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'''Equation''' is kind of mathematical problem that is formulated in the following way: some [[equality]] is given, and at least one side of this equality is dependent on some [[variable]] (or variables); it is suggested to find values of variables, at which value of equality is '''[[true(mathematics)|true]]'''.
In [[mathematics]], an '''equation''' is a relationship between quantities which are stated to be equal.  It is usually regarded as a kind of mathematical problem in which at least one side of this equality is dependent on some [[variable]] (or variables); it is required to find the ''solution'', namely the value or set of values of these variables for which the equality holds '''[[true(mathematics)|true]]'''. For example, <math>x+2 = 3</math> is an equation in which the ''solution'' is the value <math>x=1</math>.
Equation may have form  
 
(1) <math>
An '''identity''' is an equality which is stated to be universally true for all permissible values of the variables, rather than representing a condition on those values.  For example, <math>x+y = y+x</math> is an identity for real numbers, since it is true for all real values of ''x'' and ''y''.
F(x)=0
 
</math>
The equation may have the form  
:(1) <math>F(x)=0</math>
or  
or  
(2) <math>F(x)=G(x)</math>
:(2) <math>F(x)=G(x)</math>
where <math> F </math> and <math>G</math> are [[known function]]s and <math>x</math> is [[unknown variable]].
where <math> F </math> and <math>G</math> are [[known function]]s and <math>x</math> is the [[unknown variable]].
 
An equation may have solution, id est, value of variable (<math>x </math> in the example above, although variable can be denoted by any letter, including letters from exotic alphabets), at which the equality has value '''true'''.  


Usulally, it is supposed, that the variable belongs to some set <math>\mathrm A</math>; for example, it may be specified that <math>x</math> is [[real(mathematics)|real]].
An equation may have as solution somevalue of variable (<math>x </math> in the example above, although variable can be denoted by any letter, including letters from exotic alphabets), at which the equality has value '''true'''.
It is usually supposed that the variable belongs to some set <math>\mathrm A</math>; for example, it may be specified that <math>x</math> is [[real(mathematics)|real]].


The equation may have no solution, may have one solution and may have many solutions, dependently on the set <math>\mathrm A</math>.
The equation may have no solution, may have one solution and may have many solutions, depending on the form of the equations and the set <math>\mathrm A</math> of possible values.


Equations are common in language used in [[science]]; especially in [[mathematics]] and [[physics]]:
Equations are common in the language used in [[science]]; especially in [[mathematics]] and [[physics]]:
[[Newton equation]], equation of [[oscillator]], [[Schrödinger equation ]].  
[[Newton equation]], equation of [[oscillator]], [[Schrödinger equation ]].  
The termin '''equation''' is used also in [[chemistry]], indicating conservation of atomic or [[isotopic content|isotopic]] content at the [[chemical reaction]]s.
The term '''equation''' is used also in [[chemistry]], indicating conservation of atomic or [[isotopic content|isotopic]] content at the [[chemical reaction]]s.


==Examples==
==Examples==
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<math>F_b(x)</math>, <math>G_b(x)</math>.
<math>F_b(x)</math>, <math>G_b(x)</math>.


In relatively simple case, function <math> F </math> depends only on the unknown variable, and <math>G</math>
In relatively simple cases, function <math> F </math> depends only on the unknown variable, and <math>G</math>
depends on the parameter; for example,
depends on the parameter; for example,


(3) <math>F(x)=b</math> .
(3) <math>F(x)=b</math> .


In this case, the solution <math>x</math> is considered as [[inverse function]] of <math>b</math>, which can be written as
In this case, the solution <math>x</math> is considered as an [[inverse function]] of <math>b</math>, which can be written as


(4) <math> x=F^{-1}(b)</math>.
(4) <math> x=F^{-1}(b)</math>.


Dependently on function <math>F</math>, range of values of <math>b</math> and set <math>\mathbf A</math>, there may exist no inverse function,
Dependending on the function <math>F</math>, range of values of <math>b</math> and set <math>\mathbf A</math>, there may exist no inverse function,
one inverse function or several inverse functions.
one inverse function or several inverse functions.


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==[[Differential equation]]s and [[integral equation]]s==
==[[Differential equation]]s and [[integral equation]]s==
In particular, <math>x</math> may denote an element of a [[Hilbert space]], for example, set of functions of one or several variables; and
In particular, the variable may denote a function of one or several variables, so that the set of p[ossible values is some [[Hilbert space]]; the function <math> F </math> is then an [[operator]] on this space which may be expressed in terms of derivatives or integrals of the function elements.
function <math> F </math> may be expressed in terms of derivatives or integrals of <math>x</math> with respect to these variables.
In these cases, the equation is called a [[differential equation]] or an [[integral equation]].
In these cases, the equation is called [[differential equation]] of [[integral equation]].
 
