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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Equation</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, an <b>equation</b> is a <a href="Mathematical_formula" class="mw-redirect" title="Mathematical formula">mathematical formula</a> that expresses the <a href="Equality_(mathematics)" title="Equality (mathematics)">equality</a> of two <a href="Expression_(mathematics)" title="Expression (mathematics)">expressions</a>, by connecting them with the <a href="Equals_sign" title="Equals sign">equals sign</a> <span class="nounderlines" style="border: 1px solid var(--border-color-muted,#ddd); color: var(--color-base,#202122); background-color: var( --background-color-neutral-subtle, #fdfdfd); padding: 1px 1px;">=</span>.<sup id="cite_ref-:1_2-0" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The word <i>equation</i> and its <a href="Cognate" title="Cognate">cognates</a> in other languages may have subtly different meanings; for example, in <a href="French_language" title="French language">French</a> an <i>équation</i> is defined as containing one or more <a href="Variable_(mathematics)" title="Variable (mathematics)">variables</a>, while in <a href="English_language" title="English language">English</a>, any <a href="Well-formed_formula" title="Well-formed formula">well-formed formula</a> consisting of two expressions related with an equals sign is an equation.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p><a href="Equation_solving" title="Equation solving">Solving an equation</a> containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called <b>unknowns</b>, and the values of the unknowns that satisfy the equality are called <a href="Solution_(equation)" class="mw-redirect" title="Solution (equation)">solutions</a> of the equation. There are two kinds of equations: <a href="Identity_(mathematics)" title="Identity (mathematics)">identities</a> and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>The "<a href="%3D" class="mw-redirect" title="=">=</a>" symbol, which appears in every equation, was invented in 1557 by <a href="Robert_Recorde" title="Robert Recorde">Robert Recorde</a>, who considered that nothing could be more equal than parallel straight lines with the same length.<sup id="cite_ref-Whetstone_1-1" class="reference"><a href="#cite_note-Whetstone-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Description">Description</h2></div>

<p>An equation is written as two <a href="Expression_(mathematics)" title="Expression (mathematics)">expressions</a>, connected by an <a href="Equals_sign" title="Equals sign">equals sign</a> ("=").<sup id="cite_ref-:1_2-1" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The expressions on the two <a href="Sides_of_an_equation" title="Sides of an equation">sides</a> of the equals sign are called the "left-hand side" and "right-hand side" of the equation. Very often the right-hand side of an equation is assumed to be zero. This does not reduce the generality, as this can be realized by subtracting the right-hand side from both sides.

</p><p>The most common type of equation is a <a href="Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equation</a> (commonly called also an <i>algebraic equation</i>) in which the two sides are <a href="Polynomial" title="Polynomial">polynomials</a>.

The sides of a polynomial equation contain one or more <a href="Addition#Notation_and_terminology" title="Addition">terms</a>. For example, the equation

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<annotation encoding="application/x-tex">{\displaystyle Ax^{2}+Bx+C-y=0}</annotation>

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</math></span><img src="./48f35a43975baf07af45c8768074725fdcddb7aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.925ex; height:3.009ex;" alt="{\displaystyle Ax^{2}+Bx+C-y=0}" loading="lazy"></span></dd></dl>

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</math></span><img src="./dba4508f349e30d6fe1032237df53d03b7d5953e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.664ex; height:3.009ex;" alt="{\displaystyle Ax^{2}+Bx+C-y}" loading="lazy"></span>, which has four terms, and right-hand side <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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</math></span><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span>, consisting of just one term. The names of the <a href="Variable_(mathematics)" title="Variable (mathematics)">variables</a> suggest that <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>y</i></span> are unknowns, and that <span class="texhtml"><i>A</i></span>, <span class="texhtml"><i>B</i></span>, and <span class="texhtml"><i>C</i></span> are <a href="Parameter" title="Parameter">parameters</a>, but this is normally fixed by the context (in some contexts, <span class="texhtml mvar" style="font-style:italic;">y</span> may be a parameter, or <span class="texhtml"><i>A</i></span>, <span class="texhtml"><i>B</i></span>, and <span class="texhtml"><i>C</i></span> may be ordinary variables).

</p><p>An equation is analogous to a scale into which weights are placed. When equal weights of something (e.g., grain) are placed into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, an equal amount must be removed from the other pan to keep the scale in balance. More generally, an equation remains balanced if the same operation is performed on each side.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<p>Two equations or two systems of equations are <i>equivalent</i>, if they have the same set of solutions. The following operations transform an equation or a system of equations into an equivalent one – provided that the operations are meaningful for the expressions they are applied to:

</p>

<ul><li><a href="Addition" title="Addition">Adding</a> or <a href="Subtraction" title="Subtraction">subtracting</a> 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.</li>

<li><a href="Multiplication" title="Multiplication">Multiplying</a> or <a href="Division_(mathematics)" title="Division (mathematics)">dividing</a> both sides of an equation by a non-zero quantity.</li>

<li>Applying an <a href="Identity_(mathematics)" title="Identity (mathematics)">identity</a> to transform one side of the equation. For example, <a href="Polynomial_expansion" title="Polynomial expansion">expanding</a> a product or <a href="Factorization_of_polynomials" title="Factorization of polynomials">factoring</a> a sum.</li>

<li>For a system: adding to both sides of an equation the corresponding side of another equation, multiplied by the same quantity.</li></ul>

<p>If some <a href="Function_(mathematics)" title="Function (mathematics)">function</a> 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 <a href="Extraneous_solution" class="mw-redirect" title="Extraneous solution">extraneous solutions</a>. For example, the equation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}">

