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84-410: 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 of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of

168-410: A , b , c {\displaystyle a,b,c} and d {\displaystyle d} are real numbers and x , y , z {\displaystyle 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 {\displaystyle a,b,c} are the coordinates of

252-424: A curve expresses the coordinates of the points of the curve as functions of a variable , called a parameter . For example, Equation solving In mathematics , to solve an equation is to find its solutions , which are the values ( numbers , functions , sets , etc.) that fulfill the condition stated by the equation , consisting generally of two expressions related by an equals sign . When seeking

336-503: A mathematical model or computer simulation of a relatively complex system. In Euclidean 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 a x + b y + c z + d = 0 {\displaystyle ax+by+cz+d=0} , where

420-735: A conic. The use of equations allows one to call on a large area of mathematics to solve geometric questions. The Cartesian coordinate system transforms a geometric problem into an analysis problem, once the figures are transformed into equations; thus the name analytic geometry . This point of view, outlined by Descartes , 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 and linear algebra . In Cartesian geometry , equations are used to describe geometric figures . As

504-425: A function on all of the set B (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, the projection π 1  : R → R defined by π 1 ( x , y ) = x has no post-inverse, but it has a pre-inverse π 1 defined by π 1 ( x ) = ( x , 0) . Indeed,

588-425: A function on all of the set B (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, the projection π 1  : R → R defined by π 1 ( x , y ) = x has no post-inverse, but it has a pre-inverse π 1 defined by π 1 ( x ) = ( x , 0) . Indeed,

672-407: A linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution 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. In mathematics, the theory of linear systems is a fundamental part of linear algebra ,

756-402: 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 where f is a function , x 1 , ..., x n are the unknowns, and c is a constant. Its solutions are the elements of the inverse image ( fiber ) where D is the domain of the function f . The set of solutions can be

840-402: 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 where f is a function , x 1 , ..., x n are the unknowns, and c is a constant. Its solutions are the elements of the inverse image ( fiber ) where D is the domain of the function f . The set of solutions can be

924-469: A polynomial equation contain one or more terms . For example, the equation has left-hand side A x 2 + B x + C − y {\displaystyle Ax^{2}+Bx+C-y} , which has four terms, and right-hand side 0 {\displaystyle 0} , consisting of just one term. The names of the variables suggest that x and y are unknowns, and that A , B , and C are parameters , but this

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1008-432: A solution being a tuple of values, one for each unknown , that satisfies all the equations or inequalities. If the solution 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 a Diophantine equation , that is, an equation for which only integer solutions are sought. In this case,

1092-432: A solution being a tuple of values, one for each unknown , that satisfies all the equations or inequalities. If the solution 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 a Diophantine equation , that is, an equation for which only integer solutions are sought. In this case,

1176-500: A solution that is an algebraic expression , 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 one, two, three, or four; but equations of degree five or more cannot always be solved in this way, as the Abel–Ruffini theorem demonstrates. A large amount of research has been devoted to compute efficiently accurate approximations of

1260-607: 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, say x , such as can be solved using the methods of elementary algebra . Smaller systems of linear equations can be solved likewise by methods of elementary algebra. For solving larger systems, algorithms are used that are based on linear algebra . See Gaussian elimination and numerical solution of linear systems . Polynomial equations of degree up to four can be solved exactly using algebraic methods, of which

1344-607: 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, say x , such as can be solved using the methods of elementary algebra . Smaller systems of linear equations can be solved likewise by methods of elementary algebra. For solving larger systems, algorithms are used that are based on linear algebra . See Gaussian elimination and numerical solution of linear systems . Polynomial equations of degree up to four can be solved exactly using algebraic methods, of which

1428-401: A solution, one or more variables are designated as unknowns . 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, when substituted for the unknowns, the equation becomes an equality . A solution of an equation is often called a root of

1512-401: A solution, one or more variables are designated as unknowns . 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, when substituted for the unknowns, the equation becomes an equality . A solution of an equation is often called a root of

1596-408: A subject which is used in many parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra , and play a prominent role in physics , engineering , chemistry , computer science , and economics . A system of non-linear equations can often be approximated by a linear system (see linearization ), a helpful technique when making

1680-408: A symbolic solution is ( x , y ) = ( a + 1, a ) , where the variable a 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 ), and a = 1 gives ( x , y ) = (2, 1) . The distinction between known variables and unknown variables is generally made in the statement of

1764-408: A symbolic solution is ( x , y ) = ( a + 1, a ) , where the variable a 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 ), and a = 1 gives ( x , y ) = (2, 1) . The distinction between known variables and unknown variables is generally made in the statement of

