System of Equations Solver

How to Use This Solver

Write each equation in standard form: all variable terms on the left, the constant on the right. For example, the equation 3x - 2y = 7 has coefficients 3 (for x) and -2 (for y), and a right-hand side of 7. Enter those three numbers into the corresponding row. If a variable does not appear in an equation, enter 0 for its coefficient.

Worked Example (Default Values)

The default 2x2 system is:

Setting up the augmented matrix and eliminating x from equation 2: multiply equation 1 by 1/2 and subtract from equation 2. This gives the system in upper triangular form. Back substitution then yields y = 3 and x = 1. Substituting back: 2(1) + 3 = 5 and 1 + 3(3) = 10. Both equations check out.

Method: Gaussian Elimination

Gaussian elimination converts the augmented coefficient matrix [A|b] into row echelon form using three elementary row operations: swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another. These operations do not change the solution set. Once upper triangular form is reached, back substitution solves for each variable from the bottom equation upward.

This solver applies partial pivoting at each elimination step: it searches the current column for the largest absolute value and swaps that row to the pivot position before eliminating. Partial pivoting reduces rounding errors and handles near-zero pivots gracefully.

If a pivot is exactly zero after all swaps (a zero column), the system is either inconsistent (no solution) or dependent (infinitely many solutions). The solver detects this and reports accordingly.

Types of Solutions

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Sources and method: Gaussian elimination with partial pivoting as described in Golub and Van Loan, Matrix Computations (4th ed., Johns Hopkins University Press, 2013). Back substitution for upper triangular systems as per standard linear algebra curriculum.

This solver handles systems of linear equations only (all variables appear to the first power). Non-linear equations (quadratics, exponentials, trigonometric) require different methods. Results are computed using floating-point arithmetic; very large or very small coefficients may introduce rounding errors.