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  2. Indeterminate system - Wikipedia

    en.wikipedia.org/wiki/Indeterminate_system

    In the case of a linear system, the system may be said to be underspecified, in which case the presence of more than one solution would imply an infinite number of solutions (since the system would be describable in terms of at least one free variable [2]), but that property does not extend to nonlinear systems (e.g., the system with the ...

  3. Overdetermined system - Wikipedia

    en.wikipedia.org/wiki/Overdetermined_system

    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 .

  4. Cramer's rule - Wikipedia

    en.wikipedia.org/wiki/Cramer's_rule

    A system of equations is said to be inconsistent when there are no solutions and it is called indeterminate when there is more than one solution. For linear equations, an indeterminate system will have infinitely many solutions (if it is over an infinite field), since the solutions can be expressed in terms of one or more parameters that can ...

  5. Fermat's Last Theorem - Wikipedia

    en.wikipedia.org/wiki/Fermat's_Last_Theorem

    Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC). [28] Many Diophantine equations have a form similar to the equation of Fermat's Last Theorem from the point of view of algebra, in that they have no cross terms mixing two letters, without sharing its particular ...

  6. System of linear equations - Wikipedia

    en.wikipedia.org/wiki/System_of_linear_equations

    Each free variable gives the solution space one degree of freedom, the number of which is equal to the dimension of the solution set. For example, the solution set for the above equation is a line, since a point in the solution set can be chosen by specifying the value of the parameter z. An infinite solution of higher order may describe a ...

  7. Gaussian elimination - Wikipedia

    en.wikipedia.org/wiki/Gaussian_elimination

    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, [10] for a total of approximately 2n 3 /3 operations.

  8. Underdetermined system - Wikipedia

    en.wikipedia.org/wiki/Underdetermined_system

    An underdetermined linear system has either no solution or infinitely many solutions. For example, + + = + + = is an underdetermined system without any solution; any system of equations having no solution is said to be inconsistent. On the other hand, the system

  9. Fermat–Catalan conjecture - Wikipedia

    en.wikipedia.org/wiki/Fermat–Catalan_conjecture

    The first of these (1 m + 2 3 = 3 2) is the only solution where one of a, b or c is 1, according to the Catalan conjecture, proven in 2002 by Preda Mihăilescu. While this case leads to infinitely many solutions of (1) (since one can pick any m for m > 6), these solutions only give a single triplet of values (a m, b n, c k).