In

the third page of the chapter "The rule of equation, commonly called Algebers Rule."

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^{2} + ''bx'' + ''c'' = 0.
The process of finding the solutions, or, in case of parameters, expressing the unknowns in terms of the parameters, is called solving the equation. Such expressions of the solutions in terms of the parameters are also called ''solutions''.
A

polynomial equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

in one variable with

polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

equation is an equation in which both sides are polynomials (see also derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

s of the unknown functions, evaluated at multiple points, such as $f\text{'}(x)\; =\; f(x-2)$
** A difference equation is an equation where the unknown is a function ''f'' that occurs in the equation through ''f''(''x''), ''f''(''x''−1), ..., ''f''(''x''−''k''), for some whole integer ''k'' called the ''order'' of the equation. If ''x'' is restricted to be an integer, a difference equation is the same as a recurrence relation
** A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process

Winplot

General Purpose plotter that can draw and animate 2D and 3D mathematical equations.

Equation plotter

A web page for producing and downloading pdf or postscript plots of the solution sets to equations and inequations in two variables (''x'' and ''y''). {{Authority control Elementary algebra Equations,

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, an equation is a statement that asserts the equality of two expressions
Expression may refer to:
Linguistics
* Expression (linguistics), a word, phrase, or sentence
* Fixed expression, a form of words with a specific meaning
* Idiom, a type of fixed expression
* Metaphor#Common types, Metaphorical expression, a parti ...

, which are connected by the equals sign
The equals sign (, ) or equal sign (), formerly known as the equality sign, is the , which is used to indicate in some sense. In an , it is placed between two that have the same value, or for which one studies the conditions under which the ...

"=". The word ''equation'' and its cognate
In linguistics
Linguistics is the scientific study of language
A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most langu ...

s in other languages may have subtly different meanings; for example, in French
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
France (), officially the French Republic (french: link=no, République française), is a country primarily located in Western Europe, consistin ...

an ''équation'' is defined as containing one or more variables, while in English
English usually refers to:
* English language
English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, which has eventually become the World language, leading lan ...

, any equality is an equation.
''Solving'' an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions
Image:SaltInWaterSolutionLiquid.jpg, Making a saline water solution by dissolving Salt, table salt (sodium chloride, NaCl) in water. The salt is the solute and the water the solvent.
In chemistry, a solution is a special type of Homogeneous and ...

of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables.
An equation is written as two expressions
Expression may refer to:
Linguistics
* Expression (linguistics), a word, phrase, or sentence
* Fixed expression, a form of words with a specific meaning
* Idiom, a type of fixed expression
* Metaphor#Common types, Metaphorical expression, a parti ...

, connected by an equals sign
The equals sign (, ) or equal sign (), formerly known as the equality sign, is the , which is used to indicate in some sense. In an , it is placed between two that have the same value, or for which one studies the conditions under which the ...

("="). The expressions on the two sides of the equals sign are called the "left-hand side" and "right-hand side" of the equation. Very often the right-hand side of an equation is assumed to be zero. Assuming this does not reduce the generality, as this can be realized by subtracting the right-hand side from both sides.
The most common type of equation is a polynomial equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

(commonly called also an ''algebraic equation'') in which the two sides are polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s.
The sides of a polynomial equation contain one or more terms. For example, the equation
:$Ax^2\; +Bx\; +\; C\; -\; y\; =\; 0$
has left-hand side $Ax^2\; +Bx\; +\; C\; -\; y$, which has four terms, and right-hand side $0$, consisting of just one term. The names of the variables suggest that and are unknowns, and that , , and are parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whol ...

s, but this is normally fixed by the context (in some contexts, may be a parameter, or , , and may be ordinary variables).
An equation is analogous to a scale into which weights are placed. When equal weights of something (e.g., grain) are placed into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, an equal amount of grain must be removed from the other pan to keep the scale in balance. More generally, an equation remains in balance if the same operation is performed on its both sides.
In Cartesian geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...

, equations are used to describe geometric figures
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related ...

