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| {{Expand French|Équation|date=February 2013}}
| | If you might be one some of those nature lovers you will sometimes comprehend that winter season is not the best time of 12 months for any person. I bet being cooped up inside the confines of one's homes causes you to be giddy and restless. However, to those who are sensitive to cold hate this calendar year. But if you are some of those keen gardeners, you'll be busy fixing your winter the jardines de sabatini.<br><br> |
| {{other uses}}
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| {{redirect|Unknown}}
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| [[Image:First Equation Ever.png|thumb|right|300px|The first use of an equals sign, equivalent to 14''x'' + 15 = 71 in modern notation. From ''The Whetstone of Witte'' by [[Robert Recorde]] (1557).]]
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| In [[mathematics]], an '''equation''' is a [[mathematical formula|formula]] of the form ''A'' = ''B'', where ''A'' and ''B'' are [[expression (mathematics)|expressions]] that may contain one or several [[variable (mathematics)|variables]] called '''unknowns''', and "=" denotes the [[equality (mathematics)|equality]] [[binary relation]]. Although written in the form of [[proposition (mathematics)|proposition]], an equation is not a [[statement (logic)|statement]] that is either true or false, but a problem consisting of finding the values, called '''solutions''', that, when substituted for the unknowns, yield equal values of the expressions ''A'' and ''B''. For example, 2 is the unique ''solution'' of the ''equation'' ''x'' + 2 = 4, in which the ''unknown'' is ''x''.<ref>{{cite web
| | <br><br>Its hedge cutting work-time. A light trim of box and yew [http://www.hedgingplants.com/ hedges] should sufficient but my beech hedge seems to outlive a hard prune. Conifer hedges need regular attention guide them little.<br><br>First decide what require to to use your yard for the purpose of. Maybe you are a BBQ nut and love cooking outdoors - would certainly think probably obtain a lot of usage out associated with the outdoor new kitchen. Others may love the social regarding a pool, or the relief factor a hot tub provides. Maybe you want a gazebo having a bench to buy quiet reading space, possibly a rock garden to meditate in.<br><br>Pork Belly Futures the actual most illiquid, volatile, and difficult to switch. However, you can make a lots of money if you're right. This isn't a niche for the un-aware. It's very common to see Pork Belly futures go limit up and down their same day. Account margin is $1600 to control a $30,000 futures squeeze. This is similar to hogs and cattle commodity.<br><br>But due to the disease progressed, he learned to humble himself and be more associated with receiver rather than a giver. But still, my mother in order to be cautious and not do everything for him as it robbed him of his self worth. Being a fighter, he made a decision to make probably the most of a tough thing, and did his best to enhance his associated with mind until he died at age 79 in 1994. However, it wasn't Parkinson's' disease that took his life while he had a sequence of strokes, losing his mobility and subsequently lost the battle of his life to pneumonia.<br><br>When purchasing a home, together with every monthly payment, tend to be putting money into expense. On the contrary, when you rent a home you must be give money to the owner it is actually definitely an expense. Every time you pay a mortgage, a percentage goes toward your fairness. Indeed, this is similar getting money within your bank account, which you'll be able to draw upon later as required. Every year, rental rates increase and the principle regarding your mortgage is decreasing collectively payment. Further, the housing sector continues to grow and hence the valuations of living rooms.<br><br>First of all, bear in mind that you're probably dealing using a pride publication. Just because your loved underneath the Parkinson's Disease, doesn't mean they would love you to take control. Offer to help, however control the whole bunch. Ask what you can do today to make their life easy.<br><br>If you've seen Star Wars: The Clone Wars you'll remember the laser cannons and missiles and the clone turbo tank rolling through separatist opposition. And also the way Cad Bane has other plans as he sabotages them from right under the Jedi Knights in combat. This set makes it possible to relive might the characters Anakin, Ahsoka, and Aayla Secura are usually part of 1139 pieces and 5 minifigures! |
| | url = http://dictionary.reference.com/browse/equation
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| | title = Equation
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| | work = Dictionary.com
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| | publisher = Dictionary.com, LLC
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| | accessdate = 2009-11-24
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| }}</ref>
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| Historically, equations arose from the mathematical discipline of [[algebra]], but later become ubiquitous. "Equations" should not be confused with [[identity (mathematics)|"identities"]], which are presented with the same notation but have a different meaning: for example 2 + 2 = 4 and ''x'' + ''y'' = ''y'' + ''x'' are identities (which implies they are [[necessarily true]]) in [[arithmetic]], and do not constitute a values-finding problem, even when variables are present as in the latter example.
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| The term '''"equation"''' may also refer to a [[relation (mathematics)|relation]] between some variables that is presented as the equality of some expressions written in terms of those variables' values. For example the ''equation'' of the [[unit circle]] is ''x''<sup>2</sup> + ''y''<sup>2</sup> = 1, which means that a point belongs to the circle if and only if its [[coordinates]] are related by this equation. Most [[physical law]]s are expressed by equations. One of the most famous ones is [[Albert Einstein|Einstein]]'s equation [[mass–energy equivalence|''E'' = ''mc''<sup>2</sup>]].
