learn.howToCalculate
learn.whatIsHeading
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.
Formel
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)
Trin-for-trin guide
- 12×2 system: use Cramer's rule or elimination
- 2Determinant det = a₁b₂ − a₂b₁
- 3x = (c₁b₂ − c₂b₁)/det
- 4y = (a₁c₂ − a₂c₁)/det
Løste eksempler
Input
2x+3y=13, 4x−y=5
Resultat
x=2, y=3
Input
x+y=5, 2x−y=4
Resultat
x=3, y=2
Ofte stillede spørgsmål
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.
Klar til at beregne? Prøv den gratis System of Equations Solver-beregner
Prøv det selv →