How does a row of zeros make a free variable in linear systems of equations? I don't understand how a row of zeros gives a free variable when solving systems of linear equations. Here's an example matrix and let us say that we're trying to solve Ax=0:
$$\left[
        \begin{matrix}
        2 & -3 & 0 \\
        3 & 5 & 0 \\
        0 & 1 & 0 \\
        \end{matrix}
\right]$$
This makes sense to me - you have a COLUMN of numbers corresponding to the number of times the variable $x_3$ is used for each equation and since they are all zeros, $x_3$ could have been any real number because we never get to manipulate our equations to determine a value for it. Hence, it is a free variable as it is not subject to the constraints of the equations.
$$\left[
        \begin{matrix}
        2 & -3 & 5 \\
        0 & 5 & 1 \\
        0 & 0 & 0 \\
        \end{matrix}
\right]$$
Now for this second example, $x_3$ would still be a free variable. Why is this so? $x_3$ is being used in the other two equations where you could certainly come up with a finite set of answers for this variable rather than saying "It could've been anything!" right?
Also is it entirely arbitrary that $x_3$ is the free variable or could it be decided that $x_1$ or $x_2$ is free instead?
Could someone explain to me in a more layman or simplified form on why a row of zeros magically makes a free variable?
Please help me :(

It's because the row rank is equal to the column rank.  If the matrix has more rows than columns, a row of all zeros does not imply a free variable.  For example:
$$ \left[\matrix{1 & 2\cr 3 & 4\cr 0 & 0\cr}\right]$$
One way you find free variables is: put the matrix into row-echelon form, then any column that does not contain the first nonzero entry of any row corresponds to a free variable.  In your example
$$\left[\matrix{2 & -3 & 5\cr 0 & 5 & 1\cr 0 & 0 & 0\cr}\right]$$
the first nonzero entries of the rows are in columns 1 and 2, so column 3
corresponds to a free variable.
However, column 1 or 2 could also correspond to a free variable.  In the case of column $2$, just consider changing the order of the columns so $3$ comes before $2$.  In the case of column $1$, it's a bit less obvious: you'd have to do some more row operations.