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</math></span><img src="./ee42176e76ae6b56d68c42ced807e08b962a2b54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=1}" loading="lazy"></span> has the solution <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1.}">

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<annotation encoding="application/x-tex">{\displaystyle x=1.}</annotation>

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</math></span><img src="./45c68848dcaa8574feb04951e71070f80b77f752.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.237ex; height:2.176ex;" alt="{\displaystyle x=1.}" loading="lazy"></span> Raising both sides to the exponent of 2 (which means applying the function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(s)=s^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle f(s)=s^{2}}</annotation>

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</math></span><img src="./811ad531067d574222332295705ff8220cf70a78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.421ex; height:3.176ex;" alt="{\displaystyle f(s)=s^{2}}" loading="lazy"></span> to both sides of the equation) changes the equation to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=1}">

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<annotation encoding="application/x-tex">{\displaystyle x^{2}=1}</annotation>

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</math></span><img src="./51e28d7561fa4fe556f438b88380ef98c5631cf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.645ex; height:2.676ex;" alt="{\displaystyle x^{2}=1}" loading="lazy"></span>, which not only has the previous solution but also introduces the extraneous solution, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-1.}">

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<annotation encoding="application/x-tex">{\displaystyle x=-1.}</annotation>

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</math></span><img src="./a51899e84bbbb14d044de1e69f57f73f70178b4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.046ex; height:2.343ex;" alt="{\displaystyle x=-1.}" loading="lazy"></span> Moreover, if the function is not defined at some values (such as 1/<i>x</i>, which is not defined for <i>x</i> = 0), solutions existing at those values may be lost. Thus, caution must be exercised when applying such a transformation to an equation.

</p><p>The above transformations are the basis of most elementary methods for <a href="Equation_solving" title="Equation solving">equation solving</a>, as well as some less elementary ones, like <a href="Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a>.

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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Analogous_illustration">Analogous illustration</h3></div>

<p>An equation is analogous to a <a href="Weighing_scale" title="Weighing scale">weighing scale</a>, balance, or <a href="Seesaw" title="Seesaw">seesaw</a>.

</p><p>Each side of the equation corresponds to one side of the balance; to the left, and to the right. Different quantities can be placed on each side: if the weights on the two sides are equal, the scale balances, and in analogy, the equality that represents the balance is also balanced (if not, then the lack of balance corresponds to an <a href="Inequality_(mathematics)" title="Inequality (mathematics)">inequality</a> represented by an <a href="Inequation" title="Inequation">inequation</a>).

</p><p>In the illustration, <i>x</i>, <i>y</i> and <i>z</i> are all different quantities (in this case <a href="Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>) represented as circular weights, and each of <i>x</i>, <i>y</i>, and <i>z</i> 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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<div class="mw-heading mw-heading3"><h3 id="Parameters_and_unknowns">Parameters and unknowns</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Expression_(mathematics)" title="Expression (mathematics)">Expression (mathematics)</a></div>

<p>Equations often contain terms other than the unknowns. These other terms, which are assumed to be <i>known</i>, are usually called <i>constants</i>, <i>coefficients</i> or <i>parameters</i>.

</p><p>An example of an equation involving <i>x</i> and <i>y</i> as unknowns and the parameter <i>R</i> is

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<annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=R^{2}.}</annotation>

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</math></span><img src="./fe1f27078e621122772fdeb967fe85a71be259b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.003ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=R^{2}.}" loading="lazy"></span></dd></dl>

<p>When <i>R </i>is chosen to have the value of 2 (<i>R </i>= 2), this equation would be recognized in <a href="Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> as the equation for the circle of radius of 2 around the origin. Hence, the equation with <i>R</i> unspecified is the general equation for the circle.

</p><p>Usually, the unknowns are denoted by letters at the end of the alphabet, <i>x</i>, <i>y</i>, <i>z</i>, <i>w</i>, ..., while coefficients (parameters) are denoted by letters at the beginning, <i>a</i>, <i>b</i>, <i>c</i>, <i>d</i>, ... . For example, the general <a href="Quadratic_equation" title="Quadratic equation">quadratic equation</a> is usually written <i>ax</i><sup>2</sup> + <i>bx</i> + <i>c</i> = 0.

</p><p>The process of finding the solutions, or, in case of parameters, expressing the unknowns in terms of the parameters, is called <a href="Equation_solving" title="Equation solving">solving the equation</a>. Such expressions of the solutions in terms of the parameters are also called <i>solutions</i>.

</p><p>A <a href="System_of_equations" title="System of equations">system of equations</a> is a set of <i>simultaneous equations</i>, usually in several unknowns for which the common solutions are sought. Thus, a <i>solution to the system</i> 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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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}3x+5y&=2\\5x+8y&=3\end{aligned}}}</annotation>

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</math></span><img src="./64d70f591a29304ca95853d2b2ddd15fe3b67b94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.663ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}3x+5y&=2\\5x+8y&=3\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>has the unique solution <i>x</i> = −1, <i>y</i> = 1.

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<div class="mw-heading mw-heading3"><h3 id="Identities">Identities</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Identity_(mathematics)" title="Identity (mathematics)">Identity (mathematics)</a> and <a href="List_of_trigonometric_identities" title="List of trigonometric identities">List of trigonometric identities</a></div>

<p>An identity is an equation that is true for all possible values of the variable(s) it contains. Many identities are known in algebra and calculus. In the process of solving an equation, an identity is often used to simplify an equation, making it more easily solvable.

</p><p>In algebra, an example of an identity is the <a href="Difference_of_two_squares" title="Difference of two squares">difference of two squares</a>:

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<annotation encoding="application/x-tex">{\displaystyle x^{2}-y^{2}=(x+y)(x-y)}</annotation>

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<p>which is true for all <i>x</i> and <i>y</i>.

