# Dictionary Definition

equation

### Noun

1 a mathematical statement that two expressions
are equal

2 a state of being essentially equal or
equivalent; equally balanced; "on a par with the best" [syn:
equality, equivalence, par]

3 the act of regarding as equal [syn: equating]

# User Contributed Dictionary

### Noun

- An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.

#### Derived terms

rel-top Derived terms- algebraic equation
- Arrhenius equation
- chemical equation
- differential equation
- Diophantine equation
- Drake equation
- integral equation
- Kepler's equation
- Lagrange's equations
- linear equation
- Navier-Stokes equation
- parametric equation
- partial differential equation
- Pell's equation
- personal equation
- polynomial equation
- quadratic equation
- Schrödinger's wave equation
- Van der Waals equation
- wave equation

#### Related terms

#### Translations

assertion

- Croatian: jednadžba
- Czech: rovnice
- Danish: ligning
- Dutch: vergelijking
- Esperanto: ekvacio
- Finnish: yhtälö
- French: équation
- German: Gleichung
- Greek: εξίσωση
- Hungarian: egyenlet
- Icelandic: jafna
- Irish: cothromóid
- Italian: equazione
- Japanese: (hōteishiki)
- Malayalam: സമവാക്യം (samavaakyam)
- Norwegian: likning
- Polish: równanie
- Portuguese: equação
- Russian: уравнение
- Slovene: enačba
- Spanish: ecuación
- Swedish: ekvation
- Urdu: مساوات (musawaat)

# 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

equation in Arabic: معادلة

equation in Bengali: সমীকরণ

equation in Belarusian (Tarashkevitsa):
Раўнаньне

equation in Catalan: Equació

equation in Czech: Rovnice

equation in Welsh: Hafaliad

equation in Danish: Ligning

equation in German: Gleichung

equation in Estonian: Võrrand

equation in Modern Greek (1453-): Εξίσωση

equation in Emiliano-Romagnolo: Equaziån

equation in Spanish: Ecuación

equation in Esperanto: Ekvacio

equation in Persian: معادله

equation in French: Équation

equation in Galician: Ecuación

equation in Korean: 방정식

equation in Hindi: समीकरण

equation in Croatian: Jednadžba

equation in Ido: Equaciono

equation in Indonesian: Persamaan

equation in Interlingua (International Auxiliary
Language Association): Equation

equation in Icelandic: Jafna

equation in Italian: Equazione

equation in Hebrew: משוואה

equation in Georgian: განტოლება

equation in Lao: ສົມຜົນ

equation in Latin: Aequatio

equation in Lithuanian: Lygtis

equation in Lombard: Equazziun

equation in Hungarian: Egyenlet

equation in Marathi: समीकरण

equation in Dutch: Vergelijking (wiskunde)

equation in Japanese: 方程式

equation in Norwegian: Ligning
(matematikk)

equation in Norwegian Nynorsk: Likning

equation in Polish: Równanie (matematyka)

equation in Portuguese: Equação

equation in Romanian: Ecuaţie

equation in Quechua: Paqtachani

equation in Russian: Уравнение

equation in Simple English: Equation

equation in Slovak: Rovnica (matematika)

equation in Slovenian: Enačba

equation in Serbian: Једначина

equation in Serbo-Croatian: Jednačina

equation in Finnish: Yhtälö

equation in Swedish: Ekvation

equation in Tamil: சமன்பாடு

equation in Thai: สมการ

equation in Vietnamese: Phương trình

equation in Turkish: Denklem

equation in Ukrainian: Рівняння

equation in Võro: Võrrand

equation in Yiddish: גלייכונג

equation in Chinese: 方程

# Synonyms, Antonyms and Related Words

accommodation, addend, adjustment, antilogarithm, argument, balance, base, binomial, characteristic, coefficient, coequality, coextension, combination, complement, congruence, constant, coordination, correspondence, cosine, cotangent, cube, decimal, denominator, derivative, determinant, difference, differential, discriminate, dividend, divisor, e, equality, equalization, equalizing, equating, equilibration, equilibrium, equipoise, equipollence, equiponderance, equity, equivalence, equivalency, evening, evening up, evenness, exponent, exponential, factor, formula, function, i, identity, increment, index, integral, integration, justice, levelness, likeness, matrix, minuend, multiple, multiplier, norm, numerator, par, parallelism, parameter, parity, permutation, pi, poise, polynomial, power, proportion, quaternion, quotient, radical, radix, reciprocal, remainder, root, secant, sine, submultiple, subtrahend, symmetry, tangent, tensor, variable, vector, versine