Enter Your System
Formula used
A simultaneous linear system is written as A X = B. A is the coefficient matrix. X is the variable column. B is the constants column.
For a unique square system, X = A-1B. The calculator uses Gaussian elimination to avoid direct inverse work.
Row operations reduce the augmented matrix [A | B] into reduced row form. For comparison, Cramer's rule uses xi = det(Ai) / det(A) when det(A) is not zero.
How to use this calculator
- Select whether your system has two or three variables.
- Enter every coefficient, including negative signs and zero values.
- Enter the constant on the right side of each equation.
- Choose the decimal precision and step display option.
- Press Calculate to view the result above the form.
- Use CSV or PDF download buttons to save the answer.
Example data table
| Case | Equations | Expected answer | Useful check |
|---|---|---|---|
| Two variables | 2x + 3y = 7, 5x - y = 9 | x = 2, y = 1 | Both equations balance exactly. |
| Three variables | x + y + z = 6, 2x - y + 3z = 9, -x + 4y + z = 10 | x = 1, y = 2, z = 3 | Each residual equals zero. |
| No unique answer | x + y = 4, 2x + 2y = 8 | Infinitely many solutions | The second equation repeats the first. |
Why simultaneous equations matter
Simultaneous linear equations appear when several unknowns must satisfy several rules at the same time. A price problem may need two unknown prices. A geometry task may need three coordinates. An electrical network may need currents across related branches. This calculator turns those linked statements into a coefficient matrix and solves them in a controlled way.
What the calculator does
The tool accepts two by two and three by three systems. You enter each coefficient and each constant. The solver builds an augmented matrix. It then applies row operations with partial pivoting. The process reduces the system until each leading variable is isolated. When a unique answer exists, the page reports every variable, the determinant, and residual checks. Residuals show how close each original equation comes to the entered constant.
Why the method is reliable
Row reduction is flexible. It works for many ordinary classroom systems. It also detects special cases. If the coefficient rows conflict, the system has no solution. If one equation repeats information from another, the system may have infinitely many solutions. The determinant adds another useful signal. A nonzero determinant means a square system has one unique solution. A zero determinant means the system needs deeper interpretation.
Practical uses
Use this page for algebra homework, test review, model checking, and quick applied calculations. It helps when hand elimination becomes messy. It also gives steps, so learners can compare each operation with notebook work. The CSV export stores variables and diagnostics. The PDF export gives a clean summary for reports or records.
Accuracy tips
Enter coefficients carefully. Keep signs in the correct boxes. Choose more decimals when answers are sensitive. A very small determinant can magnify rounding differences. For final coursework, always review the displayed steps and verify the answer in the original equations.
Choosing between methods
Elimination is the main engine because it exposes structure. Cramer's rule is also listed when the determinant allows it. This comparison is useful for study. Both approaches should agree for a stable unique system. The extra notes help explain whether the equations are independent, dependent, or inconsistent before you rely on the numbers. Use examples first, then replace values with your own problem data carefully today.
FAQs
What are simultaneous linear equations?
They are equations solved together because the same variables must satisfy every equation. A valid solution makes all equations true at the same time.
Can this calculator solve three variable systems?
Yes. Select the three equation option. Then enter x, y, z coefficients and constants for all three rows.
What does the determinant show?
A nonzero determinant shows that a square linear system has one unique solution. A zero determinant means the system may be dependent or inconsistent.
Why are residuals included?
Residuals check the solution against the original equations. Values near zero mean the calculated variables satisfy the entered system within rounding.
What method does the calculator use?
It uses Gaussian elimination with pivoting. It also compares unique solutions with Cramer's rule when the determinant is not zero.
Can I enter negative coefficients?
Yes. Type the minus sign directly in the coefficient field. Negative values are handled in the matrix and step calculations.
What happens when no unique solution exists?
The result section reports whether the system is inconsistent or dependent. It will not invent a single answer when the equations cannot isolate variables.
Why should I export the result?
CSV is useful for spreadsheets. PDF is useful for reports, homework records, and sharing a clean calculation summary.