In mathematics, an equation is a formula of the form A = B, where A and B are expressions
that may contain one or several variables called unknowns, and "=" denotes the equality
binary relation. Although written in the form of proposition, an equation is not a 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.
Historically, equations arose from the mathematical discipline of algebra, but later became ubiquitous.
"Equations" should not be confused with "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 in arithmetic, and do not constitute a values-finding problem, even when variables
are present as in the latter example. The term "equation" may also refer to a 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 x2 + y2 = 1,
which means that a point belongs to the circle if and only if its coordinates are related
by this equation. Most physical laws are expressed by equations. One of the most famous ones
is Einstein's equation E = mc2. The = symbol was invented by Robert Recorde,
who considered that nothing could be more equal than parallel straight lines with the
same length.
Parameters and unknowns
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 ax2 + 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
has the unique solution x = −1, y = 1. Analogous illustration
A weighing scale, balance, or seesaw is often presented as an analogy to an equation.
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 represented by an inequation.
In the illustration, x, y and z are all different quantities 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.
Types of equations Equations can be classified according to the
types of operations and quantities involved. Important types include:
An algebraic equation or polynomial equation is an equation in which both sides are polynomials.
These are further classified by degree: linear equation for degree one
quadratic equation for degree two cubic equation for degree three
quartic equation for degree four quintic equation for degree five
A Diophantine equation is an equation where the unknowns are required to be integers
A transcendental equation is an equation involving a transcendental function of its unknowns
A parametric equation is an equation for which the solutions are sought as functions of some
other variables, called parameters appearing in the equations
A functional equation is an equation in which the unknowns are functions rather than simple
quantities A differential equation is a functional equation
involving derivatives of the unknown functions An integral equation is a functional equation
involving the antiderivatives of the unknown functions
An integro-differential equation is a functional equation involving both the derivatives and
the antiderivatives of the unknown functions 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
Identities
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: , which is true for all values of θ.
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:
where θ is known to be between zero and 45 degrees,
use the identity: so the above equation becomes:
Whence:
Properties 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:
Adding or 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.
Multiplying or dividing both sides of an equation by a non-zero constant.
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 has the solution Raising both
sides to the exponent of 2 changes the equation to , which not only has the previous solution
but also introduces the extraneous solution, Moreover, If the function is not defined at
some values, 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.
See also
References
External links Winplot: General Purpose plotter which can
draw and animate 2D and 3D mathematical equations. 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.
EqWorld—contains information on solutions to many different classes of mathematical
equations. EquationSolver: A webpage that can solve single
equations and linear equation systems. vCalc: A webpage that provides a one stop
math location with an extensive user modifiable equation library.