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Inmathematics, tosolve an equation is to find itssolutions, which are the values (numbers,functions,sets, etc.) that fulfill the condition stated by theequation, consisting generally of twoexpressions related by anequals sign. When seeking a solution, one or morevariables are designated asunknowns. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values (one for each unknown) such that, whensubstituted for the unknowns, the equation becomes anequality.A solution of an equation is often called aroot of the equation, particularly but not only forpolynomial equations. The set of all solutions of an equation is itssolution set.
An equation may be solved eithernumerically or symbolically. Solving an equationnumerically means that only numbers are admitted as solutions. Solving an equationsymbolically means that expressions can be used for representing the solutions.
For example, the equationx +y = 2x – 1 is solved for the unknownx by the expressionx =y + 1, because substitutingy + 1 forx in the equation results in(y + 1) +y = 2(y + 1) – 1, a true statement. It is also possible to take the variabley to be the unknown, and then the equation is solved byy =x – 1. Orx andy can both be treated as unknowns, and then there are many solutions to the equation; a symbolic solution is(x,y) = (a + 1,a), where the variablea may take any value. Instantiating a symbolic solution with specific numbers gives a numerical solution; for example,a = 0 gives(x,y) = (1, 0) (that is,x = 1,y = 0), anda = 1 gives(x,y) = (2, 1).
The distinction between known variables and unknown variables is generally made in the statement of the problem, by phrases such as "an equationinx andy", or "solveforx andy", which indicate the unknowns, herex andy.However, it is common to reservex,y,z, ... to denote the unknowns, and to usea,b,c, ... to denote the known variables, which are often calledparameters. This is typically the case when consideringpolynomial equations, such asquadratic equations. However, for some problems, all variables may assume either role.
Depending on the context, solving an equation may consist to find either any solution (finding a single solution is enough), all solutions, or a solution that satisfies further properties, such as belonging to a giveninterval. When the task is to find the solution that is thebest under some criterion, this is anoptimization problem. Solving an optimization problem is generally not referred to as "equation solving", as, generally, solving methods start from a particular solution for finding a better solution, and repeating the process until finding eventually the best solution.
One general form of an equation is
wheref is afunction,x1, ...,xn are the unknowns, andc is a constant. Its solutions are the elements of theinverse image (fiber)
whereD is thedomain of the functionf. The set of solutions can be theempty set (there are no solutions), asingleton (there is exactly one solution), finite, or infinite (there are infinitely many solutions).
For example, an equation such as
with unknownsx,y andz, can be put in the above form by subtracting21z from both sides of the equation, to obtain
In this particular case there is not justone solution, but an infinite set of solutions, which can be written usingset builder notation as
One particular solution isx = 0,y = 0,z = 0. Two other solutions arex = 3,y = 6,z = 1, andx = 8,y = 9,z = 2. There is a uniqueplane inthree-dimensional space which passes through the three points with thesecoordinates, and this plane is the set of all points whose coordinates are solutions of the equation.
Thesolution set of a given set of equations orinequalities is theset of all its solutions, a solution being atuple of values, one for eachunknown, that satisfies all the equations or inequalities.If thesolution set is empty, then there are no values of the unknowns that satisfy simultaneously all equations and inequalities.
For a simple example, consider the equation
This equation can be viewed as aDiophantine equation, that is, an equation for which onlyinteger solutions are sought. In this case, the solution set is theempty set, since 2 is not thesquare of an integer. However, if one searches forreal solutions, there are two solutions,√2 and–√2; in other words, the solution set is{√2, −√2}.
When an equation contains several unknowns, and when one has several equations with more unknowns than equations, the solution set is often infinite. In this case, the solutions cannot be listed. For representing them, aparametrization is often useful, which consists of expressing the solutions in terms of some of the unknowns or auxiliary variables. This is always possible when all the equations arelinear.
Such infinite solution sets can naturally be interpreted asgeometric shapes such aslines,curves (see picture),planes, and more generallyalgebraic varieties ormanifolds. In particular,algebraic geometry may be viewed as the study of solution sets ofalgebraic equations.
The methods for solving equations generally depend on the type of equation, both the kind of expressions in the equation and the kind of values that may be assumed by the unknowns. The variety in types of equations is large, and so are the corresponding methods. Only a few specific types are mentioned below.
