|
In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.〔The subject of this article is basic in mathematics, and is treated in a lot of textbooks. Among them, Lay 2005, Meyer 2001, and Strang 2005 contain the material of this article.〕 For example, : is a system of three equations in the three variables . A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by : since it makes all three equations valid. The word "''system''" indicates that the equations are to be considered collectively, rather than individually. In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system. Very often, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed, see Linear equation over a ring. Integer linear programming is a collection of methods for finding the "best" integer solution (when there are many). Gröbner basis theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure. ==Elementary example== The simplest kind of linear system involves two equations and two variables: : One method for solving such a system is as follows. First, solve the top equation for in terms of : : Now substitute this expression for ''x'' into the bottom equation: : This results in a single equation involving only the variable . Solving gives , and substituting this back into the equation for yields . This method generalizes to systems with additional variables (see "elimination of variables" below, or the article on elementary algebra.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「System of linear equations」の詳細全文を読む スポンサード リンク
|