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1848-429: 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 R 2 {\displaystyle \mathbb {R} ^{2}} or as the solution set of two linear equations with values in R 3 . {\displaystyle \mathbb {R} ^{3}.} A conic section

1932-427: Is x = 0, y = 0, z = 0 . Two other solutions are x = 3, y = 6, z = 1 , and x = 8, y = 9, z = 2 . There is a unique plane in three-dimensional space which passes through the three points with these coordinates , and this plane is the set of all points whose coordinates are solutions of the equation. The solution set of a given set of equations or inequalities is the set of all its solutions,

2016-427: Is x = 0, y = 0, z = 0 . Two other solutions are x = 3, y = 6, z = 1 , and x = 8, y = 9, z = 2 . There is a unique plane in three-dimensional space which passes through the three points with these coordinates , and this plane is the set of all points whose coordinates are solutions of the equation. The solution set of a given set of equations or inequalities is the set of all its solutions,

2100-482: Is integration , and the analytic methods for solving this kind of problems are now called symbolic integration . Solutions of differential equations can be implicit or explicit . Equation solving In mathematics , to solve an equation is to find its solutions , which are the values ( numbers , functions , sets , etc.) that fulfill the condition stated by the equation , consisting generally of two expressions related by an equals sign . When seeking

2184-426: 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, is a univariate algebraic (polynomial) equation with integer coefficients and is a multivariate polynomial equation over the rational numbers. Some polynomial equations with rational coefficients have

2268-414: Is analogous to a weighing scale , balance, or seesaw . 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 represented by an inequation ). In

2352-701: 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 be unsolvable by an algorithm, such as Hilbert'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 in computer 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

2436-701: 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 be unsolvable by an algorithm, such as Hilbert'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 in computer 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

2520-455: Is known as the inverse function of h . Given a function h  : A → B , the inverse function, denoted h and defined as h  : B → A , is a function such that Now, if we apply the inverse function to both sides of h ( x ) = c , where c is a constant value in B , 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

2604-455: Is known as the inverse function of h . Given a function h  : A → B , the inverse function, denoted h and defined as h  : B → A , is a function such that Now, if we apply the inverse function to both sides of h ( x ) = c , where c is a constant value in B , 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

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2688-467: Is limited to between 0 and 45 degrees, one may use the above identity for the product to give: yielding the following solution for θ: Since the sine function is a periodic function , 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. Algebra studies two main families of equations: polynomial equations and, among them,

2772-400: Is normally fixed by the context (in some contexts, y may be a parameter, or A , B , and C 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

2856-417: Is so huge that an exhaustive search is not practically feasible; this is, in fact, a requirement for strong encryption methods. As with all kinds of problem 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

2940-417: Is so huge that an exhaustive search is not practically feasible; this is, in fact, a requirement for strong encryption methods. As with all kinds of problem 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

3024-407: Is solved for the unknown x by the expression x = y + 1 , because substituting y + 1 for x in the equation results in ( y + 1) + y = 2( y + 1) – 1 , a true statement. It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1 . Or x and y can both be treated as unknowns, and then there are many solutions to the equation;

3108-407: Is solved for the unknown x by the expression x = y + 1 , because substituting y + 1 for x in the equation results in ( y + 1) + y = 2( y + 1) – 1 , a true statement. It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1 . Or x and y can both be treated as unknowns, and then there are many solutions to the equation;

3192-400: Is the difference of two squares : which is true for all x and y . Trigonometry is an area where many identities exist; these are useful in manipulating or solving trigonometric equations . Two of many that involve the sine and cosine functions are: and which are both true for all values of θ . For example, to solve for the value of θ that satisfies the equation: where θ

3276-406: Is the intersection of a cone with equation x 2 + y 2 = z 2 {\displaystyle 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

3360-609: The Newton–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 also numerical methods for systems of linear equations . Equations involving matrices and vectors of real numbers can often be solved by using methods from linear algebra . There is a vast body of methods for solving various kinds of differential equations , both numerically and analytically . A particular class of problem that can be considered to belong here

3444-553: The Newton–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 also numerical methods for systems of linear equations . Equations involving matrices and vectors of real numbers can often be solved by using methods from linear algebra . There is a vast body of methods for solving various kinds of differential equations , both numerically and analytically . A particular class of problem that can be considered to belong here

