System of Equations Solver (2x2)
Solve two linear equations in two unknowns (x and y) using Cramer’s rule.
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Formula
Examples
| Input | Result |
|---|---|
| 2x + 3y = 8; x − y = −1 | x = 1, y = 2 |
About this calculator
A 2x2 system of linear equations consists of two equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂. Cramer’s rule solves it using the determinant det = a₁b₂ − a₂b₁. When det is non-zero the system has exactly one solution given by x = (c₁b₂ − c₂b₁) ÷ det and y = (a₁c₂ − a₂c₁) ÷ det.
When the determinant equals zero the two lines are either parallel (no solution) or the same line (infinitely many solutions), so no single (x, y) pair can be returned. Geometrically, the solution is the point where the two lines intersect.
Frequently asked questions
Cramer’s rule expresses each unknown as a ratio of determinants. For a 2x2 system it gives x and y directly from the coefficients and constants without elimination.
The determinant tells you whether a unique solution exists. If it is non-zero the lines cross at one point; if it is zero they are parallel or identical and there is no single solution.
Rearrange each equation into the form ax + by = c, then enter the six coefficients a₁, b₁, c₁, a₂, b₂, c₂ in the matching fields.
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