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A system of linear equations is a set of two or more equations with the same variables. The solution is the point(s) where all equations are satisfied simultaneously.
Wzór
General: Ax = b where A is coefficient matrix, x is variable vector, b is constant vector
- A
- coefficient matrix (m × n)
- x
- variable vector (n × 1)
- b
- constant vector (m × 1)
Przewodnik krok po kroku
- 12×2 system: use Cramer's rule or elimination
- 2Determinant det = a₁b₂ − a₂b₁
- 3x = (c₁b₂ − c₂b₁)/det
- 4y = (a₁c₂ − a₂c₁)/det
Rozwiązane przykłady
Wejście
2x+3y=13, 4x−y=5
Wynik
x=2, y=3
Wejście
x+y=5, 2x−y=4
Wynik
x=3, y=2
Często zadawane pytania
How many solutions can a linear system have?
Zero (inconsistent), exactly one (unique), or infinitely many (infinitely many). It depends on rank and consistency.
When is a system inconsistent?
When the equations contradict (e.g., 0 = 1 after reduction). Geometrically: lines/planes don't all meet at a point.
Can I use elimination or substitution for large systems?
Yes, but matrix methods (Gaussian elimination, LU decomposition) are more efficient for computers.
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