System of 3 Linear Equations Solver

This solves a1x + b1y + c1z = d1, a2x + b2y + c2z = d2, and a3x + b3y + c3z = d3 by Gaussian elimination with partial pivoting: it repeatedly picks the equation with the largest remaining coefficient in the current column, swaps it into place, and subtracts multiples of it from the equations below to zero out that column — the standard method for keeping the arithmetic numerically stable, taught alongside Cramer's rule as the general-purpose way to solve a linear system.

This site's 2-variable solver can tell 'infinitely many solutions' from 'no solution' just by checking two numbers, because with only two equations that check is both necessary and sufficient. With three equations that shortcut isn't reliable — the coefficient matrix can collapse all the way to a single direction while the three constants still contradict each other. This calculator instead finishes the elimination fully: if it's left with a row that has zero coefficients but a nonzero constant, the equations contradict each other (no solution); otherwise, any leftover redundant row means the system is underdetermined (infinitely many solutions) rather than pinned to one answer.