equation
equation, in mathematics, a statement, usually written in symbols, that states the equality of two quantities or algebraic expressions, e.g., x+3=5. The quantity x+3, to the left of the equals sign (=), is called the lefthand, or first, member of the equation, that to the right (5) the righthand, or second, member. A numerical equation is one containing only numbers, e.g., 2+3=5. A literal equation is one that, like the first example, contains some letters (representing unknowns or variables). An identical equation is a literal equation that is true for every value of the variable, e.g., the equation (x+1)^{2}=x^{2}+2x+1. A conditional equation (usually referred to simply as an equation) is a literal equation that is not true for all values of the variable, e.g., only the value 2 for x makes true the equation x+3=5. To solve an equation is to find the value or values of the variable that satisfy it. Polynomial equations, containing more than one term, are classified according to the highest degree of the variable they contain. Thus the first example is a first degree (also called linear) equation. The equation ax^{2}+bx+c=0 is a second degree, or quadratic, equation in the unknown x if the letters a, b, and c are assumed to represent constants. In algebra, methods are evolved for solving various types of equations. To be valid the solution must satisfy the equation. Whether it does can be ascertained by substituting the supposed solution for the variable in the equation. The simultaneous solution of two or more equations is a set of values of the variables that satisfies each of the equations. In order that a solution may exist, the number of equations (i.e., conditions) must generally be no greater than the number of variables. In chemistry an equation (see chemical equation) is used to represent a reaction.
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equation
equation An expression that asserts the equality of two terms. To be precise, an equation has the following form. Let Σ be a signature and let t_{1}(X_{1},…, X_{n}) and t_{2}(X_{1},…, X_{n}) be two terms over Σ involving the variables X_{1},…, X_{n}. Then t_{1}(X_{1},…, X_{n}) = t_{2}(X_{1},…, X_{n})
is an equation.
Equations are a natural means of expressing possible relationships between the functions in a signature. In fact, the equations can be used to specify or define the functions uniquely using initial algebra semantics (see equational specification).
Most systems in science and engineering are described mathematically using equations. Two stages are involved: a mathematical model of the system is made using sets and functions; some functions are known and others are to be found. Equations are postulated to define the unknown functions in terms of one another and the known functions. Research has shown that the same process is possible for computing systems. Indeed, theoretically it is known that any computing system, or any physical system that can be faithfully modeled using digital computation, can be characterized by small sets of equations. See also computable algebra.
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equation
e·qua·tion / iˈkwāzhən/ • n. 1. Math. a statement that the values of two mathematical expressions are equal (indicated by the sign =). 2. the process of equating one thing with another: the equation of science with objectivity. ∎ (the equation) a situation or problem in which several factors must be taken into account: money also came into the equation. 3. Chem. a symbolic representation of the changes that occur in a chemical reaction, expressed in terms of the formulae of the molecules or other species involved.
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equation
equation Mathematical statement of variables, equal to some subset of all possible variables. The equation x^{2} = 8 − 2x is true only for certain values (solutions) of x (x = 2 and x = −4). This type of equation is contrasted with an identity, such as (x+2)^{2} = x^{2} + 4x + 4, which is true for all values of x. Equations are said to be linear, quadratic, cubic, quartic, etc., according to whether their degree (the highest power of the variable) is 1, 2, 3, 4, etc. See also simultaneous equations
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equation
equation
•abrasion, Australasian, equation, Eurasian, evasion, invasion, occasion, persuasion, pervasion, suasion, Vespasian
•adhesion, cohesion, Friesian, lesion
•circumcision, collision, concision, decision, derision, division, elision, envision, excision, imprecision, incision, misprision, precisian, precision, provision, scission, vision
•subdivision • television • Eurovision
•LaserVision
•corrosion, eclosion, erosion, explosion, implosion
•allusion, collusion, conclusion, confusion, contusion, delusion, diffusion, effusion, exclusion, extrusion, fusion, illusion, inclusion, interfusion, intrusion, obtrusion, occlusion, preclusion, profusion, prolusion, protrusion, reclusion, seclusion, suffusion, transfusion
•Monaghan • Belgian
•Bajan, Cajun, contagion, Trajan
•Glaswegian, legion, Norwegian, region
•irreligion, religion
•Injun • Harijan • oxygen • antigen
•sojourn • donjon • Georgian
•theologian, Trojan
•Rügen
•bludgeon, curmudgeon, dudgeon, gudgeon, trudgen
•dungeon • glycogen • halogen
•collagen • Imogen • carcinogen
•hallucinogen • androgen
•oestrogen (US estrogen)
•hydrogen • nitrogen
•burgeon, sturgeon, surgeon
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