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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 calledunknowns, and the values of the unknowns that satisfy the equality are calledsolutions 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.[5][6]
The "=" symbol, which appears in every equation, was invented in 1557 byRobert Recorde, who considered that nothing could be more equal than parallel straight lines with the same length.[1]
An equation is written as twoexpressions, connected by anequals sign ("=").[2] The expressions on the twosides 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. 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 apolynomial equation (commonly called also analgebraic equation) in which the two sides arepolynomials.The sides of a polynomial equation contain one or moreterms. For example, the equation
has left-hand sideFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle Ax^2 +Bx + C - y }, which has four terms, and right-hand side, consisting of just one term. The names of thevariables suggest thatx andy are unknowns, and thatA,B, andC areparameters, but this is normally fixed by the context (in some contexts,y may be a parameter, orA,B, andC 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 must be removed from the other pan to keep the scale in balance. More generally, an equation remains balanced if the same operation is performed on each side.[7]
Two equations or two systems of equations areequivalent, 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:
Adding orsubtracting 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.
Applying anidentity to transform one side of the equation. For example,expanding a product orfactoring a sum.
For a system: adding to both sides of an equation the corresponding side of another equation, multiplied by the same quantity.
If somefunction 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 calledextraneous solutions. For example, the equation has the solution Raising both sides to the exponent of 2 (which means applying the function to both sides of the equation) changes the equation to, which not only has the previous solution but also introduces the extraneous solution, Moreover, if the function is not defined at some values (such as 1/x, which is not defined forx = 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 forequation solving, as well as some less elementary ones, likeGaussian elimination.
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 aninequality represented by aninequation).
In the illustration,x,y andz are all different quantities (in this casereal numbers) represented as circular weights, and each ofx,y, andz 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.
Equations often contain terms other than the unknowns. These other terms, which are assumed to beknown, are usually calledconstants,coefficients orparameters.
An example of an equation involvingx andy as unknowns and the parameterR is
WhenRis chosen to have the value of 2 (R= 2), this equation would be recognized inCartesian coordinates as the equation for the circle of radius of 2 around the origin. Hence, the equation withR 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 generalquadratic equation is usually writtenax2 + bx + c = 0.
The process of finding the solutions, or, in case of parameters, expressing the unknowns in terms of the parameters, is calledsolving the equation. Such expressions of the solutions in terms of the parameters are also calledsolutions.
Asystem of equations is a set ofsimultaneous equations, usually in several unknowns for which the common solutions are sought. Thus, asolution 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
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.
For example, to solve for the value ofθ that satisfies the equation:
whereθ is limited to between 0 and 45 degrees, one may use the above identity for the product to give:
yielding the following solution forθ:
Since the sine function is aperiodic function, there are infinitely many solutions if there are no restrictions onθ. In this example, restrictingθ to be between 0 and 45 degrees would restrict the solution to only one number.
Algebra studies two main families of equations:polynomial equations and, among them, the special case oflinear equations. When there is only one variable, polynomial equations have the formP(x) = 0, whereP is apolynomial, and linear equations have the formax + b = 0, wherea andb areparameters. To solve equations from either family, one uses algorithmic or geometric techniques that originate fromlinear algebra ormathematical analysis. Algebra also studiesDiophantine equations where the coefficients and solutions areintegers. The techniques used are different and come fromnumber theory. 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.
whereP andQ arepolynomials with coefficients in somefield (e.g.,rational numbers,real numbers,complex numbers). An algebraic equation isunivariate if it involves only onevariable. On the other hand, a polynomial equation may involve several variables, in which case it is calledmultivariate (multiple variables, x, y, z, etc.).
For example,
is a univariate algebraic (polynomial) equation with integer coefficients and
is a multivariate polynomial equation over the rational numbers.
Some polynomial equations withrational coefficients have a solution that is analgebraic expression, with a finite number of operations involving just those coefficients (i.e., can besolved algebraically). This can be done for all such equations ofdegree one, two, three, or four; but equations of degree five or more cannot always be solved in this way, as theAbel–Ruffini theorem demonstrates.
