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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]]'''.
{{subpages}}
Equation may have form
In [[mathematics]], an '''equation''' is a statement that two quantities are equal.  It is usually regarded as a kind of mathematical problem in which you have to find a value which makes the equation true. A simple example is the question: What do you have to fill in on the dots in … + 2 = 3? The answer is 1, because 1 + 2 = 3. The unknown quantity in such problems is frequently denoted by a letter, often ''x'', so that the equation becomes ''x'' + 2 = 3. The solution of this equation is ''x'' = 1. In general, an equation may have no solution, one solution or many solutions.
(1) <math>
F(x)=0
</math>
or  
(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]].


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'''.  
By contrast, 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 number]]s, since it is true for all real values of ''x'' and ''y''.


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]].
[[Science|Scientific]] laws are often formulated as equations, especially in [[physics]] and other [[natural sciences]]. Examples are [[Newton's laws]], the equation of a [[harmonic oscillator]] and the [[Schrödinger equation]].
The term ''equation'' is used also in [[chemistry]], indicating conservation of atomic or [[isotopic content|isotopic]] content (and rarely other forms of energy, e.g. light) during [[chemical reaction]]s.


The equation may have no solution, may have one solution and may have many solutions, dependently on the set  <math>\mathrm A</math>.
==Domain of the unknown==


Equations are common in language used in [[science]]; especially in [[mathematics]] and [[physics]]:
Technically, an equation has to indicate what values the unknown variable can take. This is called the ''domain'' of the variable. For instance, the equation ''x'' + 1 = 0 has no solutions if ''x'' is supposed to be a [[natural number]], but it does have a solution
[[Newton equation]], equation of [[oscillator]], [[Schrödinger equation ]].
(namely, ''x'' = −1) if ''x'' is supposed to be an [[integer|integer number]].
The termin '''equation''' is used also in [[chemistry]], indicating conservation of atomic or [[isotopic content|isotopic]] content at the [[chemical reaction]]s.


==Examples==
Similarly, the equation <math>x^2=2</math> has no solutions if the domain is formed by the [[rational number]]s, it has two solutions (namely, <math>x=\sqrt{2}</math> and <math>x=-\sqrt{2}</math>) among the [[real number]]s, and it has only one solution
(namely, <math>x=\sqrt{2}</math>) among the [[positive]] real numbers.


For function <math>F(x)=1+x</math>, the equation (1) has no solutions among [[natural number]]s, but has solution
Often, the domain is not specified explicitly, but it is assumed that the reader knows what it is supposed to be.
<math>x=-1</math> in the set of [[integer number]]s.


Equation <math>x^2=2</math> has no solutions among [[rational number]]s, has one solution
==Inverse function==
(<math>x=\sqrt{2}</math>) among [[positive real number]]s, and has two soluitons
The equation may have the form
(<math>x=\sqrt{2}</math> and <math>x=-\sqrt{2}</math>) among  [[real number]]s.
:(1) <math>F(x)=0</math>
or
:(2) <math>F(x)=G(x)</math>
where <math> F </math> and <math>G</math> are [[known function]]s and <math>x</math> is the [[unknown variable]].


==Inverse function==
Function <math> F </math> in equation (1) ; or functions <math> F </math> and <math> G </math> in equation (2)
Function <math> F </math> in equation (1) ; or functions <math> F </math> and <math> G </math> in equation (2)
may also depend on some [[parameter]](s). In this case, the solution(s) <math> x </math> also may depend on parameter(s).
may also depend on some [[parameter]](s). In this case, the solution(s) <math> x </math> also may depend on parameter(s).
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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.