</p><p><a href="Trigonometry" title="Trigonometry">Trigonometry</a> is an area where many identities exist; these are useful in manipulating or solving <a href="Trigonometric_equation" class="mw-redirect" title="Trigonometric equation">trigonometric equations</a>. Two of many that involve the <a href="Sine_function" class="mw-redirect" title="Sine function">sine</a> and <a href="Cosine_function" class="mw-redirect" title="Cosine function">cosine</a> functions are:

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<annotation encoding="application/x-tex">{\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1}</annotation>

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</math></span><img src="./47f3203bca6dc55c36d94ee525c44dac9e1716f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.976ex; height:3.176ex;" alt="{\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1}" loading="lazy"></span></dd></dl>

<p>and

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<annotation encoding="application/x-tex">{\displaystyle \sin(2\theta )=2\sin(\theta )\cos(\theta )}</annotation>

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</math></span><img src="./49b9120d4e69a660935b978d66f352fd2e645199.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.719ex; height:2.843ex;" alt="{\displaystyle \sin(2\theta )=2\sin(\theta )\cos(\theta )}" loading="lazy"></span></dd></dl>

<p>which are both true for all values of <i>θ</i>.

</p><p>For example, to solve for the value of <i>θ</i> that satisfies the equation:

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<annotation encoding="application/x-tex">{\displaystyle 3\sin(\theta )\cos(\theta )=1\,,}</annotation>

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</math></span><img src="./ef5afc15255c1fbbb969663c81461fde3c67f197.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.998ex; height:2.843ex;" alt="{\displaystyle 3\sin(\theta )\cos(\theta )=1\,,}" loading="lazy"></span></dd></dl>

<p>where <i>θ</i> is limited to between 0 and 45 degrees, one may use the above identity for the product to give:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {3}{2}}\sin(2\theta )=1\,,}</annotation>

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<p>yielding the following solution for <i>θ:</i>

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<mi>arcsin</mi>

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<annotation encoding="application/x-tex">{\displaystyle \theta ={\frac {1}{2}}\arcsin \left({\frac {2}{3}}\right)\approx 20.9^{\circ }.}</annotation>

</semantics>

</math></span><img src="./d051c1ae231480e303ae4e8291dfe51173147613.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.89ex; height:6.176ex;" alt="{\displaystyle \theta ={\frac {1}{2}}\arcsin \left({\frac {2}{3}}\right)\approx 20.9^{\circ }.}" loading="lazy"></span></dd></dl>

<p>Since the sine function is a <a href="Periodic_function" title="Periodic function">periodic function</a>, there are infinitely many solutions if there are no restrictions on <i>θ</i>. In this example, restricting <i>θ</i> to be between 0 and 45 degrees would restrict the solution to only one number.

</p>

<div class="mw-heading mw-heading2"><h2 id="Algebra">Algebra</h2></div>

<p><a href="Algebra" title="Algebra">Algebra</a> studies two main families of equations: <a href="Polynomial_equations" class="mw-redirect" title="Polynomial equations">polynomial equations</a> and, among them, the special case of <a href="Linear_equations" class="mw-redirect" title="Linear equations">linear equations</a>. When there is only one variable, polynomial equations have the form <i>P</i>(<i>x</i>) = 0, where <i>P</i> is a <a href="Polynomial" title="Polynomial">polynomial</a>, and linear equations have the form <i>ax</i> + <i>b</i> = 0, where <i>a</i> and <i>b</i> are <a href="Parameter#Mathematical_functions" title="Parameter">parameters</a>. To solve equations from either family, one uses algorithmic or geometric techniques that originate from <a href="Linear_algebra" title="Linear algebra">linear algebra</a> or <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>. Algebra also studies <a href="Diophantine_equations" class="mw-redirect" title="Diophantine equations">Diophantine equations</a> where the coefficients and solutions are <a href="Integers" class="mw-redirect" title="Integers">integers</a>. The techniques used are different and come from <a href="Number_theory" title="Number theory">number theory</a>. These equations are difficult in general; one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions.

</p>

<div class="mw-heading mw-heading3"><h3 id="Polynomial_equations">Polynomial equations</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Polynomial_equation" class="mw-redirect" title="Polynomial equation">Polynomial equation</a></div>

<p>In general, an <i>algebraic equation</i> or <a href="Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equation</a> is an equation of the form

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<annotation encoding="application/x-tex">{\displaystyle P=0}</annotation>

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</math></span><img src="./f6f743f37b37ce0c2ddc1db0fdca0e577c19f51d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle P=0}" loading="lazy"></span>, or</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle P=Q}</annotation>

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</math></span><img src="./2abc7e2c5a78e9e6cb7a2a907279953f9b4a3f52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.682ex; height:2.509ex;" alt="{\displaystyle P=Q}" loading="lazy"></span><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup></dd></dl>

<p>where <i>P</i> and <i>Q</i> are <a href="Polynomial" title="Polynomial">polynomials</a> with coefficients in some <a href="Field_(mathematics)" title="Field (mathematics)">field</a> (e.g., <a href="Rational_number" title="Rational number">rational numbers</a>, <a href="Real_number" title="Real number">real numbers</a>, <a href="Complex_number" title="Complex number">complex numbers</a>). An algebraic equation is <i>univariate</i> if it involves only one <a href="Variable_(mathematics)" title="Variable (mathematics)">variable</a>. On the other hand, a polynomial equation may involve several variables, in which case it is called <i>multivariate</i> (multiple variables, x, y, z, etc.).

