2 × 2 example
2x + y = 5
x − y = 1
The calculator returns x = 2 and y = 1. The determinant is −3, so the coefficient matrix is non-singular.
System of Equations Calculator
Solve a 2 × 2 or 3 × 3 system of linear equations with Gaussian elimination, then inspect the determinant, solution type, and substitution residual.
What this calculator does
Enter coefficients for equations in the form ax + by (+ cz) = d. The page uses row operations to classify the system as one solution, no solution, or infinitely many solutions. It does not choose values for an underdetermined system.
Solve the system
Interactive calculator
Enter the coefficients
Use decimal or whole-number values. Negative coefficients are allowed.
Equation forma₁₁x + a₁₂y (+ a₁₃z) = b₁
Primary result
Solution typeOne solution
x2
y1
Determinant-3
Largest residual0
MethodGaussian elimination
The 2 × 2 system has one solution. The residual is the largest absolute difference after substituting the calculated solution into the original equations.
Formula and method
For a square system A·x = b, Gaussian elimination uses row swaps, scaling, and row replacement to create an upper-triangular augmented matrix. Back-substitution then finds the variables when every column has a pivot.
- Arrange the coefficients and constants as an augmented matrix.
- Use the largest available pivot in each column to reduce avoidable rounding error.
- Classify a zero-coefficient/non-zero-constant row as no solution; a missing pivot without that contradiction means infinitely many solutions.
- For a unique solution, substitute the values back into the original equations and show the largest absolute residual.
For a 2 × 2 system, a non-zero determinant is a quick uniqueness check. For a 3 × 3 system, the determinant is shown as a diagnostic; the elimination classification remains the deciding result.
Worked examples
3 × 3 example
x + y + z = 6
2x − y + z = 3
x + 2y − z = 2
The solution is x = 1, y = 2, and z = 3. This illustrates how an independent third equation can determine the remaining variable.
How to use the result
Use a unique solution when the equations represent the same unknowns and units. A no-solution result means at least one equation conflicts with the others; an infinite-solution result means the equations do not provide enough independent information.
- Sensitivity hint: constants move the solution directly, while nearly dependent coefficient rows can make the solution sensitive to small measurement changes.
- What to do next: confirm the equation signs, units, and source measurements before treating a decimal answer as meaningful.
- Common mistakes: mixing units, copying a coefficient into the constant column, or treating a rounded near-zero determinant as proof of a stable solution.
Sources and assumptions
This page follows the standard augmented-matrix approach described by OpenStax. The coefficient and constant entries are treated as exact decimal inputs for the calculation; no uncertainty interval or symbolic parameter is inferred.
- OpenStax College Algebra: Systems of Linear Equations — system setup and solution types.
- OpenStax College Algebra: Gaussian Elimination — row-reduction method.
- OpenStax College Algebra: Cramer’s Rule — determinant-based context for square systems.
- MIT OpenCourseWare: Geometry of Linear Equations — geometric interpretation of coefficient rows and solution sets.
Limitations: This tool solves numeric 2 × 2 and 3 × 3 systems only. It does not replace symbolic algebra, uncertainty analysis, domain-specific modeling, or a check that the equations describe the same real-world units.
System of equations calculator FAQs
What does a determinant of zero mean?
For a square coefficient matrix, a zero determinant means the matrix is singular. The system may have no solution or infinitely many solutions, so the elimination classification is needed.
Why does the calculator report infinitely many solutions?
At least one equation is dependent on the others and no contradiction appears. One or more variables remain free, so there is not one unique numeric answer.
Can I enter negative coefficients?
Yes. Negative coefficients and constants are valid in linear equations. Keep the sign attached to the coefficient or constant when transcribing the equation.
How accurate are the decimal results?
The page uses double-precision arithmetic and reports a residual from the unrounded calculated solution. A nearly singular system can still be sensitive to small changes in measured inputs.