What is Cramer's Rule?

Cramer's Rule is a theorem in linear algebra that expresses the solution of a system of linear equations with as many equations as unknowns using determinants. Each variable xi equals the ratio of two determinants: the determinant of a modified matrix (where column i is replaced by the constants vector) divided by the determinant of the coefficient matrix.

What happens when the main determinant D is zero?

If D = 0, the system either has no solution (inconsistent) or infinitely many solutions (dependent). In either case Cramer's Rule cannot produce a unique answer, and this tool will tell you so.

Does this tool handle decimal or fractional coefficients?

Yes. You can enter any real number including decimals (e.g. 1.5, -0.25). Results are displayed as exact fractions where the denominator is D, and as rounded decimals.

How is a 3x3 determinant expanded?

The tool uses cofactor expansion along the first row: det = a(ei-fh) - b(di-fg) + c(dh-eg), where the letters represent elements of the 3x3 matrix. Each sub-determinant is shown in the working panel.

Can this solve systems larger than 3x3?

No. Cramer's Rule becomes computationally expensive for large systems. For 4x4 and above, Gaussian elimination or LU decomposition are more practical. This tool focuses on the 2x2 and 3x3 cases commonly taught in courses.

System By Cramer's Rule

Solve a 2x2 or 3x3 system of linear equations using Cramer's Rule with full determinant step-by-step working.

System size

Enter coefficients for: a·x + b·y = d

Eq #

Eq 1

Eq 2

Enter coefficients and click Solve

Summary

This tool solves a system of 2 or 3 linear equations using Cramer's Rule. Enter the coefficients and constants, and the calculator shows the main determinant plus each variable's determinant, then derives the solution. All working is displayed so you can follow each step.

How it works

Select whether your system has 2 or 3 equations.
Enter the coefficients (a, b, c) and the constant (d) for each equation.
Click Solve to compute the main determinant D and the variable determinants Dx, Dy (and Dz for 3x3).
Each variable is found by dividing its determinant by D (e.g. x = Dx / D).
The step-by-step working panel shows every expansion so you can verify the arithmetic.

Use cases

Solve homework or exam problems for linear algebra courses.
Verify hand-calculated solutions to systems of equations.
Teach or learn Cramer's Rule with detailed step-by-step working.
Check whether a system has a unique solution, no solution, or infinitely many solutions.
Quickly compute 2x2 or 3x3 determinants as part of a larger problem.
Explore how changing coefficients affects the solution.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu

System By Cramer's Rule

Solution

Step-by-step working

Summary

How it works

Use cases

Frequently Asked Questions

System By Cramer's Rule

Use the System By Cramer's Rule

Solution

Step-by-step working

Summary

How it works

Use cases

Frequently Asked Questions