Editing Equation (section)

==Algebra==
[[Algebra]] studies two main families of equations: [[polynomial equations]] and, among them, the special case of [[linear equations]]. When there is only one variable, polynomial equations have the form ''P''(''x'')&nbsp;=&nbsp;0, where ''P'' is a [[polynomial]], and linear equations have the form ''ax''&nbsp;+&nbsp;''b''&nbsp;=&nbsp;0, where ''a'' and ''b'' are [[parameter#Mathematical functions|parameters]]. To solve equations from either family, one uses algorithmic or geometric techniques that originate from [[linear algebra]] or [[mathematical analysis]]. Algebra also studies [[Diophantine equations]] where the coefficients and solutions are [[integers]]. The techniques used are different and come from [[number theory]]. These equations are difficult in general; one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions.

===Polynomial equations===
{{main|Polynomial equation}}

[[File:Polynomialdeg2.svg|thumb|right|220px|The solutions –1 and 2 of the polynomial equation {{nowrap|1=''x''<sup>2</sup> – ''x'' + 2 = 0}} are the points where the [[graph of a function|graph]] of the [[quadratic function]] {{nowrap|1=''y'' = ''x''<sup>2</sup> – ''x'' + 2}} cuts the x-axis.]]

In general, an ''algebraic equation'' or [[polynomial equation]] is an equation of the form

:<math>P = 0</math>, or

:<math>P = Q</math>{{Efn|As such an equation can be rewritten {{math|1=''P'' – ''Q'' = 0}}, many authors do not consider this case explicitly.}}

where ''P'' and ''Q'' are [[polynomial]]s with coefficients in some [[field (mathematics)|field]] (e.g., [[Rational number|rational numbers]], [[Real number|real numbers]], [[Complex number|complex numbers]]). An algebraic equation is ''univariate'' if it involves only one [[variable (mathematics)|variable]]. On the other hand, a polynomial equation may involve several variables, in which case it is called ''multivariate'' (multiple variables, x, y, z, etc.).

For example,
:<math>x^5-3x+1=0</math>
is a univariate algebraic (polynomial) equation with integer coefficients and
:<math>y^4+\frac{xy}{2}=\frac{x^3}{3}-xy^2+y^2-\frac{1}{7}</math>
is a multivariate polynomial equation over the rational numbers.

Some polynomial equations with [[Rational number|rational coefficients]] have a solution that is an [[algebraic expression]], with a finite number of operations involving just those coefficients (i.e., can be [[Algebraic solution|solved algebraically]]). This can be done for all such equations of [[Degree of a polynomial|degree]] one, two, three, or four; but equations of degree five or more cannot always be solved in this way, as the [[Abel–Ruffini theorem]] demonstrates. 

A large amount of research has been devoted to compute efficiently accurate approximations of the [[real number|real]] or [[complex number|complex]] solutions of a univariate algebraic equation (see [[Root finding of polynomials]]) and of the common solutions of several multivariate polynomial equations (see [[System of polynomial equations]]).

===Systems of linear equations===
[[File:九章算術.gif|thumb|[[The Nine Chapters on the Mathematical Art]] is an anonymous 2nd-century Chinese book proposing a method of resolution for linear equations.]]

A [[system of linear equations]] (or ''linear system'') is a collection of [[linear equation]]s involving one or more [[variable (math)|variables]].{{efn|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,
:<math>\begin{alignat}{7}
3x &&\; + \;&& 2y             &&\; - \;&& z  &&\; = \;&& 1 & \\
2x &&\; - \;&& 2y             &&\; + \;&& 4z &&\; = \;&& -2 & \\
-x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z  &&\; = \;&& 0 &
\end{alignat}</math>
is a system of three equations in the three variables {{math|''x'', ''y'', ''z''}}. A '''solution''' to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A [[Equation solving|solution]] to the system above is given by
:<math>\begin{alignat}{2}
x &\,=\,& 1 \\
y &\,=\,& -2 \\
z &\,=\,& -2
\end{alignat}</math>
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 a fundamental part of [[linear algebra]], a subject which is used in many parts of modern mathematics. Computational [[algorithm]]s for finding the solutions are an important part of [[numerical linear algebra]], and play a prominent role in [[physics]], [[engineering]], [[chemistry]], [[computer science]], and [[economics]]. A [[Nonlinear system|system of non-linear equations]] can often be [[approximation|approximated]] by a linear system (see [[linearization]]), a helpful technique when making a [[mathematical model]] or [[computer simulation]] of a relatively complex system.