# User Contributed Dictionary

### Noun

equations- Plural of equation

# Extensive Definition

An equation is a mathematical statement, in
symbols, that two things are exactly the same (or equivalent).
Equations are written with an equal sign, as
in

- 2 + 3 = 5.

The equation above is an example of an equality:
a proposition which
states that two constants are equal. Equalities
may be true or false.

Equations are often used to state the equality of
two expressions
containing one or more variables. In the reals we can
say, for example, that for any given value of x it is true that

- x (x-1) = x^2-x.

The equation above is an example of an identity;
an equation that is true
regardless of the values of any variables that appear in it. The
following equation is not an identity:

- x^2-x = 0.

It is false for an infinite number of values of
x, and true for only two, the roots
or solutions of the equation, x=0 and x=1. Therefore, if the
equation is known to be true, it carries information about the
value of x. To solve an
equation means to find its solutions.

Many authors reserve the term equation for an
equality which is not an identity. The distinction between the two
concepts can be subtle; for example,

- (x + 1)^2 = x^2 + 2x + 1

- (x + 1)^2 = 2x^2 + x + 1

Letters from the beginning of the alphabet like
a, b, c... often denote constants in the context of the
discussion at hand, while letters from end of the alphabet, like x,
y, z..., are usually reserved for the variables, a convention
initiated by Descartes.

## Properties

If an equation in algebra is known to be true, the following operations may be used to produce another true equation:- Any quantity can be added to both sides.
- Any quantity can be subtracted from both sides.
- Any quantity can be multiplied to both sides.
- Any nonzero quantity can divide both sides.
- Generally, any function can be applied to both sides. (However, caution must be exercised to ensure that one does not encounter extraneous solutions.)

The algebraic properties (1-4) imply that
equality is a congruence
relation for a field;
in fact, it is essentially the only one.

The most well known system of numbers which
allows all of these operations is the real
numbers, which is an example of a field.
However, if the equation were based on the natural
numbers for example, some of these operations (like division
and subtraction) may not be valid as negative numbers and
non-whole
numbers are not allowed. The integers are an example of an
integral
domain which does not allow all divisions as, again, whole
numbers are needed. However, subtraction is allowed, and is the
inverse
operator in that system.

If a function that is not injective is applied to both
sides of a true equation, then the resulting equation will still be
true, but it may be less useful. Formally, one has an implication,
not an equivalence,
so the solution set may get larger. The functions implied in
properties (1), (2), and (4) are always injective, as is (3) if we
do not multiply by zero. Some
generalized products,
such as a dot product,
are never injective.

## See also

- Inequation
- Inequality
- Linear equation
- Quadratic equation
- Cubic equation
- Quartic equation
- Quintic equation
- Indeterminate equation
- Differential equation
- Integral equation
- Functional equation
- Diophantine equation
- List of equations
- Theory of equations
- Parametric equation
- Polynomial equation
- Scientific equations named after people

## External links

- Mathematical equation plotter: Plots 2D mathematical equations, computes integrals, and finds solutions online.
- Equation plotter: A web page that can plot general equations, not just functions.
- WZGrapher: A Windows freeware program that plots Cartesian and polar equations, with both integration and differentiation solvers and graphing capabilities.
- Equation Wizard: Automatic algebraic equation solver
- EqWorld — contains information on solutions to many different classes of mathematical equations.
- EquationSolver: A webpage that can solve single equations and linear equation systems.
- WebGraphing.com: Online Equation Plotter with Automatic Table of Coordinates
- Equation Solver: Automatic algebraic equation solver

equations in Arabic: معادلة

equations in Bengali: সমীকরণ

equations in Belarusian (Tarashkevitsa):
Раўнаньне

equations in Catalan: Equació

equations in Czech: Rovnice

equations in Welsh: Hafaliad

equations in Danish: Ligning

equations in German: Gleichung

equations in Estonian: Võrrand

equations in Modern Greek (1453-): Εξίσωση

equations in Emiliano-Romagnolo: Equaziån

equations in Spanish: Ecuación

equations in Esperanto: Ekvacio

equations in Persian: معادله

equations in French: Équation

equations in Galician: Ecuación

equations in Korean: 방정식

equations in Hindi: समीकरण

equations in Croatian: Jednadžba

equations in Ido: Equaciono

equations in Indonesian: Persamaan

equations in Interlingua (International
Auxiliary Language Association): Equation

equations in Icelandic: Jafna

equations in Italian: Equazione

equations in Hebrew: משוואה

equations in Georgian: განტოლება

equations in Lao: ສົມຜົນ

equations in Latin: Aequatio

equations in Lithuanian: Lygtis

equations in Lombard: Equazziun

equations in Hungarian: Egyenlet

equations in Marathi: समीकरण

equations in Dutch: Vergelijking
(wiskunde)

equations in Japanese: 方程式

equations in Norwegian: Ligning
(matematikk)

equations in Norwegian Nynorsk: Likning

equations in Polish: Równanie (matematyka)

equations in Portuguese: Equação

equations in Romanian: Ecuaţie

equations in Quechua: Paqtachani

equations in Russian: Уравнение

equations in Simple English: Equation

equations in Slovak: Rovnica (matematika)

equations in Slovenian: Enačba

equations in Serbian: Једначина

equations in Serbo-Croatian: Jednačina

equations in Finnish: Yhtälö

equations in Swedish: Ekvation

equations in Tamil: சமன்பாடு

equations in Thai: สมการ

equations in Vietnamese: Phương trình

equations in Turkish: Denklem

equations in Ukrainian: Рівняння

equations in Võro: Võrrand

equations in Yiddish: גלייכונג

equations in Chinese: 方程