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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, equation solving is the problem of finding what values (number
Number

A number is a mathematical object used in counting and measurement. A notational symbol which represents a number is called a Numeral system, but in common usage the word number is used for both the abstract object and the symbol, as well as for the numeral for the number....
s, functions
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
, sets, etc.) fulfill a condition stated as an equality (an equation
Equation

An equation is a mathematics Proposition, in table of mathematical symbols, that two things are exactly the same . Equations are written with an equal sign, as in...
). Usually, this condition involves expressions with variables (or unknowns), which are to be substituted by values in order for the equality to hold. More precisely, an equation involves some free variables.

In one general case, we have a situation such as
f(x1,...,xn)=c,
with c being a constant, which has a set of solutions S of the form
with Tn the domain of the function.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, equation solving is the problem of finding what values (number
Number

A number is a mathematical object used in counting and measurement. A notational symbol which represents a number is called a Numeral system, but in common usage the word number is used for both the abstract object and the symbol, as well as for the numeral for the number....
s, functions
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
, sets, etc.) fulfill a condition stated as an equality (an equation
Equation

An equation is a mathematics Proposition, in table of mathematical symbols, that two things are exactly the same . Equations are written with an equal sign, as in...
). Usually, this condition involves expressions with variables (or unknowns), which are to be substituted by values in order for the equality to hold. More precisely, an equation involves some free variables.

In one general case, we have a situation such as
f(x1,...,xn)=c,
with c being a constant, which has a set of solutions S of the form
with Tn the domain of the function. Note that the set of solutions can be empty (there are no solutions), singleton
Singleton

Singleton may refer to*Singleton , a set with exactly one element in mathematics*Singleton pattern, a Design pattern used in software engineering...
 (there is exactly 1 solution), finite, or infinite (there are infinitely many solutions).

For example, an expression such as
3x+2y=21z
can be solved by first modifying the equation in some way as to preserve the equality, such as subtracting both sides by 21z to obtain
3x+2y-21z=0


Now, it occurs that in solving this equation, that there is not just one solution to this equation, but an infinite set of solutions, which can be written
.


One particular solution is x = 20/3, y = 11, z = 2. In fact, this particular set of solutions describe a plane in three dimensions, which passes through the point (20/3, 11, 2).

Solution sets

If the solution set
Solution set

In mathematics, a solution set is a set of possible values that a variable can take on in order to satisfy a given set of conditions .Formally, for a collection of polynomials over some Ring , a solution set is defined to be the Set ....
 is empty, then there are no such xi such that
f(x0,...,xn)=c
becomes true for a given c.

For example, let us examine the classic one-variable case, given a function
consider the equation
f(x) = -1
The solution set is , in that no positive real number solves this equation. However note that in attempting to find solutions for this equation, if we modify the function's definition - more specifically, the function's domain, we can find solutions to this equation. So, if we were instead to define
g(x) = -1
has a solution set , where i is the imaginary unit
Imaginary unit

In mathematics, physics, and engineering, the imaginary unit is denoted by  or the Latin   or the Greek iota . It allows the real number system, to be extended to the complex number system,   Its precise definition is dependent upon the particular method of extension....
. This equation has exactly two solutions.

We have already seen that certain solutions sets can describe surfaces. For example, in studying elementary mathematics, one knows that the solution set of an equation in the form ax+by=c with a,b, and c real-valued constants, this forms a line in the vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
 R2. However, it may not always be easy to graphically depict solutions sets - for example, the solution set to an equation in the form ax+by+cz+dw=k (with a, b, c, d, and k real-valued constants) is a hyperplane
Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
.

Methods of solution

In simple cases, it is rather easy to solve an equation provided certain conditions are met. However, in more complicated cases, exact symbolic forms for solutions are often difficult to obtain or cumbersome to manipulate with, and an approximate numerical solution may be in fact sufficient for use.

Inverse functions

In the simple case of a function of one variable, say, h(x), we can solve an equation of the form
h(x)=c, c constant
by considering what is known as the inverse function
Inverse function

In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself....
 of h.

Given a function h : A → B, the inverse function, denoted h-1, defined as h-1 : B → A is a function such that

h-1(h(x)) = h(h-1(x)) = x.


Now, if we apply the inverse function to both sides of
h(x)=c, c constant
we obtain
h-1(h(x))=h-1(c)
x = h-1(c)


and we have found the solution to the equation. However, depending on the function, the inverse may be difficult to be defined, or may not be a function on all of the set B (only on some subset), and have many values at some point.

Examples
If the x is being added onto, we add the opposite number to both sides of the equation to solve for x. If the x is being multiplied, we multiply both sides of the equation by the reciprocal number. If x is an exponent in an exponential equation, we take the logarithm of the appropriate base of both sides of the equation. If x is the base of a power equation, we take the appropriate root of both sides of the equation. If x is the angle in a trigonometric equation, we take the inverse trig function of both sides of the equation.

Ex. 2x - 8 = 18

2x - 8 = 18 [+8] | 8 is added to both 2x - 8 and 18* ? 2x = 26 [÷2] | Both 2x and 26 are divided by 2* ? x = 13

Numerical methods

With more complicated equations, simple methods to solve equations can fail. Often, root-finding algorithm
Root-finding algorithm

A root-finding algorithm is a numerical method, or algorithm, for finding a value x such that f = 0, for a given function f. Such an x is called a root of the function f....
s like the Newton–Raphson method can be used to find a numerical solution to an equation, which within some applications can be entirely sufficient to solve some problem.

Taylor series
One well-studied area of mathematics involves examining whether we can create some simple function to approximate a more complex equation near a given point. In fact, polynomials in one or several variables can be used to approximate functions in this way - these are known as Taylor series
Taylor series

In mathematics, the Taylor series is a representation of a function as an Series of terms calculated from the values of its derivatives at a single point....
.

Solving other equations

It is important to note that we can create even more complex equations, involving differential operator
Differential operator

In mathematics, a differential operator is an operator defined as a function of the derivative operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another ....
s, matrices
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
, and so on. The underlying principle of solving equations by finding a value which satisfies the equation is maintained, but with vastly differing methodologies used to find them.