Result
Calculator Inputs
Example Data Table
| Case | Equations | Expected Classification | Reason |
|---|---|---|---|
| Unique solution | 2x + y - z = 8; -3x - y + 2z = -11; -2x + y + 2z = -3 | Consistent, unique | Rank equals variable count. |
| No solution | x + y = 2; 2x + 2y = 5 | Inconsistent | Augmented rank is larger. |
| Infinite solutions | x + y = 3; 2x + 2y = 6 | Consistent, dependent | Rank is smaller than variables. |
Formula Used
The calculator writes the system as A x = b. Matrix A contains the coefficients. Vector x contains the variables. Vector b contains the constants.
It compares rank(A) with rank([A|b]). If rank(A) is less than rank([A|b]), the system is inconsistent. If the ranks are equal, the system is consistent.
If rank(A) equals the number of variables, there is one solution. If rank(A) is smaller than the number of variables, there are infinitely many solutions.
For square systems, the determinant is also checked. A nonzero determinant confirms a unique solution.
How To Use This Calculator
- Select the number of variables.
- Select the number of equations.
- Enter every coefficient in standard equation form.
- Use zero when a variable is missing.
- Enter each constant from the right side.
- Choose decimal places and tolerance.
- Click Solve System.
- Download CSV or PDF when needed.
What This Calculator Does
A system of equations can describe two lines, three planes, or a larger algebra model. The main question is not always the value of x, y, and z. The first question is whether the system can be solved at all. This calculator checks that question before showing a solution. It classifies the system as consistent, inconsistent, unique, or dependent. That makes it useful for algebra classes, matrix practice, and quick homework review.
Why Consistency Matters
A consistent system has at least one solution. The equations agree at one point, one line, or one plane. An inconsistent system has no shared solution. In two variables, this often means parallel lines. In three variables, it may mean planes that never meet together. A dependent system has infinitely many solutions. The equations repeat the same relationship or do not add enough independent information.
Advanced Matrix Checks
The calculator uses the coefficient matrix and the augmented matrix. It compares their ranks. If both ranks are equal, the system is consistent. If the augmented rank is larger, the system is inconsistent. If the common rank equals the number of variables, the solution is unique. If the rank is smaller than the number of variables, there are infinitely many solutions. These checks match the Rouché-Capelli theorem.
Helpful Result Details
The result box shows the classification, determinant when the square matrix allows it, reduced row echelon steps, pivot count, and variable values when they are unique. It also reports free variables for dependent cases. The step table helps students see where row operations changed the system. The export buttons save the result as a small CSV file or a simple PDF report.
When To Use It
Use this tool after writing each equation in standard form. Put every variable on the left side. Put the constant on the right side. Use zero for missing coefficients. Check signs carefully, because a wrong sign changes the whole system. Try the example table first. Then enter your own coefficients and compare the calculator output with your manual row reduction.
Teachers can also use it for classroom demonstrations. It shows why one extra equation may confirm, contradict, or leave a solution family unchanged during guided practice sessions.
FAQs
What is a consistent system?
A consistent system has at least one solution. The equations share a point, line, plane, or higher dimensional solution set.
What is an inconsistent system?
An inconsistent system has no solution. The equations contradict each other after row reduction or rank comparison.
What does rank mean here?
Rank is the number of independent rows or columns. It shows how much unique information the equations provide.
When is the solution unique?
The solution is unique when the coefficient rank equals the number of variables and matches the augmented rank.
When are there infinitely many solutions?
There are infinitely many solutions when the system is consistent but has fewer independent equations than variables.
Can I solve two-variable systems?
Yes. Select two variables. Enter x and y coefficients, constants, and use zero where needed.
Why should I enter zero for missing terms?
Zero keeps the matrix structure correct. It tells the calculator that the variable is absent from that equation.
What does a nonzero determinant mean?
For a square coefficient matrix, a nonzero determinant means the system has exactly one solution.