System of Equations Calculator
2×2 System of Equations
Solve: a₁x + b₁y = c₁ and a₂x + b₂y = c₂
3×3 System of Equations
Solve: a₁x + b₁y + c₁z = d₁, etc.
Formula
Cramer's Rule: x = det(Ax)/det(A) | y = det(Ay)/det(A) | z = det(Az)/det(A)
Frequently Asked Questions
What is Cramer's Rule?
Cramer's Rule solves systems of linear equations using determinants. For a 2×2 system: x = det(Ax)/det(A) and y = det(Ay)/det(A), where Ax replaces the x-column with constants. It works when det(A) ≠ 0.
When does a system have no solution?
A system has no solution when the equations are inconsistent (e.g., parallel lines that never intersect). This happens when the determinant is 0 and the system is not dependent.
You may also need
+
Matrix Calculator
Free matrix calculator. Calculate the determinant, inverse, and transpose of 2×2 and 3×3 matrices.
Math+
Matrix Inverse Calculator
Free matrix inverse calculator. Find the inverse of 2x2 and 3x3 matrices with determinant check and step-by-step adjugate method.
Math+
Linear Equation Calculator
Free linear equation calculator. Solve equations of the form ax + b = c, or systems of two linear equations.
Math