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3528-534: The empty set (there are no solutions), a singleton (there is exactly one solution), finite, or infinite (there are infinitely many solutions). For example, an equation such as with unknowns x , y and z , can be put in the above form by subtracting 21 z from both sides of the equation, to obtain In this particular case there is not just one solution, but an infinite set of solutions, which can be written using set builder notation as One particular solution

3612-488: The empty set (there are no solutions), a singleton (there is exactly one solution), finite, or infinite (there are infinitely many solutions). For example, an equation such as with unknowns x , y and z , can be put in the above form by subtracting 21 z from both sides of the equation, to obtain In this particular case there is not just one solution, but an infinite set of solutions, which can be written using set builder notation as One particular solution

3696-443: The quadratic 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 as Bring radicals , although some specific cases may be solvable algebraically, for example (by using the rational root theorem ), and (by using the substitution x = z , which simplifies this to a quadratic equation in z ). In Diophantine equations

3780-443: The quadratic 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 as Bring radicals , although some specific cases may be solvable algebraically, for example (by using the rational root theorem ), and (by using the substitution x = z , which simplifies this to a quadratic equation in z ). In Diophantine equations

3864-446: The real or complex 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 linear equations (or linear system ) is a collection of linear equations involving one or more variables . For example, is a system of three equations in the three variables x , y , z . A solution to

3948-482: The Cartesian coordinate system, geometric shapes (such as curves ) 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 x + y = 4 . A parametric equation for

4032-412: 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 each side. 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

4116-526: The coefficients and solutions are integers . The techniques used are different and come from number 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. In general, an algebraic equation or polynomial equation is an equation of the form where P and Q are polynomials with coefficients in some field (e.g., rational numbers , real numbers , complex numbers ). An algebraic equation

4200-456: 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 given interval . When the task is to find the solution that is the best under some criterion, this is an optimization problem . Solving an optimization problem is generally not referred to as "equation solving", as, generally, solving methods start from

4284-456: 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 given interval . When the task is to find the solution that is the best under some criterion, this is an optimization problem . Solving an optimization problem is generally not referred to as "equation solving", as, generally, solving methods start from

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4368-432: 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 a polynomial equation (commonly called also an algebraic equation ) in which the two sides are polynomials . The sides of

4452-406: The equation π 1 ( x , y ) = c is solved by Examples of inverse functions include the n th root (inverse of x ); the logarithm (inverse of a ); the inverse trigonometric functions ; and Lambert's W function (inverse of xe ). If the left-hand side expression of an equation P = 0 can be factorized as P = QR , the solution set of the original solution consists of

4536-406: The equation π 1 ( x , y ) = c is solved by Examples of inverse functions include the n th root (inverse of x ); the logarithm (inverse of a ); the inverse trigonometric functions ; and Lambert's W function (inverse of xe ). If the left-hand side expression of an equation P = 0 can be factorized as P = QR , the solution set of the original solution consists of

4620-415: 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 is usually written ax  +  bx  +  c  = 0. The process of finding

4704-433: The equation, particularly but not only for polynomial equations . The set of all solutions of an equation is its solution set . An equation may be solved either numerically or symbolically. Solving an equation numerically means that only numbers are admitted as solutions. Solving an equation symbolically means that expressions can be used for representing the solutions. For example, the equation x + y = 2 x – 1

4788-433: The equation, particularly but not only for polynomial equations . The set of all solutions of an equation is its solution set . An equation may be solved either numerically or symbolically. Solving an equation numerically means that only numbers are admitted as solutions. Solving an equation symbolically means that expressions can be used for representing the solutions. For example, the equation x + y = 2 x – 1

4872-420: The equations that are considered, such as implicit equations or parametric 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 of algebraic geometry , an important area of mathematics. One can use the same principle to specify

4956-449: The exponent of 2 (which means applying the function f ( s ) = s 2 {\displaystyle f(s)=s^{2}} to both sides of the equation) changes the equation to x 2 = 1 {\displaystyle x^{2}=1} , which not only has the previous solution but also introduces the extraneous solution, x = − 1. {\displaystyle x=-1.} Moreover, if

5040-399: 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 , as well as some less elementary ones, like Gaussian elimination . An equation

5124-403: The illustration, x , y and z are all different quantities (in this case real numbers ) 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. Equations often contain terms other than

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5208-463: The operations are meaningful for the expressions they are applied to: If some function 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 {\displaystyle x=1} has the solution x = 1. {\displaystyle x=1.} Raising both sides to

5292-443: The polynomial equation has as rational solutions x = − ⁠ 1 / 2 ⁠ and x = 3 , and so, viewed as a Diophantine equation, it has the unique solution x = 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 form h ( x ) = c for some constant c by considering what