. As the equations that are considered, such as implicit equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s or parametric equation
In mathematics, a parametric equation defines a group of quantities as Function (mathematics), functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that m ...

s, have infinitely many solutions, the objective is now different: instead of giving the solutions explicitly or counting them, which is impossible, one uses equations for studying properties of figures. This is the starting idea of algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

, an important area of mathematics.
Algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

studies two main families of equations: polynomial equations
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term ''al ...

and, among them, the special case of linear equations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. When there is only one variable, polynomial equations have the form ''P''(''x'') = 0, where ''P'' is a polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, and linear equations have the form ''ax'' + ''b'' = 0, where ''a'' and ''b'' are parameters
A parameter (from the Ancient Greek language, Ancient Greek wikt:παρά#Ancient Greek, παρά, ''para'': "beside", "subsidiary"; and wikt:μέτρον#Ancient Greek, μέτρον, ''metron'': "measure"), generally, is any characteristic that ...

. To solve equations from either family, one uses algorithmic or geometric techniques that originate from linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (math ...

or mathematical analysis
Analysis is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...

. Algebra also studies Diophantine equations
In mathematics, a Diophantine equation is a polynomial equation, usually involving two or more unknown (mathematics), unknowns, such that the only equation solving, solutions of interest are the integer ones (an integer solution is such that all ...

where the coefficients and solutions are integers
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...

. The techniques used are different and come from number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

. These equations are difficult in general; one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions.
Differential equations
In mathematics, a differential equation is an equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...

are equations that involve one or more functions and their derivatives. They are ''solved'' by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in areas such as physics, chemistry, biology, and economics.
The " =" symbol, which appears in every equation, was invented in 1557 by Robert Recorde
Robert Recorde (c. 1512 – 1558) was a Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus sign
The plus and minus signs, and , are mathematical symbols used to represent the notions ...

, who considered that nothing could be more equal than parallel straight lines with the same length.Recorde, Robert, ''The Whetstone of Witte'' … (London, England: Kyngstone, 1557)the third page of the chapter "The rule of equation, commonly called Algebers Rule."

/ref>

Introduction

Analogous illustration

An equation is analogous to aweighing scale
A scale or balance is a device to measure weight or mass. These are also known as mass scales, weight scales, mass balances, and weight balances.
The traditional scale consists of two plates or bowls suspended at equal distances from a Lever ...

, balance, or seesaw
A seesaw (also known as a teeter-totter or teeterboard) is a long, narrow board supported by a single pivot point, most commonly located at the midpoint between both ends; as one end goes up, the other goes down. These are most commonly found at ...

.
Each side of the equation corresponds to one side of the balance. Different quantities can be placed on each side: if the weights on the two sides are equal, the scale balances, and in analogy, the equality that represents the balance is also balanced (if not, then the lack of balance corresponds to an inequality
Inequality may refer to:
Economics
* Attention inequality
Attention inequality is a term used to target the inequality of distribution of attention across users on social networks, people in general, and for scientific papers. Yun Family Foundat ...

represented by an inequation
In mathematics, an inequation is a statement that an inequality (mathematics), inequality or a non-equality holds between two values. It is usually written in the form of a pair of expression (mathematics), expressions denoting the values in ques ...

).
In the illustration, ''x'', ''y'' and ''z'' are all different quantities (in this case real numbers
Real may refer to:
* Reality, the state of things as they exist, rather than as they may appear or may be thought to be
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish col ...

