Gleichungssystemrechner (2×2) — Formel
## How to Solve Simultaneous Equations
### Cramer's Rule for 2×2 Systems
Given:
- a₁x + b₁y = c₁
- a₂x + b₂y = c₂
**Determinant D = a₁b₂ - a₂b₁**
**x = (c₁b₂ - c₂b₁) / D**
**y = (a₁c₂ - a₂c₁) / D**
If D = 0, the system has no unique solution (the lines are parallel or identical).
### Cramer's Rule for 2×2 Systems
Given:
- a₁x + b₁y = c₁
- a₂x + b₂y = c₂
**Determinant D = a₁b₂ - a₂b₁**
**x = (c₁b₂ - c₂b₁) / D**
**y = (a₁c₂ - a₂c₁) / D**
If D = 0, the system has no unique solution (the lines are parallel or identical).
Lösungsbeispiel
Solve: 2x + 3y = 8 and 4x - y = 2.
- D = (2)(-1) - (4)(3) = -2 - 12 = -14
- x = (8×(-1) - 2×3) / (-14) = (-8 - 6) / (-14) = -14 / -14 = 1
- y = (2×2 - 4×8) / (-14) = (4 - 32) / (-14) = -28 / -14 = 2
- Solution: x = 1, y = 2