</p><p>For example,

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<annotation encoding="application/x-tex">{\displaystyle x^{5}-3x+1=0}</annotation>

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</math></span><img src="./698e1afb9a1d47e492390b6a5a4612ea0dfff0cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.98ex; height:2.843ex;" alt="{\displaystyle x^{5}-3x+1=0}" loading="lazy"></span></dd></dl>

<p>is a univariate algebraic (polynomial) equation with integer coefficients and

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<annotation encoding="application/x-tex">{\displaystyle y^{4}+{\frac {xy}{2}}={\frac {x^{3}}{3}}-xy^{2}+y^{2}-{\frac {1}{7}}}</annotation>

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</math></span><img src="./b5e8a49bf6d100e7a8b65135f6faffd17e470ceb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.974ex; height:5.843ex;" alt="{\displaystyle y^{4}+{\frac {xy}{2}}={\frac {x^{3}}{3}}-xy^{2}+y^{2}-{\frac {1}{7}}}" loading="lazy"></span></dd></dl>

<p>is a multivariate polynomial equation over the rational numbers.

</p><p>Some polynomial equations with <a href="Rational_number" title="Rational number">rational coefficients</a> have a solution that is an <a href="Algebraic_expression" title="Algebraic expression">algebraic expression</a>, with a finite number of operations involving just those coefficients (i.e., can be <a href="Algebraic_solution" class="mw-redirect" title="Algebraic solution">solved algebraically</a>). This can be done for all such equations of <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> one, two, three, or four; but equations of degree five or more cannot always be solved in this way, as the <a href="Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a> demonstrates.

</p><p>A large amount of research has been devoted to compute efficiently accurate approximations of the <a href="Real_number" title="Real number">real</a> or <a href="Complex_number" title="Complex number">complex</a> solutions of a univariate algebraic equation (see <a href="Root_finding_of_polynomials" class="mw-redirect" title="Root finding of polynomials">Root finding of polynomials</a>) and of the common solutions of several multivariate polynomial equations (see <a href="System_of_polynomial_equations" title="System of polynomial equations">System of polynomial equations</a>).

</p>

<div class="mw-heading mw-heading3"><h3 id="Systems_of_linear_equations">Systems of linear equations</h3></div>

<p>A <a href="System_of_linear_equations" title="System of linear equations">system of linear equations</a> (or <i>linear system</i>) is a collection of <a href="Linear_equation" title="Linear equation">linear equations</a> involving one or more <a href="Variable_(math)" class="mw-redirect" title="Variable (math)">variables</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> For example,

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<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{7}3x&&\;+\;&&2y&&\;-\;&&z&&\;=\;&&1&\\2x&&\;-\;&&2y&&\;+\;&&4z&&\;=\;&&-2&\\-x&&\;+\;&&{\tfrac {1}{2}}y&&\;-\;&&z&&\;=\;&&0&\end{alignedat}}}</annotation>

</semantics>

</math></span><img src="./d691839a2b284331b58b0820654d32e101e26a03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:21.219ex; height:9.676ex;" alt="{\displaystyle {\begin{alignedat}{7}3x&&\;+\;&&2y&&\;-\;&&z&&\;=\;&&1&\\2x&&\;-\;&&2y&&\;+\;&&4z&&\;=\;&&-2&\\-x&&\;+\;&&{\tfrac {1}{2}}y&&\;-\;&&z&&\;=\;&&0&\end{alignedat}}}" loading="lazy"></span></dd></dl>

<p>is a system of three equations in the three variables <span class="texhtml"><i>x</i>, <i>y</i>, <i>z</i></span>. A <b>solution</b> to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A <a href="Equation_solving" title="Equation solving">solution</a> to the system above is given by

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<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}x&\,=\,&1\\y&\,=\,&-2\\z&\,=\,&-2\end{alignedat}}}</annotation>

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</math></span><img src="./4460866202afb67e822389a6b11a0b453c89c1c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:8.279ex; height:8.843ex;" alt="{\displaystyle {\begin{alignedat}{2}x&\,=\,&1\\y&\,=\,&-2\\z&\,=\,&-2\end{alignedat}}}" loading="lazy"></span></dd></dl>

<p>since it makes all three equations valid. The word "<i>system</i>" indicates that the equations are to be considered collectively, rather than individually.

</p><p>In mathematics, the theory of linear systems is a fundamental part of <a href="Linear_algebra" title="Linear algebra">linear algebra</a>, a subject which is used in many parts of modern mathematics. Computational <a href="Algorithm" title="Algorithm">algorithms</a> for finding the solutions are an important part of <a href="Numerical_linear_algebra" title="Numerical linear algebra">numerical linear algebra</a>, and play a prominent role in <a href="Physics" title="Physics">physics</a>, <a href="Engineering" title="Engineering">engineering</a>, <a href="Chemistry" title="Chemistry">chemistry</a>, <a href="Computer_science" title="Computer science">computer science</a>, and <a href="Economics" title="Economics">economics</a>. A <a href="Nonlinear_system" title="Nonlinear system">system of non-linear equations</a> can often be <a href="Approximation" title="Approximation">approximated</a> by a linear system (see <a href="Linearization" title="Linearization">linearization</a>), a helpful technique when making a <a href="Mathematical_model" title="Mathematical model">mathematical model</a> or <a href="Computer_simulation" title="Computer simulation">computer simulation</a> of a relatively complex system.

