System of Equations Calculator
Solve systems of linear equations using multiple methods with step-by-step solutions
📊 System of Equations Calculator
2x + 1y = 7
1x + -1y = 1
1x + -1y = 1
Formula
ax + by = e
cx + dy = f
Where:
- Cramer's Rule =x = Dx/D, y = Dy/D
- Elimination =Add/subtract equations to eliminate variables
- Substitution =Solve for one variable, substitute into other
- D =Determinant of coefficient matrix
- Dx, Dy =Determinants with constants replacing columns
About Systems of Equations
A system of linear equations is a collection of equations with the same variables. Solutions are the values that satisfy all equations simultaneously.
Solution Methods:
- • Elimination: Add/subtract equations to eliminate variables
- • Substitution: Solve for one variable, substitute into others
- • Cramer's Rule: Use determinants (works when D ≠ 0)
- • Matrix methods: Gaussian elimination, inverse matrices
Types of Solutions:
- • Unique solution: Lines intersect at one point
- • Infinite solutions: Lines are identical
- • No solution: Lines are parallel
Applications:
- • Business: profit/cost analysis
- • Engineering: circuit analysis
- • Economics: supply and demand
- • Physics: force equilibrium
- • Chemistry: mixture problems
Tips
- Enter the dimensions in the same unit for consistency.
- Results update automatically as you type.
- Use the unit selector to convert between different measurement systems.