Why can vectors be used to solve systems of linear equations if vector space has properties that the solution space doesn't have? I realize my question my sound a wordy, so I'll try and make it more clear here. 
Consider the following system of linear equations:
$$x - 2y + 3z = 7$$
$$2x + y + z = 4$$
$$-3x + 2y - 2z = -10$$
These solutions are real numbers and have no indication as to why they can't obey the properties of real numbers, such as having no anticommutative operations, familiarly having the classic commutative, associative and distributive properties we're familiar with when applying math in a traditional sense.
However, we solve these equations by mapping each element of the $x$, $y$, and $z$ coordinates into vectors and their solutions as vectors to find the solutions to the variables. I'm sure most people on here are familiar with this process so I can get to my point for brevity:
We're interpreting it as what transformation takes vector $\vec A$ to $\vec B$ using the matrix of the coefficients of each variables in the $3$ equations? But why can this applied to this scenario? We're dealing with real numbers here, things with properties like
$$AB = BA$$
But, in vector space (where $A$ and $B$ are matrices),
$$AB \neq BA$$
So why can we use a vector interpretation to solve these problems? It seems like to me that this can cause some unreliable results due to its properties.
 A: It seems that you are confusing three different objects that are present in the representation of a linear system as a linear transformation between vector spaces.
One kind of object are the scalars, that in your case are the real numbers. These objects are elements of a field so between them is defined an operation called multiplication that is commutative (the usual multiplication of real numbers).
Other  objects are the vectors that, in your case,  are  triples of real numbers $\vec x=[x_1,x_2,x_3]$ equipped with an addition ( $\vec x + \vec y=[x_1,x_2,x_3]+[y_1,y_2,y_3]=[x_1+y_1,x_2+y_2,x_3+y_3]$) and a scalar multiplication by the real numbers ( a \cdot \vec x=a[x_1,x_2,x_3]=[ax_1,ax_2,ax_3]$ that satisfy some properties  but , between these objects, is not defined a multiplication.
The third type of objects are the matrices, that, in your case, are arrays of $3 \times 3$ real numbers that represents a linear transformation between vectors, that is a linear function $A$ of a vector that gives another vector $A(\vec x)=\vec b$. Between these linear tranformation we can define an operation of multiplication (that is the result of applying two tranformation one after the other) and this multiplication is not commutative ($AB \ne BA$).
But the non commutativity is a properties of the linear transformations,  that are objects very different from the real numbers.  The fact that we can represents these objects by means of matrices of real numebrs   depends on a nice definition of the product of these matrice between them, that is the usual row-column rule and is not commutative.
And, using this rule, we can represent a linear system in the matrix form: $A \vec x=\vec b$