) represented as circular weights, and each of ''x'', ''y'', and ''z'' has a different weight. Addition corresponds to adding weight, while subtraction corresponds to removing weight from what is already there. When equality holds, the total weight on each side is the same.
Parameters and unknowns

Equations often contain terms other than the unknowns. These other terms, which are assumed to be ''known'', are usually called ''constants'', ''coefficients'' or ''parameters''. An example of an equation involving ''x'' and ''y'' as unknowns and the parameter ''R'' is :$x^2\; +y^2\; =\; R^2\; .$ When ''R ''is chosen to have the value of 2 (''R ''= 2), this equation would be recognized inCartesian coordinates
A Cartesian coordinate system (, ) in a plane
Plane or planes may refer to:
* Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft
Arts, entertainment and media
*Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

as the equation for the circle of radius of 2 around the origin. Hence, the equation with ''R'' unspecified is the general equation for the circle.
Usually, the unknowns are denoted by letters at the end of the alphabet, ''x'', ''y'', ''z'', ''w'', ..., while coefficients (parameters) are denoted by letters at the beginning, ''a'', ''b'', ''c'', ''d'', ... . For example, the general quadratic equation
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...

is usually written ''ax''system of equations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

is a set of ''simultaneous equations'', usually in several unknowns for which the common solutions are sought. Thus, a ''solution to the system'' is a set of values for each of the unknowns, which together form a solution to each equation in the system. For example, the system
:$\backslash begin\; 3x+5y\&=2\backslash \backslash \; 5x+8y\&=3\; \backslash end$
has the unique solution ''x'' = −1, ''y'' = 1.
Identities

An identity is an equation that is true for all possible values of the variable(s) it contains. Many identities are known in algebra and calculus. In the process of solving an equation, an identity is often used to simplify an equation, making it more easily solvable. In algebra, an example of an identity is thedifference of two squares
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

:
:$x^2\; -\; y^2\; =\; (x+y)(x-y)$
which is true for all ''x'' and ''y''.
Trigonometry
Trigonometry (from ', "triangle" and ', "measure") is a branch of that studies relationships between side lengths and s of s. The field emerged in the during the 3rd century BC from applications of to . The Greeks focused on the , while ...

is an area where many identities exist; these are useful in manipulating or solving trigonometric equations. Two of many that involve the sine
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

and cosine
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

functions are:
:$\backslash sin^2(\backslash theta)+\backslash cos^2(\backslash theta)\; =\; 1$
and
:$\backslash sin(2\backslash theta)=2\backslash sin(\backslash theta)\; \backslash cos(\backslash theta)$
which are both true for all values of ''θ''.
For example, to solve for the value of ''θ'' that satisfies the equation:
:$3\backslash sin(\backslash theta)\; \backslash cos(\backslash theta)=\; 1\backslash ,,$
where ''θ'' is limited to between 0 and 45 degrees, one may use the above identity for the product to give:
:$\backslash frac\backslash sin(2\; \backslash theta)\; =\; 1\backslash ,,$
yielding the following solution for ''θ:''
:$\backslash theta\; =\; \backslash frac\; \backslash arcsin\backslash left(\backslash frac\backslash right)\; \backslash approx\; 20.9^\backslash circ.$
Since the sine function is a periodic function
A periodic function is a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logica ...

, there are infinitely many solutions if there are no restrictions on ''θ''. In this example, restricting ''θ'' to be between 0 and 45 degrees would restrict the solution to only one number.
Properties

Two equations or two systems of equations are ''equivalent'', if they have the same set of solutions. The following operations transform an equation or a system of equations into an equivalent one – provided that the operations are meaningful for the expressions they are applied to: * or the same quantity to both sides of an equation. This shows that every equation is equivalent to an equation in which the right-hand side is zero. * or dividing both sides of an equation by a non-zero quantity. * Applying anidentity
Identity may refer to:
Social sciences
* Identity (social science), personhood or group affiliation in psychology and sociology
Group expression and affiliation
* Cultural identity, a person's self-affiliation (or categorization by others ...

to transform one side of the equation. For example, a product or factoring a sum.
* For a system: adding to both sides of an equation the corresponding side of another equation, multiplied by the same quantity.
If some function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

is applied to both sides of an equation, the resulting equation has the solutions of the initial equation among its solutions, but may have further solutions called extraneous solutions. For example, the equation $x=1$ has the solution $x=1.$ Raising both sides to the exponent of 2 (which means applying the function $f(s)=s^2$ to both sides of the equation) changes the equation to $x^2=1$, which not only has the previous solution but also introduces the extraneous solution, $x=-1.$ Moreover, if the function is not defined at some values (such as 1/''x'', which is not defined for ''x'' = 0), solutions existing at those values may be lost. Thus, caution must be exercised when applying such a transformation to an equation.
The above transformations are the basis of most elementary methods for equation solving
alt=Illustration of Newton's method, An example of using Newton–Raphson method to solve numerically the equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), ma ...

