Editing Equation solving (section)

== Methods of solution ==

The methods for solving equations generally depend on the type of equation, both the kind of expressions in the equation and the kind of values that may be assumed by the unknowns. The variety in types of equations is large, and so are the corresponding methods. Only a few specific types are mentioned below.

In general, given a class of equations, there may be no known systematic method ([[algorithm]]) that is guaranteed to work. This may be due to a lack of mathematical knowledge; some problems were only solved after centuries of effort. But this also reflects that, in general, no such method can exist: some problems are known to be [[Unsolvable problem|unsolvable]] by an algorithm, such as [[Hilbert's tenth problem]], which was proved unsolvable in 1970.

For several classes of equations, algorithms have been found for solving them, some of which have been implemented and incorporated in [[computer algebra system]]s, but often require no more sophisticated technology than pencil and paper. In some other cases, [[heuristic]] methods are known that are often successful but that are not guaranteed to lead to success.

===Brute force, trial and error, inspired guess===
If the solution set of an equation is restricted to a finite set (as is the case for equations in [[modular arithmetic]], for example), or can be limited to a finite number of possibilities (as is the case with some [[Diophantine equation]]s), the solution set can be found by [[Brute-force search|brute force]], that is, by testing each of the possible values ([[candidate solutions]]). It may be the case, though, that the number of possibilities to be considered, although finite, is so huge that an [[exhaustive search]] is not practically feasible; this is, in fact, a requirement for strong [[encryption]] methods.

As with all kinds of [[problem solving]], [[trial and error]] may sometimes yield a solution, in particular where the form of the equation, or its similarity to another equation with a known solution, may lead to an "inspired guess" at the solution. If a guess, when tested, fails to be a solution, consideration of the way in which it fails may lead to a modified guess.

===Elementary algebra===
Equations involving linear or simple rational functions of a single real-valued unknown, say {{mvar|x}}, such as

:<math>8x+7=4x+35  \quad \text{or} \quad \frac{4x + 9}{3x + 4} = 2 \, ,</math>

can be solved using the methods of [[elementary algebra]].

===Systems of linear equations===
Smaller [[System of linear equations|systems of linear equations]] can be solved likewise by methods of elementary algebra. For solving larger systems, algorithms are used that are based on [[linear algebra]]. ''See [[Gaussian elimination]] and [[numerical solution of linear systems]].''

===Polynomial equations===
{{Main|Polynomial#Solving polynomial equations|l1=Solving polynomial equations}}
{{see also|System of polynomial equations}}
[[Polynomial]] equations of degree up to four can be solved exactly using algebraic methods, of which the [[quadratic formula]] is the simplest example. Polynomial equations with a degree of five or higher require in general numerical methods (see below) or special functions such as [[Bring radical]]s, although some specific cases may be solvable algebraically, for example
:<math>4x^5 - x^3 - 3 = 0</math>
(by using the [[rational root theorem]]), and
:<math>x^6 - 5x^3 + 6 = 0 \, ,</math>
(by using the substitution {{math|''x'' {{=}} ''z''<sup>{{frac|1|3}}</sup>}}, which simplifies this to a [[quadratic equation]] in {{mvar|z}}).

===Diophantine equations===
In [[Diophantine equations]] the solutions are required to be [[integer]]s. In some cases a brute force approach can be used, as mentioned above. In some other cases, in particular if the equation is in one unknown, it is possible to solve the equation for [[Rational number|rational]]-valued unknowns (see [[Rational root theorem]]), and then find solutions to the Diophantine equation by restricting the solution set to integer-valued solutions. For example, the polynomial equation
:<math>2x^5-5x^4-x^3-7x^2+2x+3=0\,</math>
has as rational solutions {{math|''x'' {{=}} −{{sfrac|1|2}}}} and {{math|''x'' {{=}} 3}}, and so, viewed as a Diophantine equation, it has the unique solution {{math|''x'' {{=}} 3}}.

In general, however, Diophantine equations are among the most difficult equations to solve.

===Inverse functions===
{{See also|Inverse problem}}
In the simple case of a function of one variable, say, {{math|''h''(''x'')}}, we can solve an equation of the form {{math|''h''(''x'') {{=}} ''c''}} for some constant {{mvar|c}} by considering what is known as the ''[[inverse function]]'' of {{mvar|h}}.

