Calculator Form
Choose a 2×2 or 3×3 system, enter coefficients, and solve instantly.
Example Data Table
| Example | Equation Set | Expected Solution | Use Case |
|---|---|---|---|
| 2×2 System |
2x + 3y = 13 x - y = 1 |
x = 3, y = 2.3333 | Quick classroom verification and graph intersection checks |
| 3×3 System |
x + y + z = 6 2x - y + z = 3 x + 2y - z = 3 |
x = 1.2857, y = 2.1429, z = 2.5714 | Linear algebra exercises and coefficient matrix testing |
Formula Used
This calculator models the system as A · x = b, where A is the coefficient matrix, x is the solution vector, and b is the right-hand-side vector.
It uses Gaussian elimination with partial pivoting to convert the augmented matrix into row echelon form. Back substitution then recovers the variable values when the system has one unique solution.
The determinant helps identify whether the square system is singular. A nonzero determinant usually indicates one unique solution, while zero suggests either infinitely many solutions or no solution.
Residuals are computed using:
Residual for each equation = (A × x) − b
For a 2×2 system, the calculator also visualizes the equations as two lines and highlights their intersection point when one unique solution exists.
How to Use This Calculator
- Select whether you want to solve a 2×2 or 3×3 simultaneous equation system.
- Enter every coefficient carefully in the matching equation row.
- Enter the right-side values for each equation.
- Choose your preferred decimal precision and graph range.
- Click Solve Equations to see the result section above the form.
- Review the solution status, determinant, ranks, residuals, and elimination steps.
- Use the CSV button to export spreadsheet-friendly results.
- Use the PDF button to save a polished summary for sharing or printing.
Frequently Asked Questions
1. What does a unique solution mean?
A unique solution means the system intersects at exactly one point in solution space. Every variable has one fixed value, and the determinant is typically nonzero.
2. What happens when the determinant is zero?
A zero determinant means the coefficient matrix is singular. The system may have infinitely many solutions or no solution, depending on the matrix ranks after elimination.
3. Why are residuals useful?
Residuals show how closely the computed solution satisfies each original equation. Values near zero confirm that rounding and numerical operations did not distort the result significantly.
4. Can this solve three equations with three unknowns?
Yes. Change the system size to 3 variables and 3 equations. The form reveals the extra coefficient inputs and right-side value automatically.
5. Why does the graph change for 2×2 and 3×3 systems?
A 2×2 system can be drawn directly as two lines on an x-y plane. A 3×3 system is better summarized by solved variable values because its full geometry is three-dimensional.
6. What does rank comparison tell me?
Comparing coefficient and augmented matrix ranks helps classify the system. Equal ranks with full rank give one solution, while unequal ranks indicate inconsistency.
7. Is Gaussian elimination better than substitution?
For larger systems, Gaussian elimination is usually faster, more systematic, and easier to automate. It also produces step-by-step row operations that help verify the solution path.
8. Can I export the results for reports?
Yes. Use CSV for spreadsheet work and the PDF button for clean sharing, documentation, print-friendly notes, or classroom submission support.