Solving a system of three linear equations with three unknowns Is my working correct or am I completely wrong? Have I missed anything out? Any feedback is appreciated.
Question: Consider the following system of equations
$2x + 2y + z = 2$
$−x + 2y − z = −5$
$x − 3y + 2z = 8$
Form an augmented matrix, then reduce this matrix to reduced row echelon form and solve the system.
My answer/working: 
Given:
$2x + 2y + z = 2$
$-x + 2y - z = -5$
$x - 3y + 2z = 8$
Matrix form: 
$\begin{pmatrix} 2 & 2 & 1 & 2\\ -1 & 2 & -1 & -5 \\ 1& -3& 2 & 8 \end{pmatrix}$

$\begin{pmatrix}2 & 0 & 0 & 2\\ 0 & 3 & 0 & -3\\ 0 & 0 & \frac56 & \frac53\end{pmatrix}$
Solution: $x = 1; y = -1; z = 2;$
 A: You're hardly completely wrong! The process you describe is "spot on", and yes, your solution is correct. 
You could row reduce a bit further, but there was really no need here. 
You've successfully solved the system of equations.
A: You can reduce your matrix further. remember that you can multiply and/or divide each row so you end up obtaining
$$
\begin{pmatrix}
     1  &   0  &   0  &   1\\
     0  &  1   &  0  &  -1\\
     0  &   0  &   1   &  2\\ 
\end{pmatrix}
$$ 
A: You should give Sequalator a try.
It is capable of solving thousands of Linear Simultaneous Equations with great precision and speed.
On my mediocre laptop it is able to solve 1000 equations just below a second.

I am getting the same answer as yours in Sequalator with 0% error.

To verify that the answer is correct you can check the error i.e. difference between LHS and RHS.

Also in future you can do your homework on your own and check the validity of your answers by checking them in Sequalator. It will show the error for your solution set.
However do not abuse it for doing your homework!
