Cramer's Rule Calculator
Solving Linear Systems With Pure Determinants
Cramer's rule is an 18th-century method that solves a system of linear equations without ever isolating a variable by hand — it replaces one column of the coefficient matrix at a time with the constants, takes the determinant of each version, and divides. When the main determinant isn't zero, the system has exactly one solution and this method finds it directly.
The Formula
D = det(coefficient matrix)
x = Dx / D, y = Dy / D (Dx, Dy: D with that column replaced by the constants)
x = Dx / D, y = Dy / D (Dx, Dy: D with that column replaced by the constants)
Worked Example
Solving the system 2x + 3y = 8 and x − 2y = −3:
| Determinant | Calculation | Value |
|---|---|---|
| D | (2×−2) − (3×1) | −7 |
| Dx | (8×−2) − (3×−3) | −7 |
| Dy | (2×−3) − (8×1) | −14 |
| x, y | −7/−7, −14/−7 | x = 1, y = 2 |
Where This Matters
- Circuit analysis — solving simultaneous equations from Kirchhoff's laws for unknown currents or voltages.
- Mixture and allocation problems — finding quantities of two or three ingredients or resources that satisfy multiple constraints at once.
- Structural statics — resolving unknown forces or reactions from a small system of equilibrium equations.
Note: If D = 0, the system has no unique solution — either no solution exists or there are infinitely many.
How to Use This Calculator
- Choose the system size: 2 Equations, 2 Unknowns or 3 Equations, 3 Unknowns.
- Enter each equation's coefficients (a11, a12, etc.) and constant (b1, b2, etc.).
- Select Calculate to get the solved values for each variable.
Related Calculations
Prefer a step-by-step elimination approach? Try the Gaussian Elimination Calculator, or check a matrix's determinant directly with the Matrix Determinant Calculator.