Cramer's Rule

You can solve a 2×2 system by elimination — but there's an elegant alternative: each unknown is a ratio of two determinants. Swap A's i-th column for b, take that determinant, divide by det(A), and out pops xᵢ. Cramer's Rule is rarely the fastest method for a computer, but it's a stunning theorem: it tells you the solution is built from the same area-scaling numbers you just learned.