Calculator
Example Data Table
| Example | Equation Set | Main Elimination Move | Expected Result |
|---|---|---|---|
| Two-variable system | 2x + 3y = 13 5x - y = 7 |
Remove x or y using row multiples. | x = 2, y = 3 |
| Dependent system | x + y = 4 2x + 2y = 8 |
Second row becomes zero. | Infinite solutions |
| Inconsistent system | x + y = 4 2x + 2y = 10 |
Zero coefficient row keeps a nonzero constant. | No solution |
| Three-variable system | x + y + z = 6 2x - y + 3z = 14 -x + 4y + z = 2 |
Use Gaussian elimination with pivot rows. | Unique point, if determinant is nonzero |
Formula Used
For a two-variable system, write the equations as ax + by = e and cx + dy = f. The determinant is D = ad - bc. When D is not zero, x = (ed - bf) / D and y = (af - ec) / D.
For larger systems, the calculator uses row elimination. A pivot row removes one variable from lower rows. The row operation is Ri = Ri - factor × Rk, where factor = aik / akk. Back substitution then finds each variable.
Rank tests classify the answer. If rank(A) is smaller than rank(A|b), no solution exists. If both ranks match but are smaller than the variable count, infinitely many solutions exist.
How to Use This Calculator
- Choose two or three variables from the system size menu.
- Enter each coefficient beside its matching variable.
- Enter the right side constant for each equation.
- Use decimals, integers, negative values, or fractions such as 5/8.
- Select the decimal precision for your displayed result.
- Press Calculate to view the solution above the form.
- Use CSV or PDF buttons to download the same result.
Algebra Elimination Method Guide
The elimination method solves a system by removing variables. It changes the equations without changing their shared solution. Each move uses addition, subtraction, multiplication, or division. These moves keep the system equivalent. That makes the method reliable for school work, engineering checks, and quick algebra review.
Why Elimination Works
Two equations often contain the same variables. If one variable has opposite coefficients, adding the equations removes it. If the coefficients are not opposite, multiply one or both equations first. The goal is simple. Create a smaller equation with fewer unknowns. Then solve that smaller equation. Finally, substitute the value back into another equation.
Using Row Operations
This calculator writes the system as an augmented matrix. The left side holds coefficients. The right side holds constants. It then picks a pivot. A pivot is the number used to clear entries below it. Strong pivots reduce rounding errors. That is why partial pivoting is useful. It swaps rows when a better pivot exists.
Reading the Result
A unique solution means every variable has one value. The determinant is not zero in a square system. Infinite solutions mean at least one equation repeats the information of another. A free variable remains. No solution means the equations conflict. In matrix form, this appears as a zero coefficient row with a nonzero constant.
Practical Study Tips
Start with clean coefficients. Keep signs clear. Write every row operation. Check each answer in the original equations. A residual near zero means the result balances. Larger residuals usually come from rounding or entry mistakes. Use the downloaded report when you need to show work. It records the determinant, ranks, RREF, and each elimination step.
FAQs
1. What does this calculator solve?
It solves linear equation systems with two or three variables. It also shows determinant, rank, RREF, row operations, classification, and residual checks.
2. Can I enter fractions?
Yes. You can enter values like 1/2, -3/4, decimals, whole numbers, and negative coefficients. The calculator converts them into numeric values.
3. What is a unique solution?
A unique solution means the system has exactly one ordered answer. In a square system, this usually happens when the determinant is not zero.
4. What means infinite solutions?
Infinite solutions occur when equations are dependent. The ranks match, but at least one variable remains free, so many ordered answers satisfy the system.
5. What means no solution?
No solution means the equations conflict. The augmented matrix shows this when rank of the coefficient matrix is smaller than the augmented rank.
6. Why is partial pivoting used?
Partial pivoting chooses a stronger pivot by swapping rows. This helps avoid division by tiny numbers and improves numerical stability.
7. What is the residual check?
The residual is left side minus right side after inserting the answer. Values near zero show the calculated solution satisfies the original equations.
8. What is included in the downloads?
The CSV and PDF reports include the status, determinant, ranks, solution values, residuals, RREF, and the main elimination steps.