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Linear Equations Solver (Cramer's Rule)
Solve systems of linear equations using Cramer's Rule determinants (D, Dx, Dy) with complete step-by-step KaTeX evaluation and consistency analysis.
InputsChange a value to recalculate instantly
ResultUpdates as you type
FORMULA & STEPSLinear Equations Solver (Cramer's Rule) — Mathematical Derivation & Steps
2 StepsSequential Calculation Breakdown
1
Coefficient determinant
A nonzero determinant gives a unique solution.
2
Cramer values
Replace one coefficient column at a time to form Dx and Dy.
λ DooMathWork with this calculation
Local · deterministic · optionalQuick mathNatural-language-style shortcuts for deterministic arithmetic.
Calculator guide
How to use the Linear Equations Solver (Cramer's Rule)
Solve systems of linear equations using Cramer's Rule determinants (D, Dx, Dy) with complete step-by-step KaTeX evaluation and consistency analysis.
01Set inputsChoose values, units and assumptions.
02Apply the modelx = D_x / D, y = D_y / D; D = a₁b₂ - a₂b₁; D_x = c₁b₂ - c₂b₁; D_y = a₁c₂ - a₂c₁
03Read the resultCheck the output against the assumptions before using it.
What this calculator does
Enter the coefficients and constants for your system of linear equations. The solver constructs the coefficient determinant and solves for unique variable values.
Formula & method
x = D_x / D, y = D_y / D; D = a₁b₂ - a₂b₁; D_x = c₁b₂ - c₂b₁; D_y = a₁c₂ - a₂c₁
Assumptions & notes
- If D ≠ 0, the system has a unique independent solution.
- If D = 0 and Dx = 0, the system has infinitely many dependent solutions.
- If D = 0 and Dx ≠ 0, the system is inconsistent and has no solution.