How does one explain Cramer's rule without matrices? Where can I find an explanation of Cramer's rule that does not make use of matrices?
Matrices are convenient, but I think that they add a layer of opacity to the underlying mathematics.
Without them, could it be easier to reach a greater and more intuitive understanding of this rule?
 A: For intuitive understanding it's enough to consider case of 3 equations.
Solving a system of 3 linear equations of 3 variables is equivalent to solving a geometrical problem: given 4 vectors $\vec{A}, \vec{B}, \vec{C}$ and $\vec{X}$ find $a$, $b$ and $c$ such that $a*\vec{A} + b*\vec{B} + c*\vec{C} = \vec{X}$.
Let's start with finding $a$. You should take $a$ such that distance from $a*\vec{A}$ to plane $(\vec{B}, \vec{C})$ is the same as distance from vector $\vec{X}$ to this plane. Because adding $b*\vec{B} + c*\vec{C}$ would not change that distance, and you want to get $\vec{X}$ as a result.
Let's define $V(\vec{v_1}, \vec{v_2}, \vec{v_3})$ = volume of the  parallelepiped determined by these 3 vectors.
Note, that $V(a\vec{A}, \vec{B}, \vec{C}) = V(\vec{X}, \vec{B}, \vec{C})$. This is because $a*\vec{A}$ and $\vec{X}$ has same distance to plane $(\vec{B}, \vec{C})$.
So:
$a=V(\vec{X}, \vec{B}, \vec{C}) / V(\vec{A}, \vec{B}, \vec{C})$
Note, that I get this result without using words "matrix" or "determinant".
The problem now is how to calculate these volumes. Which is easy: volume of the parallelepiped determined by vectors is a determinant of a matrix, whose elements are vectors coordinates.
And so we have Cramer's rule.
This explanation is not very rigorous (what if $\vec{B}$ and $\vec{C}$ are collinear?), but for intuitive understanding it should be ok.