A large amount of research has been devoted to compute efficiently accurate approximations of thereal orcomplex solutions of a univariate algebraic equation (seeRoot finding of polynomials) and of the common solutions of several multivariate polynomial equations (seeSystem of polynomial equations).
is a system of three equations in the three variablesx,y,z. Asolution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. Asolution to the system above is given by
since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually.
The blue and red line is the set of all points (x,y) such thatx+y=5 and -x+2y=4, respectively. Theirintersection point, (2,3), satisfies both equations.
InEuclidean geometry, 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, where and are real numbers and are the unknowns that correspond to the coordinates of a point in the system given by the orthogonal grid. The values 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 or as the solution set of two linear equations with values in
Aconic section is the intersection of acone with equation 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. TheCartesian coordinate system transforms a geometric problem into an analysis problem, once the figures are transformed into equations; thus the nameanalytic geometry. This point of view, outlined byDescartes, 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 asfunctional analysis andlinear algebra.
Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is(x −a)2 + (y −b)2 =r2 wherea andb are the coordinates of the center(a,b) andr is the radius.
InCartesian geometry, equations are used to describegeometric figures. As the equations that are considered, such asimplicit equations orparametric equations, 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 ofalgebraic geometry, an important area of mathematics.
One can use the same principle to specify the position of any point in three-dimensionalspace 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 byRené Descartes revolutionized mathematics by providing the first systematic link betweenEuclidean geometry andalgebra. Using the Cartesian coordinate system, geometric shapes (such ascurves) 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 coordinatesx andy satisfy the equationx2 +y2 = 4.
are parametric equations for theunit circle, wheret is the parameter. Together, these equations are called a parametric representation of the curve.
The notion ofparametric equation has been generalized tosurfaces,manifolds andalgebraic varieties of higherdimension, 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 isone andone parameter is used, for surfaces dimensiontwo andtwo parameters, etc.).
A Diophantine equation is apolynomial equation in two or more unknowns for which only theintegersolutions 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 ofmonomials ofdegree zero or one. An example of linear Diophantine equation isax +by =c wherea,b, andc 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 analgebraic curve,algebraic surface, or more general object, and ask about thelattice points on it.
The wordDiophantine refers to theHellenistic mathematician of the 3rd century,Diophantus ofAlexandria, who made a study of such equations and was one of the first mathematicians to introducesymbolism intoalgebra. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.
Adifferential equation is amathematical equation that relates somefunction with itsderivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. 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.
Inpure mathematics, 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 ofdynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while manynumerical methods have been developed to determine solutions with a given degree of accuracy.
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 byelementary functions in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical andnumerical 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.
Aparametric equation is an equation in which the solutions for the variables are expressed as functions of some other variables, calledparameters appearing in the equations
Anintegral equation is a functional equation involving theantiderivatives 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
Anintegro-differential equation is a functional equation involving both thederivatives and theantiderivatives of the unknown functions. For functions of one variable, such an equation differs from integral and differential equations through a similar change of variable.
Adifference equation is an equation where the unknown is a functionf that occurs in the equation throughf(x),f(x−1), ...,f(x−k), for some whole integerk called theorder of the equation. Ifx is restricted to be an integer, a difference equation is the same as arecurrence relation
^As such an equation can be rewrittenP –Q = 0, many authors do not consider this case explicitly.
^The subject of this article is basic in mathematics, and is treated in a lot of textbooks. Among them, Lay 2005, Meyer 2001, and Strang 2005 contain the material of this article.
^"A statement of equality between two expressions. Equations are of two types,identities andconditional equations (or usually simply "equations")". « Equation », inMathematics Dictionary,Glenn James [de] etRobert C. James (éd.), Van Nostrand, 1968, 3 ed. 1st ed. 1948, p. 131.
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 andy).
