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The unique pair of values a, b satisfying the first two equations is (a, b) = (1, 1); since these values also satisfy the third equation, there do in fact exist a, b such that a times the original first equation plus b times the original second equation equals the original third equation; we conclude that the third equation is linearly ...
The equations x − 2y = −1, 3x + 5y = 8, and 4x + 3y = 7 are linearly dependent. For example, the equations + = + = are not independent — they are the same equation when scaled by a factor of two, and they would produce identical graphs.
On the other hand, the primes 3, 7, 11, 19, 23 and 31 are all congruent to 3 modulo 4, and none of them can be expressed as the sum of two squares. This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 (if number squared is even) or 1 (if number squared is odd) modulo 4.
This corresponds to a set of y values whose product is a square number, i.e. one whose factorization has only even exponents. The products of x and y values together form a congruence of squares. This is a classic system of linear equations problem, and can be efficiently solved using Gaussian elimination as soon as the number of rows exceeds ...
Animation depicting the process of completing the square. (Details, animated GIF version)In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form + + to the form + for some values of and . [1]
Finding a given Latin square's isomorphism class can be a difficult computational problem for squares of large order. To reduce the problem somewhat, a Latin square can always be put into a standard form known as a reduced square. A reduced square has its top row elements written in some natural order for the symbol set (for example, integers ...
[4] [5] [6] 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 (up to a constant factor independent of n ...
In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a system of linear equations.