, as well as some less elementary one, like Gaussian elimination.
Algebra

Polynomial equations

In general, an ''algebraic equation'' orpolynomial equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

is an equation of the form
:$P\; =\; 0$, or
:$P\; =\; Q$
where ''P'' and ''Q'' are polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s with coefficients in some field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

(e.g., rational numbers
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, real numbers
Real may refer to:
* Reality, the state of things as they exist, rather than as they may appear or may be thought to be
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish col ...

, complex numbers
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

). An algebraic equation is ''univariate'' if it involves only one variable. On the other hand, a polynomial equation may involve several variables, in which case it is called ''multivariate'' (multiple variables, x, y, z, etc.).
For example,
:$x^5-3x+1=0$
is a univariate algebraic (polynomial) equation with integer coefficients and
:$y^4+\backslash frac=\backslash frac-xy^2+y^2-\backslash frac$
is a multivariate polynomial equation over the rational numbers.
Some polynomial equations with rational coefficients have a solution that is an algebraic expressionIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

, with a finite number of operations involving just those coefficients (i.e., can be solved algebraically). This can be done for all such equations of degree
Degree may refer to:
As a unit of measurement
* Degree symbol (°), a notation used in science, engineering, and mathematics
* Degree (angle), a unit of angle measurement
* Degree (temperature), any of various units of temperature measurement ...

one, two, three, or four; but equations of degree five or more cannot always be solved in this way, as the Abel–Ruffini theorem
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general algebraic equation, polynomial equations of quintic equation, degree five or higher with arbitrary coef ...

demonstrates.
A large amount of research has been devoted to compute efficiently accurate approximations of the real
Real may refer to:
* Reality
Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...

or complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

solutions of a univariate algebraic equation (see Root finding of polynomials) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for ...

).
Systems of linear equations

Asystem of linear equations
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

(or ''linear system'') is a collection of linear equation
In mathematics, a linear equation is an equation that may be put in the form
:a_1x_1+\cdots +a_nx_n+b=0,
where x_1, \ldots, x_n are the variable (mathematics), variables (or unknown (mathematics), unknowns), and b, a_1, \ldots, a_n are the coeffi ...

s involving one or more variables. For example,
:$\backslash begin\; 3x\; \&\&\backslash ;\; +\; \backslash ;\&\&\; 2y\; \&\&\backslash ;\; -\; \backslash ;\&\&\; z\; \&\&\backslash ;\; =\; \backslash ;\&\&\; 1\; \&\; \backslash \backslash \; 2x\; \&\&\backslash ;\; -\; \backslash ;\&\&\; 2y\; \&\&\backslash ;\; +\; \backslash ;\&\&\; 4z\; \&\&\backslash ;\; =\; \backslash ;\&\&\; -2\; \&\; \backslash \backslash \; -x\; \&\&\backslash ;\; +\; \backslash ;\&\&\; \backslash tfrac\; y\; \&\&\backslash ;\; -\; \backslash ;\&\&\; z\; \&\&\backslash ;\; =\; \backslash ;\&\&\; 0\; \&\; \backslash end$
is a system of three equations in the three variables . A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution
Solution may refer to:
* Solution (chemistry)
Image:SaltInWaterSolutionLiquid.jpg, upMaking a saline water solution by dissolving Salt, table salt (sodium chloride, NaCl) in water. The salt is the solute and the water the solvent.
In chemistry ...

to the system above is given by
:$\backslash begin\; x\; \&\backslash ,=\backslash ,\&\; 1\; \backslash \backslash \; y\; \&\backslash ,=\backslash ,\&\; -2\; \backslash \backslash \; z\; \&\backslash ,=\backslash ,\&\; -2\; \backslash end$
since it makes all three equations valid. The word "''system''" indicates that the equations are to be considered collectively, rather than individually.
In mathematics, the theory of linear systems is a fundamental part of linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (math ...

