Linear Equation and System Solver

How to solve a system of equations with this tool

1. Type your equations in Equations (one per line), for example 2x + 3y = 7 and x − y = 1. Unknowns can be any letters (x, y, z, a, b…) or names such as x1 and x2. Coefficients may be integers, decimals (0.5) or fractions (1/2 or (1/2)x), and constants may appear on both sides: 3x + 4 = 2x − 1 works.
2. The answer appears as you type. Solve does the same, Example loads a classic 3 × 3 system, Clear empties everything and Copy copies the solution.
3. Prefer a table? Open Coefficient grid, choose the Grid size (2 × 2 to 5 × 5) and fill in the coefficients, with the constants in the column after the = sign. An empty cell counts as 0. The grid and the equations stay in sync.
4. Read the result and the explanations below it: Gauss–Jordan elimination with every row operation and the matrix after each one, Cramer’s rule for 2 × 2 and 3 × 3 systems, and Check in the original equations.

Every calculation is exact: the tool works with fractions, never with rounded decimals, so 1/3 stays 1/3. A decimal approximation is shown next to fractional answers.

Gaussian elimination, step by step

The system is first written as an augmented matrix: one row per equation, one column per unknown, and the constants after a vertical bar. Three row operations change the matrix without changing the solutions:

• swap two rows (R₁ ↔ R₂);
• multiply a row by a non-zero number (R₁ → R₁ ÷ 2);
• add a multiple of another row (R₂ → R₂ − 3R₁).

The tool works column by column. It picks a pivot, preferring an entry equal to 1 or −1 to avoid fractions, swaps it into place, divides the row so the pivot becomes 1, then clears the rest of the column in every other row. This is the Gauss–Jordan variant: at the end the matrix is in reduced row echelon form and the solution can be read directly from the last column.

One, none or infinitely many solutions

• One solution: every unknown has a pivot. The rank equals the number of unknowns.
• No solution: a row becomes 0 = 5 (zero coefficients, non-zero constant). The equations contradict each other; for two unknowns the lines are parallel.
• Infinitely many solutions: some unknowns have no pivot. They become free parameters (t, s…) and the other unknowns are expressed in terms of them, for example x = 1 + t, y = 5 − 2t, z = t. Each value of t gives one solution.

Cramer’s rule

For a square system Ax = b with det(A) ≠ 0, each unknown equals det(Aᵢ) ÷ det(A), where Aᵢ is A with column i replaced by the constants. The tool shows the determinants: ad − bc for 2 × 2, and the expansion along the first row for 3 × 3. When det(A) = 0 the rule does not apply, and the elimination tells you whether there is no solution or infinitely many. For larger systems elimination is much faster, so Cramer’s rule is only shown up to 3 × 3.