Solving a system of three linear equations with three unknowns

Question

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: $x = 1; y = -1; z = 2;$

Answer 1

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.

Answer 2

You can reduce your matrix further. remember that you can multiply and/or divide each row so you end up obtaining

$$ \begin{pmatrix} 1 & 0 & 0 & 1\\ 0 & 1 & 0 & -1\\ 0 & 0 & 1 & 2\\ \end{pmatrix} $$