Systems of Equations Calculator

Solve 2x2 or 3x3 systems of linear equations using Cramer's rule with step-by-step determinant calculations.

System Size & Coefficients

Equation 1: a1x + b1y = c1
x + y =
Equation 2: a2x + b2y = c2
x + y =

Solution

Solution
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Step-by-Step Solution (Cramer's Rule)

x = Dx/D, y = Dy/D

Solving Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of unknowns. The solution is the set of values for the unknowns that satisfies all equations simultaneously. This calculator uses Cramer's rule, which employs determinants to find the solution.

Methods for Solving

Cramer's Rule

Uses determinants of coefficient matrices. Each variable is the ratio of a modified determinant to the main determinant.

x_i = det(A_i) / det(A)

Gaussian Elimination

Row-reduce the augmented matrix to echelon form, then back-substitute to find values.

[A|b] -> [I|x] via row ops

Matrix Inverse

If A is invertible, the solution vector x = A^(-1)*b.

x = A^{-1} * b

Types of Solutions

A system of linear equations can have exactly one solution (consistent and independent), infinitely many solutions (consistent and dependent), or no solution (inconsistent). The determinant of the coefficient matrix tells us which case applies: if det(A) is nonzero, there is a unique solution.

Tips

  • If the determinant of the coefficient matrix is zero, Cramer's rule cannot be applied directly.
  • For large systems, Gaussian elimination is generally more efficient than Cramer's rule.
  • Always verify your solution by substituting back into the original equations.
  • Consistent systems with fewer equations than unknowns have infinitely many solutions.