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| The = symbol was invented by [[Robert Recorde]] (1510–1558), who considered that nothing could be more equal than parallel straight lines with the same length.
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| ==Parameters and unknowns==
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| {{see also|Expression (mathematics)}}
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| Equations often contain terms other than the unknowns. These other terms, which are assumed to be ''known'', are usually called ''constants'', ''coefficients'' or ''parameters''. Usually, the unknowns are denoted by letters at the end of the alphabet, ''x'', ''y'', ''z'', ''w'', …, while coefficients are denoted by letters at the beginning, ''a'', ''b'', ''c'', ''d'', … . For example, the general [[quadratic equation]] is usually written ''ax''<sup>2</sup> + ''bx'' + ''c'' = 0. The process of finding the solutions, or in case of parameters, expressing the unknowns in terms of the parameters is called [[Equation solving|solving the equation]]. Such expressions of the solutions in terms of the parameters are also called ''solutions''.
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| 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
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| :<math>\begin{align}
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| 3x+5y&=2\\
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| 5x+8y&=3
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| \end{align}
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| </math> | |
| has the unique solution ''x'' = −1, ''y'' = 1.
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| ==Analogous illustration==
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| [[File:Equation illustration colour.svg|thumb|Illustration of a simple equation; ''x'', ''y'', ''z'' are real numbers, analogous to weights.]]
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| A [[weighing scale]], balance, or [[seesaw]] is often presented as an analogy to an equation.
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| Each side of the balance corresponds to one side of the equation. Different quantities can be placed on each side: if the weights on the two sides are equal the scale balances, corresponding to an equality represented by an equation; if not, then the lack of balance corresponds to an [[Inequality (mathematics)|inequality]] represented by an [[inequation]].
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| 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.
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| ==Types of equations==
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| Equations can be classified according to the types of [[Operation (mathematics)|operations]] and quantities involved. Important types include:
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| * An [[algebraic equation]] or [[polynomial]] equation is an equation in which both sides are polynomials (see also [[system of polynomial equations]]). These are further classified by [[degree of a polynomial|degree]]:
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| ** [[linear equation]] for degree one
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| ** [[quadratic equation]] for degree two
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| ** [[cubic equation]] for degree three
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| ** [[quartic equation]] for degree four
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| ** [[quintic equation]] for degree five
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| * A [[Diophantine equation]] is an equation where the unknowns are required to be [[integer]]s
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| * A [[transcendental equation]] is an equation involving a [[transcendental function]] of its unknowns
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| * A [[parametric equation]] is an equation for which the solutions are sought as functions of some other variables, called [[parameter]]s appearing in the equations
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| * A [[functional equation]] is an equation in which the unknowns are [[Function (mathematics)|functions]] rather than simple quantities
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| * A [[differential equation]] is a functional equation involving [[derivative]]s of the unknown functions
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| * An [[integral equation]] is a functional equation involving the [[antiderivative]]s of the unknown functions
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| * An [[integro-differential equation]] is a functional equation involving both the [[derivative]]s and the [[antiderivative]]s of the unknown functions
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| * A [[difference equation]] is an equation where the unknown is a function ''f'' which occurs in the equation through ''f''(''x''), ''f''(''x''−1), …, ''f''(''x''−''k''), for some whole integer ''k'' called the ''order'' of the equation. If ''x'' is restricted to be an integer, a difference equation is the same as a [[recurrence relation]]
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| ==Identities==
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| {{main|Identity (mathematics)|List of trigonometric identities}}
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| An '''identity''' is a statement resembling an equation which is true for all possible values of the variable(s) it contains. Many identities are known, especially in [[trigonometry]]. Probably the best known example is: <math>\sin^2(\theta)+\cos^2(\theta)=1,</math>, which is true for all values of ''θ''.
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| In the process of solving an equation, it is often useful to combine it with an identity to produce an equation which is more easily soluble. For example, to solve the equation:
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| :<math>3\sin(\theta) \cos(\theta)= 1, </math> where ''θ'' is known to be between zero and 45 degrees,
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| use the identity: <math>\sin(2 \theta)=2\sin(\theta) \cos(\theta),</math> so the above equation becomes:
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| :<math>\frac{3}{2}\sin(2 \theta) = 1</math>
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| Whence:
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| :<math>\theta = \frac{1}{2} \arcsin\left(\frac{2}{3}\right) \approx 20.9^\circ.</math>
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| ==Properties==
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| 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 into an equivalent one:
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| * [[addition|Adding]] or [[subtraction|subtracting]] 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.
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| * [[Multiplication|Multiplying]] or [[division (mathematics)|dividing]] both sides of an equation by a non-zero constant.