Given a function {{math|''h'' : ''A'' → ''B''}}, the inverse function, denoted {{math|''h''<sup>−1</sup>}} and defined as {{math|''h''<sup>−1</sup> : ''B'' → ''A''}}, is a function such that

:<math>h^{-1}\bigl(h(x)\bigr) = h\bigl(h^{-1}(x)\bigr) = x \,.</math>

Now, if we apply the inverse function to both sides of {{math|''h''(''x'') {{=}} ''c''}}, where {{mvar|c}} is a constant value in {{mvar|B}}, we obtain

:<math>\begin{align}
h^{-1}\bigl(h(x)\bigr) &= h^{-1}(c) \\
x &= h^{-1}(c) \\
\end{align}</math>

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 {{math|B}} (only on some subset), and have many values at some point.

If just one solution will do, instead of the full solution set, it is actually sufficient if only the functional identity

:<math>h\left(h^{-1}(x)\right) = x</math>

holds. For example, the [[projection (mathematics)|projection]] {{math|π<sub>1</sub> : '''R'''<sup>2</sup> → '''R'''}} defined by {{math|1=π<sub>1</sub>(''x'', ''y'') = ''x''}} has no post-inverse, but it has a pre-inverse {{math|π{{su|b=1|p=−1}}}} defined by {{math|1=π{{su|b=1|p=−1}}(''x'') = (''x'', 0)}}. Indeed, the equation {{math|π<sub>1</sub>(''x'', ''y'') {{=}} ''c''}} is solved by

:<math>(x,y) = \pi_1^{-1}(c) = (c,0).</math>

Examples of inverse functions include the [[nth root|{{mvar|n}}th root]] (inverse of {{math|''x''<sup>''n''</sup>}}); the [[logarithm]] (inverse of {{math|''a''<sup>''x''</sup>}}); the [[inverse trigonometric function]]s; and [[Lambert's W function|Lambert's {{mvar|W}} function]] (inverse of {{math|''xe''<sup>''x''</sup>}}).

===Factorization===
If the left-hand side expression of an equation {{math|''P'' {{=}} 0}} can be [[factorization|factorized]] as {{math|''P'' {{=}} ''QR''}}, the solution set of the original solution consists of the union of the solution sets of the two equations {{math|''Q'' {{=}} 0}} and {{math|''R'' {{=}} 0}}.
For example, the equation
:<math>\tan x + \cot x = 2</math>
can be rewritten, using the identity {{math|1=tan ''x'' cot ''x'' = 1}}  as
:<math>\frac{\tan^2 x  -2 \tan x+1}{\tan x} = 0,</math>
which can be factorized into
:<math>\frac{\left(\tan x - 1\right)^2}{\tan x}= 0.</math>
The solutions are thus the solutions of the equation {{math|1=tan ''x'' = 1}}, and are thus the set
:<math>x = \tfrac{\pi}{4} + k\pi, \quad k = 0, \pm 1, \pm 2, \ldots.</math>

===Numerical methods===
With more complicated equations in real or [[complex number]]s, simple methods to solve equations can fail. Often, [[root-finding algorithm]]s like the [[Newton–Raphson method]] can be used to find a numerical solution to an equation, which, for some applications, can be entirely sufficient to solve some problem.
There are also [[Numerical_linear_algebra#Solutions_of_linear_systems|numerical methods for systems of linear equations]].

===Matrix equations===
Equations involving [[matrix (mathematics)|matrices]] and [[Vector (mathematics and physics)|vectors]] of [[real number]]s can often be solved by using methods from [[linear algebra]].

===Differential equations===
There is a vast body of methods for solving various kinds of [[differential equation]]s, both [[Numerical mathematics|numerically]] and [[Calculus|analytically]]. A particular class of problem that can be considered to belong here is [[integral|integration]], and the analytic methods for solving this kind of problems are now called [[symbolic integration]].{{citation needed|date=July 2019}} Solutions of differential equations can be ''[[implicit function|implicit]]'' or ''explicit''.<ref name="Zill2012">{{cite book|author=Dennis G. Zill|title=A First Course in Differential Equations with Modeling Applications|url=https://books.google.com/books?id=pasKAAAAQBAJ&q=solution|date=15 March 2012|publisher=Cengage Learning|isbn=978-1-285-40110-2}}</ref>