, a subject which is used in many parts of modern mathematics. Computational algorithm
In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

s for finding the solutions are an important part of numerical linear algebra
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create algorithms, computer algorithms which Algorithmic efficiency, efficiently and accurately provide approximate answers to qu ...

, and play a prominent role in physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

, chemistry
Chemistry is the scientific
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity or awareness, of someone or something, such as facts
A fact is an occurrence in the real world. T ...

, computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of computation, automation, a ...

, and economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and interact ...

. A system of non-linear equations can often be approximated by a linear system (see linearization
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

), a helpful technique when making a mathematical model
A mathematical model is a description of a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A system, surrounded and influenced by its environm ...

or computer simulation
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be determ ...

of a relatively complex system.
Geometry

Analytic geometry

InEuclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandria
Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic
Coptic may refer to:
Afro-Asia
* Copts, an ethnoreligious group mainly in the area of modern ...

, it is possible to associate a set of coordinates to each point in space, for example by an orthogonal grid. This method allows one to characterize geometric figures by equations. A plane in three-dimensional space can be expressed as the solution set of an equation of the form $ax+by+cz+d=0$, where $a,b,c$ and $d$ are real numbers and $x,y,z$ are the unknowns that correspond to the coordinates of a point in the system given by the orthogonal grid. The values $a,b,c$ are the coordinates of a vector perpendicular to the plane defined by the equation. A line is expressed as the intersection of two planes, that is as the solution set of a single linear equation with values in $\backslash mathbb^2$ or as the solution set of two linear equations with values in $\backslash mathbb^3.$
A conic section
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

is the intersection of a cone
A cone is a three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter
A parameter (from the Ancient Greek language, Ancient Greek wikt:πα ...

with equation $x^2+y^2=z^2$ and a plane. In other words, in space, all conics are defined as the solution set of an equation of a plane and of the equation of a cone just given. This formalism allows one to determine the positions and the properties of the focuses of a conic.
The use of equations allows one to call on a large area of mathematics to solve geometric questions. The Cartesian coordinate
A Cartesian coordinate system (, ) in a plane is a coordinate system
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the o ...

system transforms a geometric problem into an analysis problem, once the figures are transformed into equations; thus the name analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches ...

. This point of view, outlined by , enriches and modifies the type of geometry conceived of by the ancient Greek mathematicians.
Currently, analytic geometry designates an active branch of mathematics. Although it still uses equations to characterize figures, it also uses other sophisticated techniques such as functional analysis
200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis.
Functional analysis is a branch of mathemat ...

and linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (math ...

.
Cartesian equations

ACartesian coordinate system
A Cartesian coordinate system (, ) in a plane
Plane or planes may refer to:
* Airplane
An airplane or aeroplane (informally plane) is a fixed-wing aircraft
A fixed-wing aircraft is a heavier-than-air flying machine
Early fly ...

is a coordinate system
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

that specifies each point
Point or points may refer to:
Places
* Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point
Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...

uniquely in a plane
Plane or planes may refer to:
* Airplane
An airplane or aeroplane (informally plane) is a fixed-wing aircraft
A fixed-wing aircraft is a heavier-than-air flying machine
Early flying machines include all forms of aircraft studied ...

by a pair of coordinates, which are the signed
Signing may refer to:
* Using sign language
* Signature, placing one's name on a document
* Signature (disambiguation)
* Manual communication, signing as a form of communication using the hands in place of the voice
* Digital signature, signing as ...

distances from the point to two fixed perpendicular
In elementary geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

directed lines, that are marked using the same unit of length
A unit of length refers to any arbitrarily chosen and accepted reference standard for measurement of length. The most common units in modern use are the metric units
Metric units are units
Unit may refer to:
Arts and entertainment
* UNIT
Unit ...