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| * Applying an [[identity (mathematics)|identity]] to transform one side of the equation. For example, [[polynomial expansion|expanding]] a product or [[factorization of polynomials|factoring]] a sum.
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| * For a systems: adding to both sides of an equation the corresponding side of another, equation multiplied by the same quantity.
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| If some [[function (mathematics)|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 solution]]s. For example, the equation <math>x=1</math> has the solution <math>x=1.</math> Raising both sides to the exponent of 2 (which means applying the function <math>f(s)=s^2</math> to both sides of the equation) changes the equation to <math>x^2=1</math>, which not only has the previous solution but also introduces the extraneous solution, <math>x=-1.</math> 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.
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| The above transformations are the basis of most elementary methods for equation solving as well as some less elementary ones, like [[Gaussian elimination]].
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| {{details|Equation solving}}
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| == See also ==
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| <div style="column-count:2;-moz-column-count:2;-webkit-column-count:2"> | |
| *[[Equation (poem)]]
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| *[[Expression (mathematics)|Expression]]
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| *''[[Five Equations That Changed the World: The Power and Poetry of Mathematics]]'' (book)
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| *[[Formula]]
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| *[[Formula editor]]
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| *[[Functional equation]]
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| *[[History of algebra]]
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| *[[Inequality (mathematics)|Inequality]]
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| *[[Inequation]]
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| *[[List of equations]]
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| *[[List of scientific equations named after people]]
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| *[[Term (logic)]]
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| *[[Theory of equations]]
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| </div>
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| ==References==
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| {{reflist}}
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| == External links ==
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| * [http://math.exeter.edu/rparris/winplot.html Winplot]: General Purpose plotter which can draw and animate 2D and 3D mathematical equations.
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| * [http://www.wessa.net/math.wasp Mathematical equation plotter]: Plots 2D mathematical equations, computes integrals, and finds solutions online.
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| * [http://www.cs.cornell.edu/w8/~andru/relplot 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'' and ''y'').
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| * [http://eqworld.ipmnet.ru/ EqWorld]—contains information on solutions to many different classes of mathematical equations.
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| * [http://www.numberz.co.uk/ES.html EquationSolver]: A webpage that can solve single equations and linear equation systems.
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| [[Category:Elementary algebra]]
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| [[Category:Equations| ]]
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| {{Link FA|fr}}
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| {{Link GA|uz}}
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If you might be one some of those nature lovers you will sometimes comprehend that winter season is not the best time of 12 months for any person. I bet being cooped up inside the confines of one's homes causes you to be giddy and restless. However, to those who are sensitive to cold hate this calendar year. But if you are some of those keen gardeners, you'll be busy fixing your winter the jardines de sabatini.
Its hedge cutting work-time. A light trim of box and yew hedges should sufficient but my beech hedge seems to outlive a hard prune. Conifer hedges need regular attention guide them little.
First decide what require to to use your yard for the purpose of. Maybe you are a BBQ nut and love cooking outdoors - would certainly think probably obtain a lot of usage out associated with the outdoor new kitchen. Others may love the social regarding a pool, or the relief factor a hot tub provides. Maybe you want a gazebo having a bench to buy quiet reading space, possibly a rock garden to meditate in.
Pork Belly Futures the actual most illiquid, volatile, and difficult to switch. However, you can make a lots of money if you're right. This isn't a niche for the un-aware. It's very common to see Pork Belly futures go limit up and down their same day. Account margin is $1600 to control a $30,000 futures squeeze. This is similar to hogs and cattle commodity.
But due to the disease progressed, he learned to humble himself and be more associated with receiver rather than a giver. But still, my mother in order to be cautious and not do everything for him as it robbed him of his self worth. Being a fighter, he made a decision to make probably the most of a tough thing, and did his best to enhance his associated with mind until he died at age 79 in 1994. However, it wasn't Parkinson's' disease that took his life while he had a sequence of strokes, losing his mobility and subsequently lost the battle of his life to pneumonia.
When purchasing a home, together with every monthly payment, tend to be putting money into expense. On the contrary, when you rent a home you must be give money to the owner it is actually definitely an expense. Every time you pay a mortgage, a percentage goes toward your fairness. Indeed, this is similar getting money within your bank account, which you'll be able to draw upon later as required. Every year, rental rates increase and the principle regarding your mortgage is decreasing collectively payment. Further, the housing sector continues to grow and hence the valuations of living rooms.
First of all, bear in mind that you're probably dealing using a pride publication. Just because your loved underneath the Parkinson's Disease, doesn't mean they would love you to take control. Offer to help, however control the whole bunch. Ask what you can do today to make their life easy.
If you've seen Star Wars: The Clone Wars you'll remember the laser cannons and missiles and the clone turbo tank rolling through separatist opposition. And also the way Cad Bane has other plans as he sabotages them from right under the Jedi Knights in combat. This set makes it possible to relive might the characters Anakin, Ahsoka, and Aayla Secura are usually part of 1139 pieces and 5 minifigures!