In general, given a class of equations, there may be no known systematic method (algorithm) that is guaranteed to work. This may be due to a lack of mathematical knowledge; some problems were only solved after centuries of effort. But this also reflects that, in general, no such method can exist: some problems are known to beunsolvable by an algorithm, such asHilbert's tenth problem, which was proved unsolvable in 1970.
For several classes of equations, algorithms have been found for solving them, some of which have been implemented and incorporated incomputer algebra systems, but often require no more sophisticated technology than pencil and paper. In some other cases,heuristic methods are known that are often successful but that are not guaranteed to lead to success.
If the solution set of an equation is restricted to a finite set (as is the case for equations inmodular arithmetic, for example), or can be limited to a finite number of possibilities (as is the case with someDiophantine equations), the solution set can be found bybrute force, that is, by testing each of the possible values (candidate solutions). It may be the case, though, that the number of possibilities to be considered, although finite, is so huge that anexhaustive search is not practically feasible; this is, in fact, a requirement for strongencryption methods.
As with all kinds ofproblem solving,trial and error may sometimes yield a solution, in particular where the form of the equation, or its similarity to another equation with a known solution, may lead to an "inspired guess" at the solution. If a guess, when tested, fails to be a solution, consideration of the way in which it fails may lead to a modified guess.
Equations involving linear or simple rational functions of a single real-valued unknown, sayx, such as
can be solved using the methods ofelementary algebra.
Smallersystems of linear equations can be solved likewise by methods of elementary algebra. For solving larger systems, algorithms are used that are based onlinear algebra.SeeGaussian elimination andnumerical solution of linear systems.
Polynomial equations of degree up to four can be solved exactly using algebraic methods, of which thequadratic formula is the simplest example. Polynomial equations with a degree of five or higher require in general numerical methods (see below) or special functions such asBring radicals, although some specific cases may be solvable algebraically, for example
(by using therational root theorem), and
(by using the substitutionx =z1⁄3, which simplifies this to aquadratic equation inz).
InDiophantine equations the solutions are required to beintegers. In some cases a brute force approach can be used, as mentioned above. In some other cases, in particular if the equation is in one unknown, it is possible to solve the equation forrational-valued unknowns (seeRational root theorem), and then find solutions to the Diophantine equation by restricting the solution set to integer-valued solutions. For example, the polynomial equation
has as rational solutionsx = −1/2 andx = 3, and so, viewed as a Diophantine equation, it has the unique solutionx = 3.
In general, however, Diophantine equations are among the most difficult equations to solve.
In the simple case of a function of one variable, say,h(x), we can solve an equation of the formh(x) =c for some constantc by considering what is known as theinverse function ofh.
Given a functionh :A →B, the inverse function, denotedh−1 and defined ash−1 :B →A, is a function such that
Now, if we apply the inverse function to both sides ofh(x) =c, wherec is a constant value inB, we obtain
and we have found the solution to the equation. However, depending on the function, the inverse may be difficult to be defined, or may not be a function on all of the setB (only on some subset), and have many values at some point.
If just one solution will do, instead of the full solution set, it is actually sufficient if only the functional identity
holds. For example, theprojectionπ1 :R2 →R defined byπ1(x,y) =x has no post-inverse, but it has a pre-inverseπ−1
1 defined byπ−1
1(x) = (x, 0). Indeed, the equationπ1(x,y) =c is solved by
Examples of inverse functions include thenth root (inverse ofxn); thelogarithm (inverse ofax); theinverse trigonometric functions; andLambert'sW function (inverse ofxex).
If the left-hand side expression of an equationP = 0 can befactorized asP =QR, the solution set of the original solution consists of the union of the solution sets of the two equationsQ = 0 andR = 0.For example, the equation
can be rewritten, using the identitytanx cotx = 1 as
which can be factorized into
The solutions are thus the solutions of the equationtanx = 1, and are thus the set
With more complicated equations in real orcomplex numbers, simple methods to solve equations can fail. Often,root-finding algorithms like theNewton–Raphson method can be used to find a numerical solution to an equation, which, for some applications, can be entirely sufficient to solve some problem.There are alsonumerical methods for systems of linear equations.
Equations involvingmatrices andvectors ofreal numbers can often be solved by using methods fromlinear algebra.
There is a vast body of methods for solving various kinds ofdifferential equations, bothnumerically andanalytically. A particular class of problem that can be considered to belong here isintegration, and the analytic methods for solving this kind of problems are now calledsymbolic integration.[citation needed] Solutions of differential equations can beimplicit orexplicit.[1]