5376-443: The polynomial equation has as rational solutions x = − ⁠ 1 / 2 ⁠ and x = 3 , and so, viewed as a Diophantine equation, it has the unique solution x = 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 form h ( x ) = c for some constant c by considering what

5460-459: The position of any point in three- dimensional space 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 revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra . Using

5544-502: The problem, by phrases such as "an equation in x and y ", or "solve for x and y ", which indicate the unknowns, here x and y . However, it is common to reserve x , y , z , ... to denote the unknowns, and to use a , b , c , ... to denote the known variables, which are often called parameters . This is typically the case when considering polynomial equations , such as quadratic equations . However, for some problems, all variables may assume either role. Depending on

5628-502: The problem, by phrases such as "an equation in x and y ", or "solve for x and y ", which indicate the unknowns, here x and y . However, it is common to reserve x , y , z , ... to denote the unknowns, and to use a , b , c , ... to denote the known variables, which are often called parameters . This is typically the case when considering polynomial equations , such as quadratic equations . However, for some problems, all variables may assume either role. Depending on

5712-422: The solution set is the empty set , since 2 is not the square of an integer. However, if one searches for real 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,

5796-422: The solution set is the empty set , since 2 is not the square of an integer. However, if one searches for real 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,

5880-453: The solution set of an equation is restricted to a finite set (as is the case for equations in modular arithmetic , for example), or can be limited to a finite number of possibilities (as is the case with some Diophantine equations ), the solution set can be found by brute 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,

5964-453: The solution set of an equation is restricted to a finite set (as is the case for equations in modular arithmetic , for example), or can be limited to a finite number of possibilities (as is the case with some Diophantine equations ), the solution set can be found by brute 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,

6048-469: The solutions are required to be integers . 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 for rational -valued unknowns (see Rational root theorem ), and then find solutions to the Diophantine equation by restricting the solution set to integer-valued solutions. For example,

6132-409: The solutions are required to be integers . 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 for rational -valued unknowns (see Rational root theorem ), and then find solutions to the Diophantine equation by restricting the solution set to integer-valued solutions. For example,

6216-494: The solutions cannot be listed. For representing them, a parametrization 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 are linear . Such infinite solution sets can naturally be interpreted as geometric shapes such as lines , curves (see picture), planes , and more generally algebraic varieties or manifolds . In particular, algebraic geometry may be viewed as

6300-494: The solutions cannot be listed. For representing them, a parametrization 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 are linear . Such infinite solution sets can naturally be interpreted as geometric shapes such as lines , curves (see picture), planes , and more generally algebraic varieties or manifolds . In particular, algebraic geometry may be viewed as

6384-416: 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 system of equations 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

6468-467: The special case of linear equations . When there is only one variable, polynomial equations have the form P ( x ) = 0, where P is a polynomial , and linear equations have the form ax  +  b  = 0, where a and b are parameters . To solve equations from either family, one uses algorithmic or geometric techniques that originate from linear algebra or mathematical analysis . Algebra also studies Diophantine equations where

6552-465: The study of solution sets of algebraic 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

6636-465: The study of solution sets of algebraic 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

6720-435: The union of the solution sets of the two equations Q = 0 and R = 0 . For example, the equation can be rewritten, using the identity tan x cot x = 1 as which can be factorized into The solutions are thus the solutions of the equation tan x = 1 , and are thus the set With more complicated equations in real or complex numbers , simple methods to solve equations can fail. Often, root-finding algorithms like

6804-435: The union of the solution sets of the two equations Q = 0 and R = 0 . For example, the equation can be rewritten, using the identity tan x cot x = 1 as which can be factorized into The solutions are thus the solutions of the equation tan x = 1 , and are thus the set With more complicated equations in real or complex numbers , simple methods to solve equations can fail. Often, root-finding algorithms like

6888-476: The unknowns, which together form a solution to each equation in the system. For example, the system has the unique solution x  = −1, y  = 1. 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

6972-460: 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 When R is chosen to have the value of 2 ( R = 2), this equation would be recognized in Cartesian coordinates as the equation for the circle of radius of 2 around the origin. Hence,

7056-406: The variables. A conditional equation is only true for particular values of the variables. The " = " symbol, which appears in every equation, was invented in 1557 by Robert Recorde , who considered that nothing could be more equal than parallel straight lines with the same length. An equation is written as two expressions , connected by an equals sign ("="). The expressions on the two sides of

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