3x3 System of Equations Solver

How the 3x3 system of equations solver works

A three-variable linear system appears whenever three unknown quantities are linked by three independent equations, from balancing a small resource model to locating the intersection of three planes. This solver reads each augmented row as the coefficients of x, y, and z followed by its constant. It evaluates the coefficient determinant and three replacement determinants, then applies Cramer’s rule to recover the unique solution.

The default rows represent x plus y plus z equals 6, 2x minus y plus z equals 3, and x plus 2y minus z equals 2. Their coefficient determinant is 7. Replacing one coefficient column at a time produces the verified solution x equals 1, y equals 2, and z equals 3, which satisfies all three original equations.

Reading 3x3 system of equations solver results privately

A determinant near zero means Cramer’s rule cannot identify one unique point; the equations may be dependent or inconsistent. This page deliberately reports that condition instead of presenting unstable values. Enter exactly three rows and four finite numbers per row. For larger systems or ill-conditioned numerical work, elimination with pivot diagnostics is a more appropriate method. This 3x3 system of equations solver calculation runs entirely in your browser, so the numbers you enter never leave your device.