Systems of Equations Solver
Solve systems of 2 or 3 linear equations instantly. Enter the coefficients for each equation and get the solution using Cramer's rule, with identification of unique, infinite, or no-solution cases.
Systems of Equations Solver
a₁x + b₁y = c₁
a₂x + b₂y = c₂
How to Use the Systems of Equations Solver
Select whether you have a 2-variable or 3-variable system, then enter the coefficients for each equation. The solver uses Cramer's rule to find the solution and identifies the type of system.
Understanding Systems of Linear Equations
A system of linear equations consists of two or more equations with the same variables. The solution is the set of values that satisfies all equations simultaneously.
Types of Solutions
- Unique Solution: The system has exactly one solution where all equations intersect at a single point
- No Solution: The equations are inconsistent (parallel lines or planes that never meet)
- Infinitely Many Solutions: The equations are dependent (same line or overlapping planes)
Cramer's Rule
Cramer's rule is a method for solving systems of linear equations using determinants. For a system Ax = b:
- Calculate the determinant of the coefficient matrix A
- If det(A) ≠ 0, the system has a unique solution
- Each variable is found by replacing its column with the constant vector and dividing by det(A)
Examples
2-Variable System
Solve: 2x + 3y = 8 and x - y = -1
Solution: x = 1, y = 2
3-Variable System
Solve: x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2
Solution: x = 1, y = 2, z = 3
Applications
Systems of equations are used in:
- Engineering and physics calculations
- Economics and business modeling
- Network flow and traffic analysis
- Computer graphics and game development
- Scientific research and data analysis