Simultaneous Equations

In mathematics, simultaneous equations are a set of equations containing multiple variables. This set is often referred to as a system of equations. A solution to a system of equations is a particular specification of the values of all variables that simultaneously satisfies all of the equations. The elementary methods to solve simple systems of equations include graphical method, the matrix method, the substitution method, or the elimination method. Some textbooks refer to the elimination method as the addition method, since it involves adding equations (or constant multiples of the said equations) to one another, as detailed later in this article.

Among the systems of equations, the systems of linear equations are especially important. There were the original object of study of linear algebra. Many algorithms have been devised to solve them, which allow to solve huge systems (up to millions of variables).

To determine approximate solutions to general systems numerically on a computer, iterative methods, like Newton's method, may be used, but they can not prove that all solutions are found. In particular, if no solution is found, these methods do not allow to deduce that there is no solution.

When all equations are polynomial, a set of simultaneous equations is a system of polynomial equations. These systems were the original object of study of Algebraic geometry, but, unless in very simple case, the systems of polynomial equations may not be solved by hand computation. Thus algebraic geometry is mainly the qualitative study of the set of solutions of such systems, and the design and the study of the algorithms to solve these systems is now a subarea of computer algebra. Algorithms are now available that allow to solve systems of polynomial equations having several hundreds of solutions.