<s>[INST] Solve the following in vector form: 
So i did a substitution to solve the system normally, and got
$x=17.67$
$y=9.67$
$z=10.67$
Where I am stuck is how to represent something like this in a vector form, maybe my solution was wrong in the first place.
 [/INST] For example
\begin{align}
(1,2,-3)^T \cdot x &= 5 \\
(2,1,-3)^T \cdot x &= 13 \\
(-1,1,0)^T \cdot x &= -8
\end{align}
where $x = (x_1,x_2,x_3)^T$.
But more usual is using a matrix equation
$$
A x = b
$$
with $A$ using the above first argument vectors as row vectors and 
$b = (5, 13, -8)^T$.
The solution would be $x = A^{-1} b$, if $A$ is invertible.
In your case it is not. Your solution works, so there must be infinite many solutions.
$$
(A | b) = \\
\left(
\begin{array}{rrr|r}
 1 & 2 & -3 & 5 \\
 2 & 1 & -3 & 13 \\
-1 & 1 &  0 & -8
\end{array}
\right)
\to \\
\left(
\begin{array}{rrr|r}
1 &  2 & -3 &  5 \\
0 & -3 &  3 &  3\\
0 &  3 & -3 & -3
\end{array}
\right)
\to \\
\left(
\begin{array}{rrr|r}
1 & 0 & -1 &  7 \\
0 & 1 & -1 & -1 \\
0 & 0 &  0 &  0
\end{array}
\right)
$$
This gives the general solution $x = (7 + x_3,-1 + x_3, x_3)^T$ with arbitrary $x_3 \in \mathbb{R}$. 

</s>