Calculator Inputs
Example Data Table
| Equation | x1 | x2 | x3 | Right Side |
|---|---|---|---|---|
| 1 | 2 | 1 | -1 | 8 |
| 2 | -3 | -1 | 2 | -11 |
| 3 | -2 | 1 | 2 | -3 |
This example gives x1 = 2, x2 = 3, and x3 = -1.
Formula Used
The calculator represents the system as Ax = b. A is the coefficient matrix. x is the variable vector. b is the constant vector.
Gaussian elimination and Gauss-Jordan elimination use elementary row operations. These operations keep the solution set unchanged.
For a unique square system, Cramer’s rule uses xj = det(Aj) / det(A). Aj is formed by replacing column j of A with b.
The rank test compares rank(A) with rank([A|b]). If both ranks match, the system is consistent. If the common rank equals the variable count, the solution is unique.
How to Use This Calculator
- Select the number of equations and variables.
- Enter each coefficient in the matching equation card.
- Enter each right side value.
- Press the solve button.
- Read the classification above the form.
- Review values, RREF, determinant, rank, and residuals.
- Use CSV or PDF download buttons for records.
About This Linear System Solver
Why Linear Systems Matter
A system of linear equations appears when several unknowns must satisfy related rules. Each equation adds one condition. Together, the conditions define one point, many points, or no valid point. This calculator helps you see that structure without hiding the mathematics.
What This Tool Checks
The calculator reads the coefficient matrix and the constant vector. It builds an augmented matrix. Then it reduces that matrix with row operations. The reduced form shows pivots, free variables, and contradictions. It also estimates rank, determinant, residual error, and solution type.
Unique, Infinite, or No Solution
A unique solution appears when every variable has a pivot. The determinant is nonzero for square systems. Infinite solutions appear when at least one variable is free, while the equations remain consistent. No solution appears when a row says zero equals a nonzero value.
Why More Than One Method Helps
Gaussian elimination is stable for daily study and engineering checks. Gauss-Jordan elimination gives a clear reduced matrix. Cramer’s rule is useful when the determinant is not zero. The inverse method confirms the same answer when the matrix is invertible.
Reading the Results
Start with the classification line. It tells you whether the system is consistent. Next, review the variable values or parameter form. Then check residuals. A small residual means the values satisfy the original equations. Large residuals warn about bad inputs or rounding.
Practical Uses
Linear systems support circuit analysis, balancing equations, economics, mixture planning, computer graphics, and geometry. They also help test models. When many quantities interact, a matrix gives a compact view. This calculator turns that view into clear steps and exportable reports.
Input Tips
Use decimals, fractions converted to decimals, or negative numbers. Keep units consistent before entering values. If one equation is a multiple of another, the system may have free variables. If the equations conflict, the calculator will show an inconsistent row.
Good Checking Habits
Always compare the displayed equations with your source problem. One copied sign can change the answer. Save the CSV file for spreadsheets. Use the PDF file for sharing. Recalculate after changing dimension, method notes, or rounding. These habits make the result easier to trust. They also support cleaner reviews during later study sessions.
FAQs
What is a system of linear equations?
It is a group of linear equations using the same variables. The solution must satisfy every equation at the same time.
How many variables can this calculator solve?
This version supports two to six variables. You can extend the limit by changing the maximum size value in the file.
What does a unique solution mean?
It means every variable has exactly one value. The coefficient matrix has full rank, and the determinant is not zero.
What does infinite solutions mean?
It means the equations are consistent, but at least one variable is free. The answer is shown using parameters.
What does no solution mean?
It means the equations contradict each other. In row form, this appears as zero on the left and a nonzero value on the right.
Why is the determinant useful?
For a square system, a nonzero determinant confirms an invertible matrix. That usually means a unique solution exists.
What is a residual?
A residual is the difference between Ax and b after using the solution. Smaller residuals show better numerical agreement.
Can I export my results?
Yes. After solving, use the CSV button for spreadsheet work. Use the PDF button for a simple printable report.