System of Linear Equations Solver

Solve systems of 2 or 3 linear equations using Cramer's rule. Get the determinant, variable values, and step-by-step solution for any system of linear equations.

Coefficients / Constants

a₁x + b₁y = c₁ , a₂x + b₂y = c₂

How to Use the System of Linear Equations Solver

Enter the numbers or values in the input fields.
The result is calculated and displayed automatically.
Review the step-by-step solution or detailed breakdown.
Copy the result or adjust inputs for a new calculation.

快速参考

从	到
2 + 3	5
12 × 12	144
√144	12
2¹⁰	1,024
π	3.14159
e	2.71828

使用场景

•Checking homework or exam answers quickly and accurately.
•Verifying manual calculations in professional or academic work.
•Learning mathematical concepts with instant visual feedback.
•Performing quick computations during meetings or presentations.

公式

Cramer's rule solves a system of linear equations using determinants. For a 2×2 system: x = Dx/D, y = Dy/D, where D is the coefficient determinant, Dx replaces the x-column with constants, and Dy replaces the y-column. If D = 0, the system has no unique solution.

常见问题

What is Cramer's rule?

Cramer's rule is a method for solving systems of linear equations using determinants. Each variable equals the ratio of a modified determinant (where one column is replaced by the constants) to the main coefficient determinant.

When does a system have no solution?

A system has no solution when the coefficient determinant D = 0 and at least one of the numerator determinants (Dx, Dy, Dz) is non-zero. This means the equations are contradictory.

When does a system have infinitely many solutions?

A system has infinitely many solutions when D = 0 and all numerator determinants are also zero. This means the equations are dependent — they describe the same line or plane.

Can I solve a 3×3 system with this calculator?

Yes. Switch to 3×3 mode to enter 9 coefficients and 3 constants. The calculator uses 3×3 determinants via Cramer's rule to find x, y, and z.