How this instrument works
A system of two linear equations, a₁x + b₁y = c₁ and a₂x + b₂y = c₂, describes two straight lines, and solving the system means finding the single point (x, y) where those two lines cross. Cramer's Rule solves for both coordinates directly with two ratio formulas: x = (c₁b₂ − c₂b₁) ⁄ (a₁b₂ − a₂b₁) and y = (a₁c₂ − a₂c₁) ⁄ (a₁b₂ − a₂b₁), reaching the same answer that substitution or elimination would, but without showing either of those intermediate step-by-step processes.
The shared denominator, a₁b₂ − a₂b₁, is the system's determinant — when it equals zero, the two lines are either perfectly parallel (no crossing point exists at all) or are actually the exact same line (infinitely many crossing points), and Cramer's Rule can't produce a single unique answer in either case.
Cramer's Rule generalizes cleanly to systems with three or more equations and unknowns, using larger determinants built the same conceptual way, though the arithmetic grows quickly more involved — for a plain two-equation, two-unknown system, it stays this compact and direct.
- Enter the first equation's coefficients into the Equation 1 fields (coefficient of x, coefficient of y, and constant).
- Enter the second equation's coefficients into the matching Equation 2 fields.
- Read x and y: the sheet applies Cramer's Rule to solve both simultaneously.
Worked example — 2x + 3y = 13 and 5x − y = 7
The system 2x + 3y = 13 and 5x − y = 7 solves to x = 2, y = 3 via Cramer's Rule — checking directly: 2×2 + 3×3 = 4+9 = 13 ✓ and 5×2 − 3 = 10−3 = 7 ✓, confirming both original equations hold at that single point.
The system x + y = 10 and x − y = 2 solves to x = 6, y = 4 — a simple sum-and-difference pair, easy to verify by hand: 6+4=10 and 6−4=2, the same answer Cramer's Rule reaches through its determinant-ratio formulas instead of ordinary elimination.
Questions
What is Cramer's Rule?
A method for solving a system of linear equations using ratios of determinants, giving each variable's value directly without the step-by-step substitution or elimination process — for a 2-by-2 system, it reduces to two compact formulas built from the equations' coefficients.
What does it mean if the denominator is zero?
The two equations describe either perfectly parallel lines (no solution exists, since they never cross) or the exact same line (infinitely many solutions, since every point on that line satisfies both equations) — Cramer's Rule can't produce a single unique answer in either case.
Does Cramer's Rule work for more than two equations?
Yes — it generalizes to any number of equations and unknowns, using larger determinants built the same conceptual way, though the arithmetic grows substantially more involved as the system grows larger.
How is this different from graphing the two lines?
Graphing finds the same crossing point visually, by drawing both lines and reading off where they intersect — Cramer's Rule reaches the identical answer algebraically, with exact values rather than an estimate read off a graph.
What does the solution (x, y) actually represent?
It's the one point that satisfies BOTH equations simultaneously — geometrically, the single point where the two lines described by the equations cross paths on the coordinate plane.