</p>

<div class="mw-heading mw-heading2"><h2 id="Geometry">Geometry</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Analytic_geometry">Analytic geometry</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Analytic_geometry" title="Analytic geometry">Analytic geometry</a></div>

<p>In <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, it is possible to associate a set of coordinates to each point in space, for example by an orthogonal grid. This method allows one to characterize geometric figures by equations. A plane in three-dimensional space can be expressed as the solution set of an equation of the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax+by+cz+d=0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle ax+by+cz+d=0}</annotation>

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</math></span><img src="./746e9e9c5e6a562485cbc5ed6a9375ec67bad26f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.805ex; height:2.509ex;" alt="{\displaystyle ax+by+cz+d=0}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}">

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<annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation>

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</math></span><img src="./f13f068df656c1b1911ae9f81628c49a6181194d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

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</math></span><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"></span> are real numbers and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation>

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</math></span><img src="./bbeca34b28f569a407ef74a955d041df9f360268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}" loading="lazy"></span> are the unknowns that correspond to the coordinates of a point in the system given by the orthogonal grid. The values <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation>

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</math></span><img src="./f13f068df656c1b1911ae9f81628c49a6181194d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}" loading="lazy"></span> are the coordinates of a vector perpendicular to the plane defined by the equation. A line is expressed as the intersection of two planes, that is as the solution set of a single linear equation with values in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

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</math></span><img src="./e150115ab9f63023215109595b76686a1ff890fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" loading="lazy"></span> or as the solution set of two linear equations with values in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}.}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}.}</annotation>

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</math></span><img src="./b00b2b4fd27c2cbffa02df568472f77b194a6db9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}.}" loading="lazy"></span>

</p><p>A <a href="Conic_section" title="Conic section">conic section</a> is the intersection of a <a href="Cone" title="Cone">cone</a> with equation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}=z^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=z^{2}}</annotation>

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</math></span><img src="./24defb565dbcba7850e1bfb51176bcf574b4e56b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.682ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=z^{2}}" loading="lazy"></span> and a plane. In other words, in space, all conics are defined as the solution set of an equation of a plane and of the equation of a cone just given. This formalism allows one to determine the positions and the properties of the focuses of a conic.

</p><p>The use of equations allows one to call on a large area of mathematics to solve geometric questions. The <a href="Cartesian_coordinate" class="mw-redirect" title="Cartesian coordinate">Cartesian coordinate</a> system transforms a geometric problem into an analysis problem, once the figures are transformed into equations; thus the name <a href="Analytic_geometry" title="Analytic geometry">analytic geometry</a>. This point of view, outlined by <a href="Descartes" class="mw-redirect" title="Descartes">Descartes</a>, enriches and modifies the type of geometry conceived of by the ancient Greek mathematicians.

</p><p>Currently, analytic geometry designates an active branch of mathematics. Although it still uses equations to characterize figures, it also uses other sophisticated techniques such as <a href="Functional_analysis" title="Functional analysis">functional analysis</a> and <a href="Linear_algebra" title="Linear algebra">linear algebra</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Cartesian_equations">Cartesian equations</h3></div>

<p>In <a href="Cartesian_geometry" class="mw-redirect" title="Cartesian geometry">Cartesian geometry</a>, equations are used to describe <a href="Geometric_figures" class="mw-redirect" title="Geometric figures">geometric figures</a>. As the equations that are considered, such as <a href="Implicit_equation" class="mw-redirect" title="Implicit equation">implicit equations</a> or <a href="Parametric_equation" title="Parametric equation">parametric equations</a>, have infinitely many solutions, the objective is now different: instead of giving the solutions explicitly or counting them, which is impossible, one uses equations for studying properties of figures. This is the starting idea of <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, an important area of mathematics.

</p><p>One can use the same principle to specify the position of any point in three-<a href="Dimension" title="Dimension">dimensional</a> <a href="Space_(mathematics)" title="Space (mathematics)">space</a> by the use of three Cartesian coordinates, which are the signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines).

</p><p>The invention of Cartesian coordinates in the 17th century by <a href="Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> revolutionized mathematics by providing the first systematic link between <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> and <a href="Algebra" title="Algebra">algebra</a>. Using the Cartesian coordinate system, geometric shapes (such as <a href="Curve" title="Curve">curves</a>) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 in a plane, centered on a particular point called the origin, may be described as the set of all points whose coordinates <i>x</i> and <i>y</i> satisfy the equation <span class="nowrap"><i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = 4</span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Parametric_equations">Parametric equations</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Parametric_equation" title="Parametric equation">Parametric equation</a></div>

<p>A <a href="Parametric_equation" title="Parametric equation">parametric equation</a> for a <a href="Curve" title="Curve">curve</a> expresses the <a href="Coordinates" class="mw-redirect" title="Coordinates">coordinates</a> of the points of the curve as functions of a <a href="Variable_(mathematics)" title="Variable (mathematics)">variable</a>, called a <a href="Parameter" title="Parameter">parameter</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> For example,

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</math></span><img src="./627bb5a6d2b57174bd596f1fc7990592b8e6076b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.517ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}x&=\cos t\\y&=\sin t\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>are parametric equations for the <a href="Unit_circle" title="Unit circle">unit circle</a>, where <i>t</i> is the parameter. Together, these equations are called a parametric representation of the curve.

</p><p>The notion of <i>parametric equation</i> has been generalized to <a href="Surface_(topology)" title="Surface (topology)">surfaces</a>, <a href="Manifold" title="Manifold">manifolds</a> and <a href="Algebraic_variety" title="Algebraic variety">algebraic varieties</a> of higher <a href="Dimension_of_a_manifold" class="mw-redirect" title="Dimension of a manifold">dimension</a>, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is <i>one</i> and <i>one</i> parameter is used, for surfaces dimension <i>two</i> and <i>two</i> parameters, etc.).

