System of Equations Solver
Solve 2×2 and 3×3 systems of linear equations with Cramer's rule, determinant steps, and input matrix echoes.
Category: Mathematics
About this tool
What Is Cramer's rule? System of Equations Solver. The system solver handles linear systems of two or three equations in the same number of unknowns using Cramer's rule, printing the coefficient determinant and the solution set. For a square system, each unknown xᵢ equals the ratio of two determinants: the matrix with column i replaced by the constants, divided by the coefficient determinant. When that determinant is zero the system has either no solution or infinitely many. The solver also detects the degenerate case: a near-zero determinant produces a clear warning instead of a misleading number, so singular and inconsistent systems are not silently mistreated. xᵢ = det(A with column i replaced by b) ÷ det(A). How to Use Choose 2×2 or 3×3 system size. Fill the coefficient fields and the constants on the right side of each equation. Read the determinant, each unknown, and a warning if the system is singular. Worked Example For {2x + y = 7, x − y = −1} the tool returns x = 2, y = 3. For the 3×3 system {x + y + z = 6, 2x + y − z = 1, x − 2y + 3z = 5} it returns x = 13/7, y = 9/7, z = 20/7, given exactly as fractions. Common Use Cases mixture and price problems in word form circuit current loops with Kirchhoff's laws balancing chemical equations quickly econometrics and network flow setups Important Notes Answers are shown as reduced fractions and decimals. Terms use literal keys