.
One can use the same principle to specify the position of any point in three-dimension
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

al space
Space is the boundless three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter
A parameter (from the Ancient Greek language, Ancient Gre ...

by the use of three Cartesian coordinates, which are the signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines).
The invention of Cartesian coordinates in the 17th century by René Descartes
René Descartes ( or ; ; Latinized
Latinisation or Latinization can refer to:
* Latinisation of names, the practice of rendering a non-Latin name in a Latin style
* Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

(Latinized
Latinisation or Latinization can refer to:
* Latinisation of names, the practice of rendering a non-Latin name in a Latin style
* Latinisation in the Soviet Union, the campaign in the USSR during the 1920s and 1930s to replace traditional writing sy ...

name: ''Cartesius'') revolutionized mathematics by providing the first systematic link between Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandria
Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic
Coptic may refer to:
Afro-Asia
* Copts, an ethnoreligious group mainly in the area of modern ...

and algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

. Using the Cartesian coordinate system, geometric shapes (such as curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

s) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 in a plane, centered on a particular point called the origin, may be described as the set of all points whose coordinates ''x'' and ''y'' satisfy the equation .
Parametric equations

Aparametric equation
In mathematics, a parametric equation defines a group of quantities as Function (mathematics), functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that m ...

for a curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

expresses the coordinates
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

of the points of the curve as functions of a variable, called a parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whol ...

.Weisstein, Eric W. "Parametric Equations." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ParametricEquations.html For example,
:$\backslash begin\; x\&=\backslash cos\; t\backslash \backslash \; y\&=\backslash sin\; t\; \backslash end$
are parametric equations for the unit circle
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

, where ''t'' is the parameter. Together, these equations are called a parametric representation of the curve.
The notion of ''parametric equation'' has been generalized to surfaces
Water droplet lying on a damask. Surface tension">damask.html" ;"title="Water droplet lying on a damask">Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile.
A surface, as the term is most gener ...

, manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

s and algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...

of higher dimension
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is ''one'' and ''one'' parameter is used, for surfaces dimension ''two'' and ''two'' parameters, etc.).
Number theory

Diophantine equations

A Diophantine equation is apolynomial equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

in two or more unknowns for which only the integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

solutions
Image:SaltInWaterSolutionLiquid.jpg, Making a saline water solution by dissolving Salt, table salt (sodium chloride, NaCl) in water. The salt is the solute and the water the solvent.
In chemistry, a solution is a special type of Homogeneous and ...

are sought (an integer solution is a solution such that all the unknowns take integer values). A linear Diophantine equation is an equation between two sums of monomials of degree
Degree may refer to:
As a unit of measurement
* Degree symbol (°), a notation used in science, engineering, and mathematics
* Degree (angle), a unit of angle measurement
* Degree (temperature), any of various units of temperature measurement ...

zero or one. An example of linear Diophantine equation is where ''a'', ''b'', and ''c'' are constants. An exponential Diophantine equation is one for which exponents of the terms of the equation can be unknowns.
Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

, algebraic surface
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, or more general object, and ask about the lattice point
In geometry and group theory, a lattice in Real coordinate space, \mathbb^n is a subgroup of the additive group \mathbb^n which is isomorphic to the additive group Free abelian group, \mathbb^n, and which linear span, spans the real number, rea ...

s on it.
The word ''Diophantine'' refers to the of the 3rd century, Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the autho ...

of Alexandria
Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakodī; el, Αλεξάνδρεια ''Alexandria'') is the List of cities and towns in Egypt, third-largest city in Egypt after Cairo and Giza, ...

