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Text of the page (random words):
mation cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikimedia commons wikibooks wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia redirected from unknown algebra mathematical formula expressing equality for other uses see equation disambiguation you can help expand this article with text translated from the corresponding article in french click show for important translation instructions machine translation like deepl or google translate is a useful starting point for translations but translators must follow the llm translation guideline revise errors as necessary and confirm that the translation is accurate rather than simply copy pasting machine translated text into the english wikipedia consider adding a topic to this template there are already 1 328 articles in the main category and specifying topic will aid in categorization do not translate text that appears unreliable or low quality if possible verify the text with references provided in the foreign language article you must provide copyright attribution in the edit summary accompanying your translation by providing an interlanguage link to the source of your translation a model attribution edit summary is content in this edit is translated from the existing french wikipedia article at fr équation see its history for attribution you may also add the template translated fr équation to the talk page for more guidance see wikipedia translation the first use of an equals sign equivalent to 14 x 15 71 in modern notation from the whetstone of witte by robert recorde of wales 1557 1 in mathematics an equation is a mathematical formula that expresses the equality of two expressions by connecting them with the equals sign 2 3 the word equation and its cognates in other languages may have subtly different meanings for example in french an équation is defined as containing one or more variables while an égalité does not require variables but in english any well formed formula consisting of two expressions related with an equals sign is an equation 4 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 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 by robert recorde who considered that nothing could be more equal than parallel straight lines with the same length 1 description edit an equation is written as two expressions connected by an equals sign 2 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 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 a polynomial equation contain one or more terms for example the equation a x 2 b x c y 0 displaystyle ax 2 bx c y 0 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 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 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 properties edit 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 adding or subtracting the same quantity to both sides of an equation citation needed this shows that every equation is equivalent to an equation in which the right hand side is zero multiplying or dividing both sides of an equation by a non zero quantity citation needed applying an identity to transform one side of the equation for example expanding 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 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 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 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 examples edit analogous illustration edit illustration of a simple equation x y z are real numbers analogous to weights an equation is analogous to a weighing scale balance or seesaw each side of the equation corresponds to one side of the balance to the left and to the right 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 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 parameters and unknowns edit see also expression mathematics 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 displaystyle x 2 y 2 r 2 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 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 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 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 the unknowns which together form a solution to each equation in the system for example the system 3 x 5 y 2 5 x 8 y 3 displaystyle begin aligned 3x 5y 2 5x 8y 3 end aligned has the unique solution x 1 y 1 identities edit main articles identity mathematics and list of trigonometric 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 the difference of two squares x 2 y 2 x y x y displaystyle x 2 y 2 x y x y 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 sin 2 θ cos 2 θ 1 displaystyle sin 2 theta cos 2 theta 1 and sin 2 θ 2 sin θ cos θ displaystyle sin 2 theta 2 sin theta cos theta which are both true for all values of θ for example to solve the equation 3 sin θ cos θ 1 displaystyle 3 sin theta cos theta 1 for a value of θ that is between 0 and 45 degrees one may use the above identity for the product to give 3 2 sin 2 θ 1 displaystyle frac 3 2 sin 2 theta 1 yielding the following solution for θ θ 1 2 arcsin 2 3 20 9 displaystyle theta frac 1 2 arcsin left frac 2 3 right approx 20 9 circ 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 edit algebra studies two main families of equations polynomial equations and among them 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 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 polynomial equations edit main article polynomial equation the solutions 1 and 2 of the polynomial equation x 2 x 2 0 are the points where the graph of the quadratic function y x 2 x 2 cuts the x axis in general an algebraic equation or polynomial equation is an equation of the form p 0 displaystyle p 0 or p q displaystyle p q a where p and q are polynomials with coefficients in some field e g rational numbers real numbers complex numbers 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 3 x 1 0 displaystyle x 5 3x 1 0 is a univariate algebraic polynomial equation with integer coefficients and y 4 x y 2 x 3 3 x y 2 y 2 1 7 displaystyle y 4 frac xy 2 frac x 3 3 xy 2 y 2 frac 1 7 is a multivariate polynomial equation over the rational numbers some polynomial equations with rational coefficients have 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 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 systems of linear equations edit the nine chapters on the mathematical art is an anonymous 2nd century chinese book proposing a method of resolution for linear equations a system of linear equations or linear system is a collection of linear equations involving one or more variables b for example 3 x 2 y z 1 2 x 2 y 4 z 2 x 1 2 y z 0 displaystyle begin alignedat 7 3x 2y z 1 2x 2y 4z 2 x tfrac 1 2 y z 0 end alignedat is a system of three equations in the three variables x y z a solution to 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 x 1 y 2 z 2 displaystyle begin alignedat 2 x 1 y 2 z 2 end alignedat 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 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 a mathematical model or computer simulation of a relatively complex system geometry edit analytic geometry edit the blue and red line is the set of all points x y such that x y 5 and x 2 y 4 respectively their intersection point 2 3 satisfies both equations main article analytic geometry 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 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 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 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 ...