Calculator
Use decimals, integers, or fractions such as 1/2. Enter zero for any missing variable.
Example Data Table
| Case | Equations | Expected Type | Notes |
|---|---|---|---|
| 2 Variable | 2x + y = 7 x - y = 2 |
Unique | Good starter example. |
| 3 Variable | 2x + y - z = 8 -3x - y + 2z = -11 -2x + y + 2z = -3 |
Unique | Default sample values. |
| Dependent | x + y = 4 2x + 2y = 8 |
Infinite | Second equation repeats the first. |
| Conflict | x + y = 4 x + y = 7 |
No Solution | Same left side, different constants. |
Formula Used
A linear system is written as Ax = b. The matrix A contains coefficients. The vector x contains unknown variables. The vector b contains constants.
This calculator uses Gaussian elimination with partial pivoting. Row operations convert the augmented matrix [A | b] into reduced row echelon form. If the coefficient rank equals the augmented rank and also equals the number of variables, the system has one unique solution.
If rank(A) < rank([A | b]), the system is inconsistent. If both ranks match but are lower than the number of variables, the system has infinitely many solutions. The determinant is also shown. A nonzero determinant confirms one unique solution for square coefficient matrices.
How to Use This Calculator
- Select whether your system has two, three, or four variables.
- Enter each coefficient in the matching variable box.
- Enter the constant value on the right side of each equation.
- Use zero when a variable is not present in an equation.
- Choose the decimal precision for rounded output.
- Click Calculate to view the answer above the form.
- Use CSV or PDF buttons to save the current calculation.
Complete Guide to Linear Systems
Why Systems Matter
A system of equations connects several unknown values through shared rules. This calculator helps students, tutors, and analysts solve linear systems without losing each algebra step. It supports two, three, and four variable models, so you can test small homework problems or larger planning cases. The white layout keeps the inputs clear. Each equation has a row for coefficients and a constant term.
Common Uses
Linear systems appear in budgeting, mixture problems, geometry, physics, and business planning. A pair of equations may describe two prices. A three variable model may describe ingredient blends. A four variable model can compare production limits across several resources. Because every value affects the final answer, careful coefficient entry matters. This tool groups values in a responsive grid. Large screens show three fields per row. Smaller screens reduce the columns for easier typing.
How the Solver Works
The calculator uses matrix elimination. It first builds an augmented matrix from the coefficient table and constants. Then it checks ranks to identify unique, infinite, or inconsistent systems. When a unique solution exists, it uses pivoting and row operations to isolate each variable. The result panel shows the determinant, rank details, equations, and solution values. Optional step output helps you follow the row reduction process.
Saving and Checking Work
The export tools are useful for records. Use CSV when you want spreadsheet data. Use PDF when you need a clean printable summary. The example table below shows common systems and expected solution types. You can copy those values into the form to test the workflow.
Best Practice
For best results, enter coefficients exactly as they appear in your equations. Use negative signs for subtraction. Use zero when a variable is missing from an equation. Choose the number of variables before calculating. Increase decimal precision when answers contain fractions or repeating decimals. Review the determinant and rank notes before using the result in formal work. A zero determinant means the system does not have one unique solution. The calculator will then explain whether the equations conflict or describe many possible solutions.
You can also use this page to compare manual answers. Enter your own work, calculate again, and check each row operation against the displayed matrix steps online before submitting assignments.
FAQs
What is a system of equations?
A system of equations is a group of equations solved together. The same variables appear across the equations. A solution must satisfy every equation at the same time.
How many variables can this calculator solve?
This version solves two, three, and four variable linear systems. Choose the system size first, then enter the coefficients and constants for each equation.
Can I enter fractions?
Yes. You can enter fractions like 1/2 or -3/4. The calculator converts them into decimal values before solving the matrix.
What does determinant mean?
The determinant helps identify whether a square linear system has one unique solution. A nonzero determinant means the solution is unique.
What does infinite solutions mean?
Infinite solutions mean the equations are dependent. They describe the same line, plane, or relationship, so many values can satisfy the system.
What does no solution mean?
No solution means the equations conflict. Their left sides and constants cannot be true together, so no variable values satisfy every equation.
Why should I use residual checks?
Residual checks compare calculated answers against the original equations. Values near zero show that the solution fits the entered system accurately.
Can I download the result?
Yes. Use the CSV button for spreadsheet work. Use the PDF button when you need a printable report with equations, ranks, and steps.