Cramer's Rule for y (2×2 System)

The companion to Cramer's rule for x: keep the same denominator D = a₁b₂ − b₁a₂, but now replace the y-column of the coefficient matrix with the right-hand sides, giving a₁c₂ − c₁a₂ on top. Worked example: 2x + 3y = 12 and x − y = 1 have D = −5 and numerator 2(1) − 12(1) = −10, so y = −10/−5 = 2, matching the x = 3 the companion page returns.

The pattern generalises cleanly — for three unknowns you replace the third column and divide by a 3×3 determinant — which is what gives the rule its enduring textbook place even though it is computationally hopeless at scale: solving an n×n system this way costs on the order of n! operations against roughly n³ for Gaussian elimination, so a 20×20 system would take longer than the age of the universe. Cramer set it out in 1750 in a book about fitting algebraic curves through given points, which is exactly the kind of small dense system it suits. The traps are clerical: subscripts must stay with their own equation, the denominator is the same for x and y so compute it once, and D = 0 signals parallel lines rather than an arithmetic mistake. A quick check costs nothing — substitute both answers back into the equation you used least.