Does one always use augmented matrices to solve systems of linear equations? The homework tag is to express that I am a student with no working knowledge of math.
I know how to use elimination to solve systems of linear equations. I set up the matrix, perform row operations until I can get the resulting matrix into row echelon form or reduced row echelon form, and then I back substitute and get the values for each of my variables.
Just some random equations: $a + 3b + c = 5 \\ a + 0 + 5c = -2$
My question is, don't you always want to be using an augmented matrix to solve systems of linear equations? By augmenting the matrix, you're performing row operations to both the left and right side of those equations. By not augmenting the matrix, aren't you missing out on performing row operations on the right side of the equations (the $5$ and $-2$)?
The sort of problems I'm talking about are homework/test-level questions (as opposed to real world harder data and more complex solving methods?) where they give you $Ax = b$ and ask you to solve it using matrix elimination.
Here is what I mean mathematically:
$[A] =
        \begin{bmatrix}
        1 & 3 & 1 \\
        1 & 0 & 5 \\
        \end{bmatrix}
$ 
$ [A|b] =
\left[\begin{array}{ccc|c}
        1 & 3 & 1 & 5\\
        1 & 0 & 5 & -2\\
    \end{array}\right]
$
So, to properly solve the systems of equations above, you want to be using $[A|b]$ to perform Elementary Row Operations and you do not want to only use $[A]$ right?
The answer is: yes, if you use this method of matrices to solve systems of linear equations, you must use the augmented form $[A|b]$ in order to perform EROs to both $A$ and $b$.

Does one always use augmented matrices to solve systems of linear equations?
No. In the case of an $n \times n$ system of linear equations which can be expressed by the matrix equation $A x = b$, assuming $A$ has a nonzero determinant, you can multiply both sides by $A^{-1} $ to get the unique solution $x = A^{-1} b$ . I used no augmented matrix here. For non-square systems of linear equations, or systems of linear equations with more than one solution, row reducing the augmented matrix $[A ~ \mid b ~]$ to $ [ D ~ \mid b~' ~] $ , where $D$ is a reduced row echelon matrix and $b~'$ is getting closer to the solution $x$, would be advised.
