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Cramer's rule solves a system of linear equations directly from determinants, without the row-reduction steps that elimination or substitution require.
This calculator solves a 2×2 up to 6×6 system of linear equations using Cramer's rule, showing each variable's determinant.
How it works
Enter the coefficient matrix and the constants on the right-hand side of your equations. The calculator finds the determinant of the coefficient matrix, then for each variable, replaces that variable's column with the constants and finds that determinant too — each variable equals its modified determinant divided by the original.
- Click Calculate to see your results.
Examples
A 2×2 system
For 2x + y = 8 and x + 3y = 13, Cramer's rule gives x = 2.2 and y = 3.6.
Who should use it
- Solving small systems of linear equations for coursework.
- Learning or teaching Cramer's rule as an alternative to elimination or substitution.
Industry applications
- Linear algebra and mathematics education
- Engineering and applied mathematics
Advantages
- Shows every determinant used to find each variable individually.
- A direct, formula-based method with no row-reduction steps.
Limitations
- Limited to 2×2 up to 6×6 systems, and computationally inefficient for larger systems.
Common mistakes to avoid
- Replacing the wrong column when computing each variable's determinant — column 1 for x, column 2 for y, and so on.
- Not checking whether the coefficient determinant is zero before applying the rule, which would make the result undefined.
Best practices
- Double check that you've entered the constants (the right-hand side of each equation) into the b1, b2, b3 fields, not mixed into the coefficient matrix.
Tips
- If det(A) turns out to be zero, the system doesn't have a unique solution — go back and check your original equations, since this often signals two equations that are multiples of each other or an inconsistent system.