System of Linear Equations Solver
Solve systems of 2 or 3 linear equations with multiple unknowns.
2×2 System
3×3 System
Solution
—
Enter equations and calculate
x
—
y
—
z
—
Determinant
—
Status
—
Type
—
Solving Systems of Equations
A system of linear equations has a unique solution if the coefficient matrix is non-singular (determinant ≠ 0).
Cramer's Rule (2×2)
x = Dₓ/D, y = Dᵧ/D
Where D is the determinant of the coefficient matrix, Dₓ replaces the first column with constants, and Dᵧ replaces the second column.
Solution Types
| Condition | Solution |
|---|---|
| D ≠ 0 | Unique solution |
| D = 0, Dₓ = 0, Dᵧ = 0 | Infinite solutions |
| D = 0, Dₓ ≠ 0 or Dᵧ ≠ 0 | No solution |
Geometric Interpretation
- 2×2: Intersection point of two lines
- 3×3: Intersection point of three planes
Frequently Asked Questions
What if the system has no solution?
If D = 0 but Dₓ or Dᵧ ≠ 0, the equations are inconsistent (parallel lines that never intersect).
What are infinite solutions?
When all determinants are zero, the equations are dependent (same line or plane), resulting in infinitely many solutions.