, who made a study of such equations and was one of the first mathematicians to introduce symbolism
Symbolism or symbolist may refer to:
Arts
* Symbolism (arts), a 19th-century movement rejecting Realism
** Symbolist movement in Romania, symbolist literature and visual arts in Romania during the late 19th and early 20th centuries
** Russian symb ...

into algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.
Algebraic and transcendental numbers

Analgebraic number
An algebraic number is any complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, ev ...

is a number that is a solution of a non-zero rational
Rationality is the quality or state of being rational – that is, being based on or agreeable to reason
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογι ...

coefficients (or equivalently — by clearing denominatorsIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

— with integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

coefficients). Numbers such as that are not algebraic are said to be transcendental. Almost all In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...

real
Real may refer to:
* Reality
Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...

and complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

numbers are transcendental.
Algebraic geometry

Algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, classically studying solutions of polynomial equations
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term ''al ...

. Modern algebraic geometry is based on more abstract techniques of abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...

, especially commutative algebra
Commutative algebra is the branch of algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...

, with the language and the problems of geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

.
The fundamental objects of study in algebraic geometry are algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...

, which are geometric manifestations of solutions
Image:SaltInWaterSolutionLiquid.jpg, Making a saline water solution by dissolving Salt, table salt (sodium chloride, NaCl) in water. The salt is the solute and the water the solvent.
In chemistry, a solution is a special type of Homogeneous and ...

of systems of polynomial equations
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for ...

. Examples of the most studied classes of algebraic varieties are: plane algebraic curve
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s, which include lines
Long interspersed nuclear elements (LINEs) (also known as long interspersed nucleotide elements or long interspersed elements) are a group of non-LTR (long terminal repeat
A long terminal repeat (LTR) is a pair of identical sequences of DNA
...

, circle
A circle is a shape
A shape or figure is the form of an object or its external boundary, outline, or external surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

s, parabola
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

s, ellipse
In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

s, hyperbola
In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it ...

s, cubic curve
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s like elliptic curve
In , an elliptic curve is a , , of one, on which there is a specified point ''O''. An elliptic curve is defined over a ''K'' and describes points in ''K''2, the of ''K'' with itself. If the field's is different from 2 and 3, then the curv ...

s and quartic curves like , and Cassini oval
A Cassini oval is a quartic plane curve defined as the Set (mathematics), set (or Locus (mathematics), locus) of points in the plane (mathematics), plane such that the product of the distances to two fixed points is constant. This may be contrasted ...

s. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection point
In differential calculus
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathem ...

s and the points at infinity
150px, The real line with the point at infinity; it is called the real projective line.
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic ...

. More advanced questions involve the topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of the curve and relations between the curves given by different equations.
Differential equations

Adifferential equation
In mathematics, a differential equation is an equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

is a mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

equation that relates some function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

with its derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

s. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

, economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and interact ...

, and biology
Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Development ...

.
In pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...

, differential equations are studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.
If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometrical space. Examples include the mathematical models that describe the ...

puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods
Numerical analysis is the study of algorithm
In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as ...

have been developed to determine solutions with a given degree of accuracy.
Ordinary differential equations

Anordinary differential equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

or ODE is an equation containing a function of one independent variable
Dependent and Independent variables are variables in mathematical modeling
A mathematical model is a description of a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to f ...

and its derivatives. The term "''ordinary''" is used in contrast with the term partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...

, which may be with respect to ''more than'' one independent variable.
Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined and understood, and exact closed-form solutions are obtained. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analytic solutions.
Partial differential equations

Apartial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...

(PDE) is a differential equation
In mathematics, a differential equation is an equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

that contains unknown multivariable functions and their partial derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s. (This is in contrast to ordinary differential equations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model
Computer simulation is the process of mathematical modelling
A mathematical model is a description of a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unifi ...

.
PDEs can be used to describe a wide variety of phenomena such as sound
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

, heat
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these ...

, electrostatics
Electrostatics is a branch of physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related enti ...

, electrodynamics
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is carried by electromagneti ...

, fluid flow
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and h ...

, elasticity
Elasticity often refers to:
*Elasticity (physics), continuum mechanics of bodies that deform reversibly under stress
Elasticity may also refer to:
Information technology
* Elasticity (data store), the flexibility of the data model and the clu ...

