The formula
x = Δx ÷ Δ · y = Δy ÷ Δ, with Δ = a₁b₂ − a₂b₁
Why it works
Each equation describes a line, so solving the system means finding where the two lines meet. The determinant tells you which of three things happens: a non-zero determinant means they cross at exactly one point, and a zero determinant means they are parallel — either the same line or never meeting at all.
How to solve it by hand
- Work out the determinant: a₁b₂ − a₂b₁
- Replace the x column with the constants for Δx
- Replace the y column with the constants for Δy
- Divide each by the determinant
What is worth knowing
The geometry makes the three cases obvious. Two lines in a plane either cross once, lie on top of each other, or run parallel and never meet — there is no fourth option, which is why a system of two equations can never have exactly two solutions. The determinant being zero is the algebraic signal that the lines have the same slope; whether they coincide or not then depends on the constants. Cramer's rule is elegant and fine for two or three unknowns, but it is far too slow for large systems, where elimination is used instead.