==[[Operator equation]]s==
==[[Operator equation]]s==
Equations can be used for objects of any origen, as soon, as the operation of [[equality]] is defined. In particular, in [[Quantum mechanics]], the
Equations can be used for objects of any origin, as soon, as the operation of [[equality]] is defined. In particular, in [[Quantum mechanics]], the [[Heisenberg equation]] deals with non-[[commutativity|commuting]] objects ([[operator(quantum mechanics)|operator]]s).
[[Heisenberg equation]] deals with non-commuting objects ([[operator(quantum mechanics)|operator]]s).

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In mathematics, an equation is a relationship between quantities which are stated to be equal. It is usually regarded as a kind of mathematical problem in which at least one side of this equality is dependent on some variable (or variables); it is required to find the solution, namely the value or set of values of these variables for which the equality holds true. For example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+2 = 3} is an equation in which the solution is the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=1} .

An identity is an equality which is stated to be universally true for all permissible values of the variables, rather than representing a condition on those values. For example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+y = y+x} is an identity for real numbers, since it is true for all real values of x and y.

The equation may have the form

(1) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)=0}

or

(2) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)=G(x)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} are known functions and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is the unknown variable.

An equation may have as solution somevalue of variable (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } in the example above, although variable can be denoted by any letter, including letters from exotic alphabets), at which the equality has value true. It is usually supposed that the variable belongs to some set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm A} ; for example, it may be specified that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is real.

The equation may have no solution, may have one solution and may have many solutions, depending on the form of the equations and the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm A} of possible values.

Equations are common in the language used in science; especially in mathematics and physics: Newton equation, equation of oscillator, Schrödinger equation . The term equation is used also in chemistry, indicating conservation of atomic or isotopic content at the chemical reactions.

Examples

For function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)=1+x} , the equation (1) has no solutions among natural numbers, but has solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=-1} in the set of integer numbers.

Equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2=2} has no solutions among rational numbers, has one solution (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\sqrt{2}} ) among positive real numbers, and has two soluitons (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\sqrt{2}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=-\sqrt{2}} ) among real numbers.

Inverse function

Function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F } in equation (1) ; or functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G } in equation (2) may also depend on some parameter(s). In this case, the solution(s) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } also may depend on parameter(s). Indicating the function, the parameter, say, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} , can be specified as a second argument, writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x,b)} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(x,b)} or as subscript, writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_b(x)} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_b(x)} .

In relatively simple cases, function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F } depends only on the unknown variable, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} depends on the parameter; for example,

(3) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)=b} .

In this case, the solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is considered as an inverse function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} , which can be written as

(4) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=F^{-1}(b)} .

Dependending on the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} , range of values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} and set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A} , there may exist no inverse function, one inverse function or several inverse functions.

Graphical solution of equations

FIg.1. Example of graphic solution of equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\log_b(x)} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}} (two solutions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=4} ), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/\rm e)} (one solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\rm e} ), and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=2} (no real solutions).

Solving equations, it may worth to begin with graphic solution of the equation, which allows the quick and dirty estimates. One plots both functions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} at the same graphic, and watch the point (s) of the intersection of the curves. In figire 1, functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=F(x)=x} is plotted with black line, and function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=G(x)=\log_{\sqrt{2}}(x)} is plotted with red curve. The intersections with black curve indicate values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} which are solutions.

At the same figure, the cases Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(x)=\log_{\exp(1/\rm e)}(x)} (only one solution, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\rm e} ) and (no solutions among real numbers) are shown with green and blue curves.

System of equations

In the equations (1) or (2), may denote several numbers at once, ; and functions and may return values from multidimentional space . In this case, one says that there is system of equations. For example, there is well developed theory of systems of linear equations, while unknown variables are real or complex numbers.

Differential equations and integral equations

In particular, the variable may denote a function of one or several variables, so that the set of p[ossible values is some Hilbert space; the function is then an operator on this space which may be expressed in terms of derivatives or integrals of the function elements. In these cases, the equation is called a differential equation or an integral equation.

Operator equations

Equations can be used for objects of any origin, as soon, as the operation of equality is defined. In particular, in Quantum mechanics, the Heisenberg equation deals with non-commuting objects (operators).

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