Math

Two-variable Linear System Solver

Solve two simultaneous linear equations ax + by = c and dx + ey = f.

✓ Browser-local processing

Result

Loading tool…

How to use Two-variable Linear System Solver

Cramer’s rule solves a two-by-two linear system using its determinant. All coefficients are scaled together before multiplication.

Enter a in ax + by = c, b in ax + by = c, c in ax + by = c, d in dx + ey = f, e in dx + ey = f, f in dx + ey = f.
Check that the input follows the stated limits and uses consistent units.
Run the calculator, review the result, then copy or download it.

Singular and numerically near-singular systems are rejected because they do not provide a reliable unique solution. General matrices are not supported.
Decimal inputs have at most 120 characters and magnitude from 1e-100 to 1e15, or zero. Floating-point results are rounded to 12 significant digits; extreme derived results may underflow or be rejected. Exact integer results are identified separately.

What calculation does Two-variable Linear System Solver perform?

Cramer’s rule solves a two-by-two linear system using its determinant. All coefficients are scaled together before multiplication.

What does a singular system mean?

The equations may be dependent or inconsistent. In either case, there is no unique pair of x and y for this tool to return.

Summarize a numeric list with mean, median, modes, range, count, and sum.

Add, subtract, multiply, or divide two fractions and reduce the exact result.

Compute exact reactions, left/right shear and moment extrema for a declared simply supported span with point and partial uniform loads.

Optimize a numeric two-variable linear objective within an explicit box and inspect exact feasible vertices.

Distinguish unique intersections, collinear overlaps, parallel lines and degenerate planar segments on a declared grid.

Add or multiply bounded matrices, or compute a square matrix determinant or inverse.