==Graphical solution of equations==
==Graphical solution of equations==
[[Image:ExampleEquationLog01.png|300px|right|thumb|FIg.1. Example of graphic solution of equation <math>x=\log_b(x)</math> for
{{Image|ExampleEquationLog01.png|right|300px|Fig.1. Example of graphic solution of equation <math>x=\log_b(x)</math> for
<math>b=\sqrt{2}</math> (two solutions, <math>x=2</math> and <math>x=4</math>),
<math>b=\sqrt{2}</math> (two solutions, <math>x=2</math> and <math>x=4</math>),
<math>b=\exp(1/\rm e)</math> (one solution <math>x=\rm e</math>), and  
<math>b=\exp(1/\rm e)</math> (one solution <math>x=\rm e</math>), and  
<math>b=2</math> (no real solutions).]]
<math>b=2</math> (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,
Solving equations, it may worth to begin with [[graphic solution]] of the equation, which allows the [[quick and dirty]] estimates. One plots both functions,
<math>F</math> and <math>G</math> at the same graphic, and watch the point (s) of the intersection of the curves. In figire 1, functions
<math>F</math> and <math>G</math> at the same graphic, and watch the point (s) of the intersection of the curves. In figure 1, functions
<math>y=F(x)=x</math> is plotted with black line,
<math>y=F(x)=x</math> is plotted with black line,
and function <math>y=G(x)=\log_{\sqrt{2}}(x)</math> is plotted with red curve. The intersections with black curve indicate values of <math>x</math> which are solutions.
and function <math>y=G(x)=\log_{\sqrt{2}}(x)</math> is plotted with red curve. The intersections with black curve indicate values of <math>x</math> which are solutions.
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==System of equations==
==System of equations==
In the equations (1) or (2), <math> x </math> may denote several numbers at once,  <math>x=\{x_0,x_2,.. x_{n-1}\}</math>; and
In the equations (1) or (2), <math> x </math> may denote several numbers at once,  <math>x=\{x_0,x_2,.. x_{n-1}\}</math>; and
functions <math>F</math> and <math>G</math> may return values from multidimentional space <math>\{F_0,F_1,F_2,..F_{m-1} \}</math>.
functions <math>F</math> and <math>G</math> may return values from multidimensional space <math>\{F_0,F_1,F_2,..F_{m-1} \}</math>.
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 <math>x</math> are [[real number|real]] or [[complex number|complex]] numbers.
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 <math>x</math> are [[real number|real]] or [[complex number|complex]] numbers.


==[[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 possible values is some functional space, usually a [[Banach space|Banach]] or even [[Hilbert space|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).[[Category:Suggestion Bot Tag]]
[[Heisenberg equation]] deals with non-commuting objects ([[operator(quantum mechanics)|operator]]s).
==See also==
*[[linear equation]]
*[[quadratic equation]]
*[[Solving cubic equations]]
*[[algebraic equation]]
*[[transcendental equation]]
*[[differential equation]]
*[[integral equation]]
*[[integro-differential equation]]
*[[analytic solution]]
*[[numeric solution]]
*[[Newton equation]]

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In mathematics, an equation is a statement that two quantities are equal. It is usually regarded as a kind of mathematical problem in which you have to find a value which makes the equation true. A simple example is the question: What do you have to fill in on the dots in … + 2 = 3? The answer is 1, because 1 + 2 = 3. The unknown quantity in such problems is frequently denoted by a letter, often x, so that the equation becomes x + 2 = 3. The solution of this equation is x = 1. In general, an equation may have no solution, one solution or many solutions.

By contrast, 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.

Scientific laws are often formulated as equations, especially in physics and other natural sciences. Examples are Newton's laws, the equation of a harmonic oscillator and the Schrödinger equation. The term equation is used also in chemistry, indicating conservation of atomic or isotopic content (and rarely other forms of energy, e.g. light) during chemical reactions.

Domain of the unknown

Technically, an equation has to indicate what values the unknown variable can take. This is called the domain of the variable. For instance, the equation x + 1 = 0 has no solutions if x is supposed to be a natural number, but it does have a solution (namely, x = −1) if x is supposed to be an integer number.

Similarly, the 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 if the domain is formed by the rational numbers, it has two solutions (namely, 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 the real numbers, and it has only one solution (namely, 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 the positive real numbers.

Often, the domain is not specified explicitly, but it is assumed that the reader knows what it is supposed to be.

Inverse function

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.

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

© Image: Dmitrii Kouznetsov
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 figure 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 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_2(x)} (no solutions among real numbers) are shown with green and blue curves.

System of equations

In the equations (1) 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 x } may denote several numbers at once, 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=\{x_0,x_2,.. x_{n-1}\}} ; and 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} may return values from multidimensional space 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_0,F_1,F_2,..F_{m-1} \}} . 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 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} 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 possible values is some functional space, usually a Banach or even Hilbert space; 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 } 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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