</p>

<div class="mw-heading mw-heading2"><h2 id="Number_theory">Number theory</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Diophantine_equations">Diophantine equations</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Diophantine_equation" title="Diophantine equation">Diophantine equation</a></div>

<p>A Diophantine equation is a <a href="Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equation</a> in two or more unknowns for which only the <a href="Integer" title="Integer">integer</a> <a href="Zero_of_a_function#Polynomial_roots" title="Zero of a function">solutions</a> are sought (an integer solution is a solution such that all the unknowns take integer values). A linear Diophantine equation is an equation between two sums of <a href="Monomials" class="mw-redirect" title="Monomials">monomials</a> of <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> zero or one. An example of linear Diophantine equation is <span class="texhtml"><i>ax</i> + <i>by</i> = <i>c</i></span> where <i>a</i>, <i>b</i>, and <i>c</i> are constants. An exponential Diophantine equation is one for which exponents of the terms of the equation can be unknowns.

</p><p>Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an <a href="Algebraic_curve" title="Algebraic curve">algebraic curve</a>, <a href="Algebraic_surface" title="Algebraic surface">algebraic surface</a>, or more general object, and ask about the <a href="Lattice_point" class="mw-redirect" title="Lattice point">lattice points</a> on it.

</p><p>The word <i>Diophantine</i> refers to the <a href="Greek_mathematics" class="mw-redirect" title="Greek mathematics">Hellenistic mathematician</a> of the 3rd century, <a href="Diophantus" title="Diophantus">Diophantus</a> of <a href="Alexandria" title="Alexandria">Alexandria</a>, who made a study of such equations and was one of the first mathematicians to introduce <a href="Mathematical_symbol" class="mw-redirect" title="Mathematical symbol">symbolism</a> into <a href="Algebra" title="Algebra">algebra</a>. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.

</p>

<div class="mw-heading mw-heading3"><h3 id="Algebraic_and_transcendental_numbers">Algebraic and transcendental numbers</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Algebraic_number" title="Algebraic number">Algebraic number</a> and <a href="Transcendental_number" title="Transcendental number">Transcendental number</a></div>

<p>An <a href="Algebraic_number" title="Algebraic number">algebraic number</a> is a number that is a solution of a non-zero <a href="Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equation</a> in one variable with <a href="Rational_number" title="Rational number">rational</a> coefficients (or equivalently — by <a href="Clearing_denominators" title="Clearing denominators">clearing denominators</a> — with <a href="Integer" title="Integer">integer</a> coefficients). Numbers such as <a href="Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> that are not algebraic are said to be <a href="Transcendental_number" title="Transcendental number">transcendental</a>. <a href="Almost_all" title="Almost all">Almost all</a> <a href="Real_number" title="Real number">real</a> and <a href="Complex_number" title="Complex number">complex</a> numbers are transcendental.

</p>

<div class="mw-heading mw-heading3"><h3 id="Algebraic_geometry">Algebraic geometry</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a></div>

<p><a href="Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a> is a branch of <a href="Mathematics" title="Mathematics">mathematics</a>, classically studying solutions of <a href="Polynomial_equations" class="mw-redirect" title="Polynomial equations">polynomial equations</a>. Modern algebraic geometry is based on more abstract techniques of <a href="Abstract_algebra" title="Abstract algebra">abstract algebra</a>, especially <a href="Commutative_algebra" title="Commutative algebra">commutative algebra</a>, with the language and the problems of <a href="Geometry" title="Geometry">geometry</a>.

</p><p>The fundamental objects of study in algebraic geometry are <a href="Algebraic_variety" title="Algebraic variety">algebraic varieties</a>, which are geometric manifestations of <a href="Solution_set" title="Solution set">solutions</a> of <a href="Systems_of_polynomial_equations" class="mw-redirect" title="Systems of polynomial equations">systems of polynomial equations</a>. Examples of the most studied classes of algebraic varieties are: <a href="Plane_algebraic_curve" class="mw-redirect" title="Plane algebraic curve">plane algebraic curves</a>, which include <a href="Line_(geometry)" title="Line (geometry)">lines</a>, <a href="Circle" title="Circle">circles</a>, <a href="Parabola" title="Parabola">parabolas</a>, <a href="Ellipse" title="Ellipse">ellipses</a>, <a href="Hyperbola" title="Hyperbola">hyperbolas</a>, <a href="Cubic_curve" class="mw-redirect" title="Cubic curve">cubic curves</a> like <a href="Elliptic_curve" title="Elliptic curve">elliptic curves</a> and quartic curves like <a href="Lemniscate_of_Bernoulli" title="Lemniscate of Bernoulli">lemniscates</a>, and <a href="Cassini_oval" title="Cassini oval">Cassini ovals</a>. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the <a href="Singular_point_of_a_curve" title="Singular point of a curve">singular points</a>, the <a href="Inflection_point" title="Inflection point">inflection points</a> and the <a href="Point_at_infinity" title="Point at infinity">points at infinity</a>. More advanced questions involve the <a href="Topology" title="Topology">topology</a> of the curve and relations between the curves given by different equations.

</p>

<div class="mw-heading mw-heading2"><h2 id="Differential_equations">Differential equations</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Differential_equation" title="Differential equation">Differential equation</a></div>

<p>A <a href="Differential_equation" title="Differential equation">differential equation</a> is a <a href="Mathematics" title="Mathematics">mathematical</a> equation that relates some <a href="Function_(mathematics)" title="Function (mathematics)">function</a> with its <a href="Derivative" title="Derivative">derivatives</a>. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. They are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in areas such as physics, chemistry, biology, and economics.

</p><p>In <a href="Pure_mathematics" title="Pure mathematics">pure mathematics</a>, differential equations are studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.

