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# equation

[ih-kwey-zhuhn, -shuhn]
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 left-hand, or first, member of the equation, that to the right (5) the right-hand, 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=x2+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 ax2+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.
equation, chemical: see chemical equation.

Any of a class of equations that relate the pressure math.P, volume math.V, and temperature math.T of a given substance in thermodynamic equilibrium. For example, the equation math.Pmath.V = math.nmath.Rmath.T, where math.n is the number of moles of gas and math.R is the universal gas constant, relates the pressure, volume, and temperature of a perfect gas. Real gases, solids, and liquids have more complicated equations of state. Seealso thermodynamics.

Algebraic equation of particular importance in optimization. A more descriptive name is second-degree polynomial equation. Its standard form is math.amath.x2 + math.bmath.x + math.c = 0, and its solution is given by the quadratic formula which guarantees two real-number solutions, one real-number solution, or two complex-number solutions, depending on whether the discriminate, math.b2 − 4math.amath.c, is greater than, equal to, or less than 0.

In mathematics, an equation that contains partial derivatives, expressing a process of change that depends on more than one independent variable. It can be read as a statement about how a process evolves without specifying the formula defining the process. Given the initial state of the process (such as its size at time zero) and a description of how it is changing (i.e., the partial differential equation), its defining formula can be found by various methods, most based on integration. Important partial differential equations include the heat equation, the wave equation, and Laplace's equation, which are central to mathematical physics.

Equation containing derivatives of a function of a single variable. Its order is the order of the highest derivative it contains (e.g., a first-order differential equation involves only the first derivative of the function). Because the derivative is a rate of change, such an equation states how a function changes but does not specify the function itself. Given sufficient initial conditions, however, such as a specific function value, the function can be found by various methods, most based on integration.

Mathematical formula that describes the motion of a body relative to a given frame of reference, in terms of the position, velocity, or acceleration of the body. In classical mechanics, the basic equation of motion is Newton's second law (see Newton's laws of motion), which relates the force on a body to its mass and acceleration. When the force is described in terms of the time interval over which it is applied, the velocity and position of the body can be derived. Other equations of motion include the position-time equation, the velocity-time equation, and the acceleration-time equation of a moving body.

Relationship between mass (math.m) and energy (math.E) in Albert Einstein's special theory of relativity, expressed math.E = math.mmath.c2, where math.c equals 186,000 mi/second (300,000 km/second), the speed of light. Whereas mass and energy were viewed as distinct in earlier physical theories, in special relativity a body's mass can be converted into energy in accordance with Einstein's formula. Such a release of energy decreases the body's mass (see conservation law).

In mathematics, an equation with an unknown function within an integral. An example is where math.f(math.x) is known and phiv(math.t) is to be found, given certain conditions on math.f. Such equations are useful in solving differential equations.

Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions. Differential equations are very common in physics, engineering, and all fields involving quantitative study of change. They are used whenever a rate of change is known but the process giving rise to it is not. The solution of a differential equation is generally a function whose derivatives satisfy the equation. Differential equations are classified into several broad categories. The most important are ordinary differential equations (ODEs), in which change depends on a single variable, and partial differential equations (PDEs), in which change depends on several variables. Seealso differentiation.

Equation involving differences between successive values of a function of a discrete variable (i.e., one defined for a sequence of values that differ by the same amount, usually 1). A function of such a variable is a rule for assigning values in sequence to it. For example, math.f(math.x + 1) = math.xmath.f(math.x) is a difference equation. Methods developed for solving such equations have much in common with methods for solving linear differential equations, which difference equations are often used to approximate.

Mathematical statement of equality between algebraic expressions. An expression is algebraic if it involves a finite combination of numbers and variables and algebraic operations (addition, subtraction, multiplication, division, raising to a power, and extracting a root). Two important types of such equations are linear equations, in the form math.y = math.amath.x + math.b, and quadratic equations, in the form math.y = math.amath.x2 + math.bmath.x + math.c. A solution is a numerical value that makes the equation a true statement when substituted for a variable. In some cases it may be found using a formula; in others the equation may be rewritten in simpler form. Algebraic equations are particularly useful for modeling real-life phenomena.

An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent). Equations are written with an equal sign, as in

$2 + 3 = 5,.$

The equation above is an example of an equality: a proposition which states that two constants are equal. Equalities may be true or false.

Equations are often used to state the equality of two expressions containing one or more variables. In the reals we can say, for example, that for any given value of $x$ it is true that

$x \left(x-1\right) = x^2-x,.$

The equation above is an example of an identity, that is, an equation that is true regardless of the values of any variables that appear in it. The following equation is not an identity:

$x^2-x = 0,.$

It is false for an infinite number of values of $x$, and true for only two, the roots or solutions of the equation, $x=0$ and $x=1$. Therefore, if the equation is known to be true, it carries information about the value of $x.$ To solve an equation means to find its solutions.

Many authors reserve the term equation for an equality which is not an identity. The distinction between the two concepts can be subtle; for example,

$\left(x + 1\right)^2 = x^2 + 2x + 1,$
is an identity, while
$\left(x + 1\right)^2 = 2x^2 + x + 1,$
is an equation, whose roots are $x=0$ and $x=1$. Whether a statement is meant to be an identity or an equation, carrying information about its variables can usually be determined from its context.

Letters from the beginning of the alphabet like a, b, c... often denote constants in the context of the discussion at hand, while letters from end of the alphabet, like x, y, z..., are usually reserved for the variables, a convention initiated by Descartes.

## Properties

If an equation in algebra is known to be true, the following operations may be used to produce another true equation:

1. Any quantity can be added to both sides.
2. Any quantity can be subtracted from both sides.
3. Any quantity can be multiplied to both sides.
4. Any nonzero quantity can divide both sides.
5. Generally, any function can be applied to both sides. (However, caution must be exercised to ensure that one does not encounter extraneous solutions.)

The algebraic properties (1-4) imply that equality is a congruence relation for a field; in fact, it is essentially the only one.

The most well known system of numbers which allows all of these operations is the real numbers, which is an example of a field. However, if the equation were based on the natural numbers for example, some of these operations (like division and subtraction) may not be valid as negative numbers and non-whole numbers are not allowed. The integers are an example of an integral domain which does not allow all divisions as, again, whole numbers are needed. However, subtraction is allowed, and is the inverse operator in that system.

If a function that is not injective is applied to both sides of a true equation, then the resulting equation will still be true, but it may be less useful. Formally, one has an implication, not an equivalence, so the solution set may get larger. The functions implied in properties (1), (2), and (4) are always injective, as is (3) if we do not multiply by zero. Some generalized products, such as a dot product, are never injective.

• Mathematical equation plotter: Plots 2D mathematical equations, computes integrals, and finds solutions online.
• 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).
• WZGrapher: A Windows freeware program that plots Cartesian and polar equations, with both integration and differentiation solvers and graphing capabilities.
• EqWorld — contains information on solutions to many different classes of mathematical equations.
• EquationSolver: A webpage that can solve single equations and linear equation systems.
• WebGraphing.com: Online Equation Plotter with Automatic Table of Coordinates
• Online Equation Solver

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