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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).
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.
Fourier–Motzkin elimination, also known as the FME method, is a mathematical algorithm for eliminating variables from a system of linear inequalities. It can output real solutions. The algorithm is named after Joseph Fourier [ 1 ] who proposed the method in 1826 and Theodore Motzkin who re-discovered it in 1936.
A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I].
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.
Variable elimination (VE) is a simple and general exact inference algorithm in probabilistic graphical models, such as Bayesian networks and Markov random fields. [1] It can be used for inference of maximum a posteriori (MAP) state or estimation of conditional or marginal distributions over a subset of variables.
Example with infinitely many solutions: 3x + 3y = 3, 2x + 2y = 2, x + y = 1. Example with no solution: 3 x + 3 y + 3 z = 3, 2 x + 2 y + 2 z = 2, x + y + z = 1, x + y + z = 4. These results may be easier to understand by putting the augmented matrix of the coefficients of the system in row echelon form by using Gaussian elimination .
On the contrary, the lexicographical order is, almost always, the most difficult to compute, and using it makes impractical many computations that are relatively easy with graded reverse lexicographic order (grevlex), or, when elimination is needed, the elimination order (lexdeg) which restricts to grevlex on each block of variables.