Calculator Input
Plotly Graph
The chart compares determinants and solved variables. It helps identify scale, sign, and stability patterns in the current system.
Example Data Table
| System Type | Equation Set | Main Determinant | Expected Solution |
|---|---|---|---|
| 2 × 2 | 2x + 3y = 13, x − y = 1 | -5 | x = 3.2, y = 2.2 |
| 3 × 3 | 2x − y + 3z = 9, x + y + z = 6, 3x − y + 2z = 8 | -1 | x = 1, y = 2, z = 3 |
| 3 × 3 | x + 2y + z = 9, 2x − y + 3z = 13, 3x + y + 2z = 13 | 2 | x = 1, y = 2, z = 4 |
Formula Used
Cramer’s Rule solves a square linear system when the main determinant is not zero.
For 2 × 2 systems: If A = [[a, b], [c, d]], then D = ad − bc.
Variable formulas: x = Dx / D and y = Dy / D, where each replaced determinant uses the constants column.
For 3 × 3 systems: Compute D, then replace one coefficient column at a time to get Dx, Dy, and Dz.
Final values: x = Dx / D, y = Dy / D, and z = Dz / D.
Condition: If D = 0, the method does not yield a unique solution.
How to Use This Calculator
- Select either a 2 × 2 or 3 × 3 linear system.
- Enter all coefficients for each equation row.
- Enter the constants on the right side.
- Click Solve System to compute determinants and variables.
- Read the main determinant before trusting the solution.
- Review residuals to confirm equation balance.
- Use the chart to compare determinant magnitude and variable values.
- Export the result using CSV or PDF buttons.
Frequently Asked Questions
1. What does Cramer’s Rule calculate?
It solves square systems of linear equations by using determinants. Each variable comes from a replaced determinant divided by the system’s main determinant.
2. When should I use this method?
Use it for 2 × 2 or 3 × 3 systems when you want exact determinant-based steps. It is especially helpful in teaching, validation, and small-system analysis.
3. What happens when the main determinant equals zero?
A zero main determinant means the system lacks a unique Cramer solution. The equations may be dependent, inconsistent, or require another solving approach.
4. Why are residuals shown?
Residuals measure how closely the computed values satisfy the original equations. Values near zero indicate that the solution balances the system properly.
5. Can I solve fractional or negative coefficients?
Yes. The form accepts decimal, fractional-equivalent, positive, and negative numeric entries. The calculation logic works for real-number coefficients and constants.
6. Why include a graph for a linear solver?
The graph helps compare determinant size and variable magnitude quickly. It adds a visual check for sign direction, scale differences, and sensitivity patterns.
7. Is Cramer’s Rule efficient for large systems?
Not usually. It is elegant for small systems, but determinant expansion becomes expensive as size grows. Matrix elimination methods scale better for larger systems.
8. What is the advantage of exporting results?
Exporting lets you document determinant values, variable outputs, and validation checks. That is useful for reports, homework logs, worksheets, and audit trails.