I assume the third equation is supposed to be [tex]6x+4y+2z=-44[/tex]. We can divide both sides by 2 right away to simplify it a bit, [tex]3x+2y+z=-22[/tex].
To start, the system in augmented-matrix form is
[tex]\begin{bmatrix}1&-1&4&23\\2&-1&1&-1\\3&2&1&-22\end{bmatrix}[/tex]
Subtract 2 times row 1 from row 2, and 3 times row 1 from row 3:
[tex]\begin{bmatrix}1&-1&4&23\\0&1&-7&-47\\0&5&-11&-91\end{bmatrix}[/tex]
Subtract 5 times row 2 from row 3:
[tex]\begin{bmatrix}1&-1&4&23\\0&1&-7&-47\\0&0&24&144\end{bmatrix}[/tex]
Multiply row 3 by 1/24:
[tex]\begin{bmatrix}1&-1&4&23\\0&1&-7&-47\\0&0&1&6\end{bmatrix}[/tex]
Add 7 times row 3 to row 2:
[tex]\begin{bmatrix}1&-1&4&23\\0&1&0&-5\\0&0&1&6\end{bmatrix}[/tex]
Add row 2 and -4 times row 3 to row 1:
[tex]\begin{bmatrix}1&0&0&-6\\0&1&0&-5\\0&0&1&6\end{bmatrix}[/tex]
Then the solution to the system is
[tex](x,y,z)=(-6,-5,6)[/tex]
