Answer:
The system has unique solution.
,
and 
Step-by-step explanation:
Given system of equation is
2x+3y+z=5
2x+9y+0.z=5
12x+y+3z=5
The augmented matrix is
![\left[\begin{array}{ccc}2&3&1\\2&9&0\\12&1&3\end{array}\right|\left\begin{array}{c}5\\5\\5\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%263%261%5C%5C2%269%260%5C%5C12%261%263%5Cend%7Barray%7D%5Cright%7C%5Cleft%5Cbegin%7Barray%7D%7Bc%7D5%5C%5C5%5C%5C5%5Cend%7Barray%7D%5Cright%5D)
Subtract row 1 from row 2 (
)
![\left[\begin{array}{ccc}2&3&1\\0&6&-1\\12&1&3\end{array}\right|\left\begin{array}{c}5\\0\\5\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%263%261%5C%5C0%266%26-1%5C%5C12%261%263%5Cend%7Barray%7D%5Cright%7C%5Cleft%5Cbegin%7Barray%7D%7Bc%7D5%5C%5C0%5C%5C5%5Cend%7Barray%7D%5Cright%5D)
subtract row 1 multiplied by 6 from row 3 (
)
![\left[\begin{array}{ccc}2&3&1\\0&6&-1\\0&-17&-3\end{array}\right|\left\begin{array}{c}5\\0\\-25\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%263%261%5C%5C0%266%26-1%5C%5C0%26-17%26-3%5Cend%7Barray%7D%5Cright%7C%5Cleft%5Cbegin%7Barray%7D%7Bc%7D5%5C%5C0%5C%5C-25%5Cend%7Barray%7D%5Cright%5D)
Divide row 1 by 2 (
)
![\left[\begin{array}{ccc}1&\frac32&\frac12\\0&6&-1\\0&-17&-3\end{array}\right|\left\begin{array}{c}\frac52\\0\\-25\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26%5Cfrac32%26%5Cfrac12%5C%5C0%266%26-1%5C%5C0%26-17%26-3%5Cend%7Barray%7D%5Cright%7C%5Cleft%5Cbegin%7Barray%7D%7Bc%7D%5Cfrac52%5C%5C0%5C%5C-25%5Cend%7Barray%7D%5Cright%5D)
Divide row 2 by 6 
![\left[\begin{array}{ccc}1&\frac32&\frac12\\ \\0&1&\frac{-1}{6}\\ \\0&-17&-3\end{array}\right|\left\begin{array}{c}\frac52\\ \\0\\ \\-25\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26%5Cfrac32%26%5Cfrac12%5C%5C%20%5C%5C0%261%26%5Cfrac%7B-1%7D%7B6%7D%5C%5C%20%5C%5C0%26-17%26-3%5Cend%7Barray%7D%5Cright%7C%5Cleft%5Cbegin%7Barray%7D%7Bc%7D%5Cfrac52%5C%5C%20%5C%5C0%5C%5C%20%5C%5C-25%5Cend%7Barray%7D%5Cright%5D)
Subtract row 2 multiplied by
from row 1 
![\left[\begin{array}{ccc}1&0&\frac34\\ \\0&1&\frac{-1}{6}\\ \\0&-17&-3\end{array}\right|\left\begin{array}{c}\frac52\\ \\0\\ \\-25\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%26%5Cfrac34%5C%5C%20%5C%5C0%261%26%5Cfrac%7B-1%7D%7B6%7D%5C%5C%20%5C%5C0%26-17%26-3%5Cend%7Barray%7D%5Cright%7C%5Cleft%5Cbegin%7Barray%7D%7Bc%7D%5Cfrac52%5C%5C%20%5C%5C0%5C%5C%20%5C%5C-25%5Cend%7Barray%7D%5Cright%5D)
Add row 2 multiplied by 17 to row 3 
![\left[\begin{array}{ccc}1&0&\frac34\\ \\0&1&-\frac{1}{6}\\ \\0&0&-\frac{35}{6}\end{array}\right|\left\begin{array}{c}\frac52\\ \\0\\ \\-25\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%26%5Cfrac34%5C%5C%20%5C%5C0%261%26-%5Cfrac%7B1%7D%7B6%7D%5C%5C%20%5C%5C0%260%26-%5Cfrac%7B35%7D%7B6%7D%5Cend%7Barray%7D%5Cright%7C%5Cleft%5Cbegin%7Barray%7D%7Bc%7D%5Cfrac52%5C%5C%20%5C%5C0%5C%5C%20%5C%5C-25%5Cend%7Barray%7D%5Cright%5D)
Multiply row 3 by

![\left[\begin{array}{ccc}1&0&\frac34\\ \\0&1&-\frac{1}{6}\\ \\0&0&1\end{array}\right|\left\begin{array}{c}\frac52\\ \\0\\ \\\frac{30}7\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%26%5Cfrac34%5C%5C%20%5C%5C0%261%26-%5Cfrac%7B1%7D%7B6%7D%5C%5C%20%5C%5C0%260%261%5Cend%7Barray%7D%5Cright%7C%5Cleft%5Cbegin%7Barray%7D%7Bc%7D%5Cfrac52%5C%5C%20%5C%5C0%5C%5C%20%5C%5C%5Cfrac%7B30%7D7%5Cend%7Barray%7D%5Cright%5D)
Subtract row 3 multiplied by
from row 1 
![\left[\begin{array}{ccc}1&0&0\\ \\0&1&-\frac{1}{6}\\ \\0&0&1\end{array}\right|\left\begin{array}{c}-\frac57\\ \\0\\ \\\frac{30}7\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%260%5C%5C%20%5C%5C0%261%26-%5Cfrac%7B1%7D%7B6%7D%5C%5C%20%5C%5C0%260%261%5Cend%7Barray%7D%5Cright%7C%5Cleft%5Cbegin%7Barray%7D%7Bc%7D-%5Cfrac57%5C%5C%20%5C%5C0%5C%5C%20%5C%5C%5Cfrac%7B30%7D7%5Cend%7Barray%7D%5Cright%5D)
Add row 3 multiplied by
to row 2 
![\left[\begin{array}{ccc}1&0&0\\ \\0&1&0\\ \\0&0&1\end{array}\right|\left\begin{array}{c}-\frac57\\ \\\frac57\\ \\\frac{30}7\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%260%5C%5C%20%5C%5C0%261%260%5C%5C%20%5C%5C0%260%261%5Cend%7Barray%7D%5Cright%7C%5Cleft%5Cbegin%7Barray%7D%7Bc%7D-%5Cfrac57%5C%5C%20%5C%5C%5Cfrac57%5C%5C%20%5C%5C%5Cfrac%7B30%7D7%5Cend%7Barray%7D%5Cright%5D)
,
and 
Unique solution: The system has one specific solution.
The system has unique solution.