Cramer's Rule 3x3 System Solver
Solve a system of three linear equations in three unknowns with Cramer’s rule, showing the main determinant and each substituted determinant.
How it works
x = Dx / D · y = Dy / D · z = Dz / D, where D is the coefficient determinant and Dx, Dy, Dz replace that column with the right-hand sides
Cramer's rule solves a square linear system by determinants. Compute D, the determinant of the coefficient matrix; then for each unknown, replace the matching column with the right-hand-side vector and take that determinant. Dividing each substituted determinant by D gives the value of that variable. The method only works when D is non-zero — a zero D signals no unique solution.
Worked example
For 2x + y − z = 8, −3x − y + 2z = −11, −2x + y + 2z = −3: the coefficient determinant D = 1. Substituting the right-hand column gives Dx = 2, Dy = 3, Dz = −1, so x = 2, y = 3, z = −1.
Sources: Cramer's rule (Wikipedia)
Frequently asked questions
When does Cramer’s rule fail?
When the coefficient determinant D equals zero. That means the coefficient matrix is singular, and the system has either no solution or infinitely many — neither of which Cramer’s rule can express, since it would require dividing by zero.
Is Cramer’s rule practical for large systems?
For 2×2 and 3×3 systems it is clean and quick. For larger systems it becomes computationally expensive compared with Gaussian elimination, which is why it is mainly a teaching and small-system tool.
What do Dx, Dy and Dz represent?
Each is the determinant of the coefficient matrix with one column swapped for the vector of right-hand-side constants: the x-column for Dx, the y-column for Dy, the z-column for Dz.
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This calculator is for educational purposes. Double-check important results before acting on them.