Cramer's Rule for x (2×2 System)

Cramer's rule solves a system of two linear equations by determinants alone, with no elimination and no substitution. Write the system as a₁x + b₁y = c₁ and a₂x + b₂y = c₂; the denominator is the coefficient determinant D = a₁b₂ − b₁a₂, and the numerator is the same determinant with the x-column replaced by the right-hand sides. Worked example: 2x + 3y = 12 and x − y = 1 give D = 2(−1) − 3(1) = −5 and numerator 12(−1) − 3(1) = −15, so x = −15/−5 = 3 — and substituting back, 2(3) + 3y = 12 puts y at 2, which satisfies 3 − 2 = 1.

Gabriel Cramer published the rule in an appendix to his 1750 treatise on algebraic curves, though Colin Maclaurin had written essentially the same thing in a manuscript of the 1720s printed posthumously in 1748. Its appeal is that it gives a formula for the answer rather than a procedure, which is why it survives in physics derivations and in symbolic algebra even though Gaussian elimination beats it badly for speed on anything larger than 3×3. The trap is a zero denominator: D = 0 means the two lines are parallel, so the system has either no solution or infinitely many, and the rule simply cannot distinguish the two cases. Keep the columns straight, too — swapping which column you replace gives you y instead of x.