What is Cramer's rule?

Cramer's rule expresses each variable in a linear system as a ratio of two determinants: the denominator is the determinant of the coefficient matrix (D), and the numerator is D with the constant column substituted for the variable's column.

What happens when D = 0?

When the coefficient matrix determinant is zero, the system is either inconsistent (no solution) or dependent (infinitely many solutions). The solver will flag this case and explain the situation.

Does this tool handle decimal coefficients?

Yes. You can enter any real number — integers, decimals, and negatives are all supported. Results are displayed as fractions when they are not whole numbers.

Is there a limit on coefficient size?

There is no hard limit, but very large or very small floating-point values may accumulate rounding errors. For exact fraction arithmetic, keep coefficients as small integers when possible.

Can I solve a 1-variable equation here?

No. This tool is designed for systems of 2 or 3 equations. A single-variable equation is trivially solved by dividing both sides by the coefficient.

Linear System Solver

Name: Linear System Solver
Availability: InStock
Author: Nham Vu

Enter coefficients for 2 or 3 linear equations and get exact solutions with full Cramer's rule step-by-step workings.

System Size

Enter coefficients for: ax + by = d

Equation 1

a₁

b₁

d₁

Equation 2

a₂

b₂

d₂

Enter coefficients and click Solve
to see the step-by-step solution.

Summary

Enter coefficients for 2 or 3 linear equations and get exact solutions with full Cramer's rule step-by-step workings.

How it works

Select whether you want to solve a 2-variable or 3-variable system.
Enter the coefficients (a, b, c) and the constant (d) for each equation.
Click "Solve" to compute the coefficient matrix determinant D.
The solver computes Dx, Dy (and Dz for 3-variable systems) by substituting the constants column.
Variables are found by x = Dx/D, y = Dy/D, z = Dz/D (Cramer's rule).
All intermediate determinants are shown so you can follow every step.

Use cases

Checking algebra homework involving simultaneous equations.
Verifying hand-computed Cramer's rule answers.
Teaching determinant-based linear algebra concepts.
Solving engineering or physics problems with multiple unknowns.
Quickly finding intersection points of two or three planes.
Debugging spreadsheet models with linear constraints.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu