System of Equations Solver with Steps

Three equations in three unknowns is where solving by hand starts to go wrong: one sign slip in the second elimination and every answer after it is off. Type the equations one per line and this System of Equations Solver reduces them with exact fractions. It reports the answer, then shows every row operation that produced it.

Equations do not need tidying first. You can write 2(a + b) = 18, a - b/2 = 3 or put variables on both sides; the parser expands brackets and moves every term to the left. Any single letter works as a variable, with an optional number (x1, x_2).

Worked Example: Three Equations, Three Unknowns

The page opens on this system:

• 2x + 3y - z = 4
• x - y + 2z = 7
• 3x + y + z = 10

The result is One solution: x = 2, y = 1, z = 3, with rank 3. Open Show Gauss–Jordan steps and there are nine row operations, starting with R1 → (1/2)R1. The green check list substitutes the answer back into each original line, so the first reads 4 = 4. Load Sample moves on to a bracketed pair, a dependent system, a contradictory one, a 4×4 and a pair with decimal coefficients.

Row Reduction Method, Pivot by Pivot

For each variable in turn, the solver finds the first row at or below the current one with a non-zero entry, swaps it up if needed and scales it so the pivot is 1. It then clears that column in every other row. The finished matrix is in reduced row echelon form. Every value is a fraction of two arbitrary-precision integers, so 1/3 stays 1/3 through a 12×12 system instead of drifting to 0.333333.

The Decimals switch changes only how numbers are printed. Results are rounded to 6 decimal places and marked with ≈ when the exact value does not terminate. Copy Steps writes the whole working as plain text, matrix by matrix, in whichever format is showing.

Edge Cases: Dependent and Contradictory Systems

When a row reduces to all zeros, one equation was a multiple or mixture of the others. With x + 2y - z = 4 and its double, z becomes a free variable and the answer is written as x = 2 - (1/3)z, y = 1 + (2/3)z. When a row reduces to 0 = a non-zero number, the equations contradict each other and the tool says which row.

Only linear systems are accepted. Typing xy = 3 stops with “Line 1 multiplies two variables together (x·y), which makes the system nonlinear.” A ^ or a variable in a denominator is refused the same way. Minus signs and × pasted from a PDF are read as ordinary operators, and a line starting with # is skipped, which is handy for labelling equations.