System of Equations Calculator

Solution classification and interpretation

A unique solution displays x and y. Dependent or contradictory equations instead show a classification and leave the coordinates undetermined. The coefficient determinant supports that distinction.

To find the slope and equation through two fixed points, use the Slope Calculator and follow its own input conventions.

The formula

Cramer’s rule computes x and y when the coefficient determinant is nonzero. For a zero determinant, the calculator checks zero-coefficient contradictions and whether the augmented equation rows agree.

How to use this calculator

1. Rearrange the first equation into ax + by = c and enter its three coefficients.
2. Rearrange the second equation into dx + ey = f and enter its three coefficients.
3. Read the solution classification; substitute any unique x and y into both original equations.

System of Equations input conventions

Move all constant terms to the right side before entering each equation. The coefficients must describe the same x and y variables in both rows. A missing variable has coefficient zero, while a lone negative variable has coefficient −1. The displayed determinant decides whether the coefficient matrix can produce a unique solution.

Interpreting solution classification

When the determinant is zero, the equations can be contradictory or describe the same constraint. The page checks zero-coefficient equations and proportional augmented rows before reporting a classification. Close-to-parallel lines may have a very distant intersection and be sensitive to small measurement errors. A numerical answer does not establish that an approximate physical model is well conditioned or that rounded inputs are exact.

Assumptions & limitations

What this calculation assumes

• Both rows describe real linear equations in the same x and y variables.

What to keep in mind

• Two real linear equations only; no nonlinear equations, symbolic coefficients, or conditioning estimate.

Common questions