Gaussian Elimination Calculator

When to use?

Use it for systems of equations in algebra, linear algebra homework, circuit analysis, or small equilibrium models.

How to use

• Choose the number of variables.
• Enter coefficients and constants in the matrix.
• The solution is calculated with row reduction.

Input Explanation

Each row represents one equation. The last column is the right-hand side constant. For example, 2x + y - z = 8 becomes 2, 1, -1, 8.

Calculation Basis

It uses Gauss-Jordan elimination with partial pivoting and reports no solution or infinitely many solutions when the matrix is inconsistent or underdetermined.

Usage Examples

• Math and engineering homework - Solve and verify small systems of equations.
• Circuit analysis - Solve linear equations from Kirchhoff laws.

Examples

• 2x+y-z=8, -3x-y+2z=-11, -2x+y+2z=-3 → x=2, y=3, z=-1
• x+y=1 and x+y=2 is inconsistent, so there is no solution
• 2x+2y=4 and x+y=2 represents infinitely many solutions

Cautions

• Nearly singular matrices or very large coefficients can produce floating-point rounding error.

FAQ

When does a system have no solution?

When equations contradict each other, such as x + y = 1 and x + y = 2.

Why can rounding error appear?

Floating-point arithmetic and ill-conditioned matrices can amplify small numerical differences.

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