I am tutoring a high school student in Algebra 2. Her class has just covered systems of linear equations, mainly via the "substitution" and "elimination" methods. Her textbook concludes that unit with Cramer's rule (which is only used for 2 x 2 and 3 x 3 systems).
My student likes using Cramer's rule, and asks me if there is any easy way to explain why it works. I'm having a hard time coming up with an explanation that doesn't require undergraduate-level linear algebra. Is there a simple way to explain it?