</p><p>If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of <a href="Dynamical_systems" class="mw-redirect" title="Dynamical systems">dynamical systems</a> puts emphasis on qualitative analysis of systems described by differential equations, while many <a href="Numerical_methods" class="mw-redirect" title="Numerical methods">numerical methods</a> have been developed to determine solutions with a given degree of accuracy.

</p>

<div class="mw-heading mw-heading3"><h3 id="Ordinary_differential_equations">Ordinary differential equations</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a></div>

<p>An <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> or ODE is an equation containing a function of one <a href="Independent_variable" class="mw-redirect" title="Independent variable">independent variable</a> and its derivatives. The term "<i>ordinary</i>" is used in contrast with the term <a href="Partial_differential_equation" title="Partial differential equation">partial differential equation</a>, which may be with respect to <i>more than</i> one independent variable.

</p><p>Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined and understood, and exact closed-form solutions are obtained. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by <a href="Elementary_functions" class="mw-redirect" title="Elementary functions">elementary functions</a> in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical and <a href="Numerical_ordinary_differential_equations" class="mw-redirect" title="Numerical ordinary differential equations">numerical</a> methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analytic solutions.

</p>

<div class="mw-heading mw-heading3"><h3 id="Partial_differential_equations">Partial differential equations</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></div>

<p>A <a href="Partial_differential_equation" title="Partial differential equation">partial differential equation</a> (PDE) is a <a href="Differential_equation" title="Differential equation">differential equation</a> that contains unknown <a href="Multivariable_calculus" title="Multivariable calculus">multivariable functions</a> and their <a href="Partial_derivative" title="Partial derivative">partial derivatives</a>. (This is in contrast to <a href="Ordinary_differential_equations" class="mw-redirect" title="Ordinary differential equations">ordinary differential equations</a>, which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant <a href="Computer_model" class="mw-redirect" title="Computer model">computer model</a>.

</p><p>PDEs can be used to describe a wide variety of phenomena such as <a href="Sound" title="Sound">sound</a>, <a href="Heat" title="Heat">heat</a>, <a href="Electrostatics" title="Electrostatics">electrostatics</a>, <a href="Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</a>, <a href="Fluid_flow" class="mw-redirect" title="Fluid flow">fluid flow</a>, <a href="Elasticity_(physics)" title="Elasticity (physics)">elasticity</a>, or <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional <a href="Dynamical_systems" class="mw-redirect" title="Dynamical systems">dynamical systems</a>, partial differential equations often model <a href="Multidimensional_systems" class="mw-redirect" title="Multidimensional systems">multidimensional systems</a>. PDEs find their generalisation in <a href="Stochastic_partial_differential_equations" class="mw-redirect" title="Stochastic partial differential equations">stochastic partial differential equations</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Types_of_equations">Types of equations</h2></div>

<p>Equations can be classified according to the types of <a href="Operation_(mathematics)" title="Operation (mathematics)">operations</a> and quantities involved. Important types include:

</p>

<ul><li>An <a href="Algebraic_equation" title="Algebraic equation">algebraic equation</a> or <a href="Polynomial" title="Polynomial">polynomial</a> equation is an equation in which both sides are polynomials (see also <a href="System_of_polynomial_equations" title="System of polynomial equations">system of polynomial equations</a>). These are further classified by <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a>:

<ul><li><a href="Linear_equation" title="Linear equation">linear equation</a> for degree one</li>

<li><a href="Quadratic_equation" title="Quadratic equation">quadratic equation</a> for degree two</li>

<li><a href="Cubic_equation" title="Cubic equation">cubic equation</a> for degree three</li>

<li><a href="Quartic_equation" title="Quartic equation">quartic equation</a> for degree four</li>

<li><a href="Quintic_equation" class="mw-redirect" title="Quintic equation">quintic equation</a> for degree five</li>

<li><a href="Sextic_equation" title="Sextic equation">sextic equation</a> for degree six</li>

<li><a href="Septic_equation" title="Septic equation">septic equation</a> for degree seven</li>

<li><a href="Octic_equation" class="mw-redirect" title="Octic equation">octic equation</a> for degree eight</li></ul></li>

<li>A <a href="Diophantine_equation" title="Diophantine equation">Diophantine equation</a> is an equation where the unknowns are required to be <a href="Integer" title="Integer">integers</a></li>

<li>A <a href="Transcendental_equation" title="Transcendental equation">transcendental equation</a> is an equation involving a <a href="Transcendental_function" title="Transcendental function">transcendental function</a> of its unknowns</li>

<li>A <a href="Parametric_equation" title="Parametric equation">parametric equation</a> is an equation in which the solutions for the variables are expressed as functions of some other variables, called <a href="Parameter" title="Parameter">parameters</a> appearing in the equations</li>

<li>A <a href="Functional_equation" title="Functional equation">functional equation</a> is an equation in which the unknowns are <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> rather than simple quantities</li>

<li>Equations involving derivatives, integrals and finite differences:

<ul><li>A <a href="Differential_equation" title="Differential equation">differential equation</a> is a functional equation involving <a href="Derivative" title="Derivative">derivatives</a> of the unknown functions, where the function and its derivatives are evaluated at the same point, such as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)=x^{2}}">

<msup>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f'(x)=x^{2}}</annotation>

</semantics>

</math></span><img src="./bc93d64f68cd72d9f3cd1a4af2f8c6c8490cdeec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.627ex; height:3.176ex;" alt="{\displaystyle f'(x)=x^{2}}" loading="lazy"></span>. Differential equations are subdivided into <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a> for functions of a single variable and <a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a> for functions of multiple variables</li>

<li>An <a href="Integral_equation" title="Integral equation">integral equation</a> is a functional equation involving the <a href="Antiderivative" title="Antiderivative">antiderivatives</a> of the unknown functions. For functions of one variable, such an equation differs from a differential equation primarily through a change of variable substituting the function by its derivative, however this is not the case when the integral is taken over an open surface</li>

