/tex]
Multipy through row 2 by -1:
[tex]\left[\begin{array}{ccc|c}1&0&0&1\\0&1&0&-2\\0&0&1&1\end{array}\right][/tex]
The solution to the system is then
[tex]\boxed{x=1,y=-2,z=1}[/tex]
