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Two linear systems using the same set of variables are equivalent if each of the equations in the second system can be derived algebraically from the equations in the first system, and vice versa. Two systems are equivalent if either both are inconsistent or each equation of each of them is a linear combination of the equations of the other one.
For example, to solve a system of n equations for n unknowns by performing row operations on the matrix until it is in echelon form, and then solving for each unknown in reverse order, requires n(n + 1)/2 divisions, (2n 3 + 3n 2 − 5n)/6 multiplications, and (2n 3 + 3n 2 − 5n)/6 subtractions, [9] for a total of approximately 2n 3 /3 operations.
The field of elimination theory was motivated by the need of methods for solving systems of polynomial equations. One of the first results was Bézout's theorem, which bounds the number of solutions (in the case of two polynomials in two variables at Bézout time).
Drug elimination, clearance of a drug or other foreign agent from the body; Elimination, the destruction of an infectious disease in one region of the world as opposed to its eradication from the entire world; Hazard elimination, the most effective type of hazard control; Elimination (pharmacology), processes by which a drug is eliminated from ...
Cramer's rule, implemented in a naive way, is computationally inefficient for systems of more than two or three equations. [7] In the case of n equations in n unknowns, it requires computation of n + 1 determinants, while Gaussian elimination produces the result with the same computational complexity as the computation of a single determinant.
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