, or quantum mechanics
Quantum mechanics is a fundamental theory
A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...

. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometrical space. Examples include the mathematical models that describe the ...

, partial differential equations often model multidimensional systems
In mathematical systems theory, a multidimensional system or m-D system is a system in which not only one independent variable exists (like time), but there are several independent variables.
Important problems such as factorization and Stability t ...

. PDEs find their generalisation in stochastic partial differential equations.
Types of equations

Equations can be classified according to the types ofoperations
Operation or Operations may refer to:
Science and technology
* Surgical operation
Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical or dental specialty that ...

and quantities involved. Important types include:
* An algebraic equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

or system of polynomial equations
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for ...

). These are further classified by degree
Degree may refer to:
As a unit of measurement
* Degree symbol (°), a notation used in science, engineering, and mathematics
* Degree (angle), a unit of angle measurement
* Degree (temperature), any of various units of temperature measurement ...

:
** linear equation
In mathematics, a linear equation is an equation that may be put in the form
:a_1x_1+\cdots +a_nx_n+b=0,
where x_1, \ldots, x_n are the variable (mathematics), variables (or unknown (mathematics), unknowns), and b, a_1, \ldots, a_n are the coeffi ...

for degree one
** quadratic equation
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...

for degree two
** cubic equation
roots
A root is the part of a plant that most often lies below the surface of the soil but can also be aerial or aerating, that is, growing up above the ground or especially above water.
Root or roots may also refer to:
Art, entertainment, a ...

for degree three
** quartic equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

for degree four
** quintic equation
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

for degree five
** sextic equation
233px, Graph of a sextic function, with 6 real number, real root of a function, roots (crossings of the axis) and 5 critical point (mathematics), critical points. Depending on the number and vertical locations of minimum, minima and maxima, the se ...

for degree six
** septic equation
roots (crossings of the axis) and 6 critical points. Depending on the number and vertical location of the minima and maxima, the septic could have 7, 5, 3, or 1 real root counted with their multiplicity; the number of complex non-real roots i ...

for degree seven
** for degree eight
* A Diophantine equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

is an equation where the unknowns are required to be integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s
* A transcendental equation
A transcendental equation is an equation
In mathematics, an equation is a statement that asserts the equality (mathematics), equality of two Expression (mathematics), expressions, which are connected by the equals sign "=". The word ''equation ...

is an equation involving a transcendental function of its unknowns
* A parametric equation
In mathematics, a parametric equation defines a group of quantities as Function (mathematics), functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that m ...

is an equation in which the solutions for the variables are expressed as functions of some other variables, called parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whol ...

s appearing in the equations
* A functional equation is an equation in which the unknowns are Function (mathematics), functions rather than simple quantities
* Equations involving derivatives, integrals and finite differences:
** A differential equation
In mathematics, a differential equation is an equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

is a functional equation involving derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

s of the unknown functions, where the function and its derivatives are evaluated at the same point, such as $f\text{'}(x)\; =\; x^2$. Differential equations are subdivided into ordinary differential equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s for functions of a single variable and partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...

s for functions of multiple variables
** An integral equation is a functional equation involving the antiderivatives of the unknown functions. For functions of one variable, such an equation differs from a differential equation primarily through a change of variable substituting the function by its derivative, however this is not the case when the integral is taken over an open surface
** An integro-differential equation is a functional equation involving both the derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

s and the antiderivatives of the unknown functions. For functions of one variable, such an equation differs from integral and differential equations through a similar change of variable.
** A functional differential equation of delay differential equation is a function equation involving See also

* Formula * History of algebra * Indeterminate equation * List of equations * List of scientific equations named after people * Term (logic) * Theory of equations * Cancelling outNotes

References

External links

Winplot

General Purpose plotter that can draw and animate 2D and 3D mathematical equations.

Equation plotter

A web page for producing and downloading pdf or postscript plots of the solution sets to equations and inequations in two variables (''x'' and ''y''). {{Authority control Elementary algebra Equations,