<li>An <a href="Integro-differential_equation" title="Integro-differential equation">integro-differential equation</a> is a functional equation involving both the <a href="Derivative" title="Derivative">derivatives</a> and the <a href="Antiderivative" title="Antiderivative">antiderivatives</a> of the unknown functions. For functions of one variable, such an equation differs from integral and differential equations through a similar change of variable.</li>

<li>A <a href="Functional_differential_equation" title="Functional differential equation">functional differential equation</a> of <a href="Delay_differential_equation" title="Delay differential equation">delay differential equation</a> is a function equation involving <a href="Derivative" title="Derivative">derivatives</a> of the unknown functions, evaluated at multiple points, such as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)=f(x-2)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f'(x)=f(x-2)}</annotation>

</semantics>

</math></span><img src="./551d5554d0f762372d809cfeec19057805b4793c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.663ex; height:3.009ex;" alt="{\displaystyle f'(x)=f(x-2)}" loading="lazy"></span></li>

<li>A <a href="Difference_equation" class="mw-redirect" title="Difference equation">difference equation</a> is an equation where the unknown is a function <i>f</i> that occurs in the equation through <i>f</i>(<i>x</i>), <i>f</i>(<i>x</i>−1), ..., <i>f</i>(<i>x</i>−<i>k</i>), for some whole integer <i>k</i> called the <i>order</i> of the equation. If <i>x</i> is restricted to be an integer, a difference equation is the same as a <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a></li>

<li>A <a href="Stochastic_differential_equation" title="Stochastic differential equation">stochastic differential equation</a> is a differential equation in which one or more of the terms is a <a href="Stochastic_process" title="Stochastic process">stochastic process</a></li></ul></li></ul>

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<ul><li><a href="Formula" title="Formula">Formula</a></li>

<li><a href="History_of_algebra" title="History of algebra">History of algebra</a></li>

<li><a href="Indeterminate_equation" class="mw-redirect" title="Indeterminate equation">Indeterminate equation</a></li>

<li><a href="List_of_equations" title="List of equations">List of equations</a></li>

<li><a href="List_of_scientific_equations_named_after_people" title="List of scientific equations named after people">List of scientific equations named after people</a></li>

<li><a href="Term_(logic)" title="Term (logic)">Term (logic)</a></li>

<li><a href="Theory_of_equations" title="Theory of equations">Theory of equations</a></li>

<li><a href="Cancelling_out" title="Cancelling out">Cancelling out</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">As such an equation can be rewritten <span class="texhtml"><i>P</i> – <i>Q</i> = 0</span>, many authors do not consider this case explicitly.</span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">The subject of this article is basic in mathematics, and is treated in a lot of textbooks. Among them, Lay 2005, Meyer 2001, and Strang 2005 contain the material of this article.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-Whetstone-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Whetstone_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Whetstone_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Recorde, Robert, <i>The Whetstone of Witte</i> ... (London, England: Jhon Kyngstone, 1557), <a rel="nofollow" class="external text" href="https://archive.org/stream/TheWhetstoneOfWitte#page/n237/mode/2up">the third page of the chapter "The rule of equation, commonly called Algebers Rule."</a></span>

</li>

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathopenref.com/equation.html">"Equation - Math Open Reference"</a>. <i>www.mathopenref.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-09-01</span></span>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/algebra/equation-formula.html">"Equations and Formulas"</a>. <i>www.mathsisfun.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-09-01</span></span>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFMarcusWatt" class="citation web cs1"><a href="Solomon_Marcus" title="Solomon Marcus">Marcus, Solomon</a>; Watt, Stephen M. <a rel="nofollow" class="external text" href="https://www.academia.edu/3287674">"What is an Equation?"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-02-27</span></span>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFLachaud2025" class="citation book cs1 cs1-prop-foreign-lang-source">Lachaud, Gilles (29 January 2025). <a rel="nofollow" class="external text" href="http://www.universalis.fr/encyclopedie/NT01240/EQUATION_mathematique.htm">"Équation, mathématique"</a>. <i>Encyclopædia Universalis</i> (in French).</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">"A statement of equality between two expressions. Equations are of two types, <b>identities</b> and <b>conditional equations</b> (or usually simply "equations")". « <i>Equation</i> », in <i><span title="English-language text"><span lang="en">Mathematics Dictionary</span></span></i>, Glenn James et <a href="Robert_C._James" title="Robert C. James">Robert C. James</a> (éd.), Van Nostrand, 1968, 3 ed. 1st ed. 1948, p. 131.</span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://blendedlearningmath.com/pages/mathematical-equations-guide/">"Math equations guide with rules and interesting examples"</a>. <i>blendedlearningmath</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-12-02</span></span>.</cite></span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Thomas, George B., and Finney, Ross L., <i>Calculus and Analytic Geometry</i>, Addison Wesley Publishing Co., fifth edition, 1979, p. 91.</span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Weisstein, Eric W. "Parametric Equations." From MathWorld--A Wolfram Web Resource. <a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/ParametricEquations.html">http://mathworld.wolfram.com/ParametricEquations.html</a></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20090816161008/http://math.exeter.edu/rparris/winplot.html">Winplot</a>: General Purpose plotter that can draw and animate 2D and 3D mathematical equations.</li>

<li><a rel="nofollow" class="external text" href="http://www.cs.cornell.edu/w8/~andru/relplot">Equation plotter</a>: A web page for producing and downloading pdf or postscript plots of the solution sets to equations and inequations in two variables (<i>x</i> and <i>y</i>).</li></ul>

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