Enter Simultaneous Equations
Use two or three equations. Each row should match one complete equation.
Example Data Table
These examples show common system types and expected calculator behavior.
| System | Equations | Expected Result | Meaning |
|---|---|---|---|
| Two variables | 2x + 3y = 13, x - y = 1 | x = 3.2, y = 2.2 | One unique intersection point. |
| Dependent | x + y = 4, 2x + 2y = 8 | Infinitely many solutions | Both equations describe the same line. |
| Inconsistent | x + y = 4, x + y = 9 | No solution | The equations contradict each other. |
Formula Used
Matrix form: A × X = B
Determinant test: det(A) ≠ 0 means a unique solution exists.
Cramer relation: xᵢ = det(Aᵢ) / det(A)
Residual check: r = A × X - B
The calculator first writes the system as a coefficient matrix. It then applies Gauss-Jordan elimination. This produces row-reduced echelon form. The ranks of the coefficient matrix and augmented matrix are compared. If both ranks equal the number of unknowns, the answer is unique. If the augmented rank is larger, no answer exists. If both ranks match but are smaller than the unknown count, many answers exist.
How to Use This Calculator
- Select whether your system has two or three variables.
- Enter every coefficient beside its matching variable.
- Enter each constant from the right side of the equation.
- Choose the decimal precision for the final output.
- Press the calculate button to solve the system.
- Review the determinant, ranks, residuals, and solution table.
- Use the CSV or PDF button to save the result.
About Simultaneous Equation Solving
Why Simultaneous Equations Matter
A system of simultaneous equations links several unknown values. Each equation gives one rule. The solution is the set of values that satisfies every rule together. This idea appears in budgets, mixtures, physics, supply planning, and classroom algebra. A calculator helps when the arithmetic is long. It also reduces copying mistakes. You can still see the method, because the page shows matrix checks, ranks, residuals, and clear output.
Matrix Thinking
The matrix method is clean and reliable. Coefficients are placed inside matrix A. Unknown values form vector X. Constants form vector B. The problem becomes A times X equals B. This structure works for two-variable and three-variable systems. It also supports decimals, negative values, and zero coefficients. When the determinant is not zero, the system has one exact solution.
Rank and Consistency
Some systems do not have a single answer. Parallel equations may never meet. Repeated equations may describe the same line or plane. The rank test handles these cases. It compares the coefficient matrix with the augmented matrix. If the augmented rank is larger, the system is inconsistent. If both ranks are lower than the number of unknowns, free variables exist. That means many solutions are possible.
Practical Accuracy
Decimal systems can create rounding concerns. For that reason, the calculator includes residual checks. A residual shows how much error remains after substituting the answer. A value near zero is a strong verification. The precision selector controls display length. It does not hide the logic. Use more decimals when values are close together. Use exported files when you need records for homework, reports, or planning work.
FAQs
1. What does this simultaneous calculator solve?
It solves two-variable and three-variable linear equation systems. It also identifies unique, inconsistent, and dependent systems using determinant and rank checks.
2. Can I enter decimal coefficients?
Yes. You can enter whole numbers, decimals, negative values, and zero coefficients. The precision menu controls how many decimals appear in the final answer.
3. What does a zero determinant mean?
A zero determinant means the system does not have one guaranteed unique solution. The calculator then checks ranks to decide if there is no solution or many solutions.
4. What method is used here?
The calculator uses Gauss-Jordan elimination. It also shows determinant values, Cramer determinant checks, matrix ranks, and residual verification for stronger review.
5. What is a residual check?
A residual is the difference after putting the calculated answer back into an equation. A residual near zero means the solution fits that equation well.
6. Can this calculator solve nonlinear equations?
No. This page is designed for linear simultaneous equations only. Each unknown must appear with a coefficient, not powers, roots, or products.
7. Can I download my results?
Yes. After calculation, you can download a CSV file or create a PDF report containing the solution, determinant, ranks, residuals, and RREF table.
8. Why do I see infinitely many solutions?
This happens when equations are dependent. The system has fewer independent equations than unknowns, so at least one variable can vary freely.