# 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

## English

### Pronunciation

• /ɪˈkweɪʒən/|/ɪˈkweɪʃən/,
• /I"keIZ@n/|/I"keIS@n/
• Rhymes with: -eɪʒən

### Noun

1. 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.

#### Translations

assertion
• 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
is an identity, while
(x + 1)^2 = 2x^2 + x + 1
is an equation, whose roots are x=0 and x=1. Whether a statement is meant to be an identity or an equation, carrying information about its variables can usually be determined from its context.
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:
1. Any quantity can be added to both sides.
2. Any quantity can be subtracted from both sides.
3. Any quantity can be multiplied to both sides.
4. Any nonzero quantity can divide both sides.
5. 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.

equation in Arabic: معادلة
equation in Bengali: সমীকরণ
equation in Belarusian (Tarashkevitsa): Раўнаньне
equation in Catalan: Equació
equation in Czech: Rovnice
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 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