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  2. System of equations - Wikipedia

    en.wikipedia.org/wiki/System_of_equations

    In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single equations, namely as a: System of linear equations, System of nonlinear equations,

  3. Overdetermined system - Wikipedia

    en.wikipedia.org/wiki/Overdetermined_system

    The concept can also be applied to more general systems of equations, such as systems of polynomial equations or partial differential equations. In the case of the systems of polynomial equations, it may happen that an overdetermined system has a solution, but that no one equation is a consequence of the others and that, when removing any ...

  4. System of linear equations - Wikipedia

    en.wikipedia.org/wiki/System_of_linear_equations

    The simplest method for solving a system of linear equations is to repeatedly eliminate variables. This method can be described as follows: In the first equation, solve for one of the variables in terms of the others. Substitute this expression into the remaining equations. This yields a system of equations with one fewer equation and unknown.

  5. Consistent and inconsistent equations - Wikipedia

    en.wikipedia.org/wiki/Consistent_and...

    The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...

  6. List of unsolved problems in mathematics - Wikipedia

    en.wikipedia.org/wiki/List_of_unsolved_problems...

    Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.

  7. Relaxation (iterative method) - Wikipedia

    en.wikipedia.org/wiki/Relaxation_(iterative_method)

    Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. [ 2 ] [ 3 ] [ 4 ] Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation , it resembles repeated application of a local ...

  8. Linear algebra - Wikipedia

    en.wikipedia.org/wiki/Linear_algebra

    Systems of linear equations form a fundamental part of linear algebra. Historically, linear algebra and matrix theory have been developed for solving such systems. In the modern presentation of linear algebra through vector spaces and matrices, many problems may be interpreted in terms of linear systems.

  9. System of polynomial equations - Wikipedia

    en.wikipedia.org/wiki/System_of_polynomial_equations

    A system with infinitely many solutions is said to be positive-dimensional. A zero-dimensional system with as many equations as variables is sometimes said to be well-behaved. [3] Bézout's theorem asserts that a well-behaved system whose equations have degrees d 1, ..., d n has at most d 1 ⋅⋅⋅d n solutions. This bound is sharp.

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