# equation

Oldest printed equation (1557), in today's notation "14x + 15 = 71"

In mathematics, an equation is understood to mean a statement about the equality of two terms , which is symbolized with the help of the equal sign ("="). Formally, an equation has the form

${\ displaystyle T_ {1} = T_ {2}}$,

where the term is called the left side and the term is called the right side of the equation. Equations are either true or satisfied (for example ) or false (for example ). If at least one of the terms of variables depends, is just a form of expression before; whether the equation is true or false then depends on the specific values ​​used. The values ​​of the variables for which the equation is satisfied are called solutions of the equation. If two or more equations are given, one also speaks of a system of equations${\ displaystyle T_ {1}}$${\ displaystyle T_ {2}}$${\ displaystyle 1 = 1}$${\ displaystyle 1 = 2}$${\ displaystyle T_ {1}, T_ {2}}$, a solution to it must satisfy all equations simultaneously.

## Types of equations

Equations are used in many contexts; accordingly there are different ways of dividing the equations according to different aspects. The respective classifications are largely independent of one another; an equation can fall into several of these groups. For example, it makes sense to speak of a system of linear partial differential equations.

### Classification according to validity

#### Identity equations

Equations can be generally valid, i.e. they can be true by inserting all variable values ​​from a given basic set or at least from a previously defined subset thereof. The general validity can either be proved with other axioms or it can itself be assumed as an axiom.

Examples are:

• the Pythagorean theorem : is true for right triangles , if the side opposite the right angle ( hypotenuse ) and the cathetus denote${\ displaystyle a ^ {2} + b ^ {2} = c ^ {2}}$${\ displaystyle c}$${\ displaystyle a, b}$
• the associative law : is true for all natural numbers and generally for any elements of a group (as an axiom)${\ displaystyle (a + b) + c = a + (b + c)}$ ${\ displaystyle a, b, c}$${\ displaystyle a, b, c}$
• the first binomial formula : is true for all real numbers${\ displaystyle (a + b) ^ {2} = a ^ {2} + 2ab + b ^ {2}}$ ${\ displaystyle a, b}$
• the Euler's identity : is true for all real${\ displaystyle e ^ {i \ varphi} = \ cos \ left (\ varphi \ right) + i \ sin \ left (\ varphi \ right)}$${\ displaystyle \ varphi}$

In this context one speaks of a mathematical proposition or law. To distinguish between equations that are not generally valid, the congruence sign ("≡") is used for identities instead of the equal sign.

#### Determining equations

Often one task is to determine all variable assignments for which the equation becomes true. This process is known as solving the equation . To distinguish between identity equations, such equations are referred to as determining equations . The set of variable assignments for which the equation is true is called the solution set of the equation. If the solution set is the empty set , the equation is called unsolvable or unsatisfiable.

Whether an equation is solvable or not can depend on the basic set considered, for example:

• the equation is unsolvable as an equation over the natural or the rational numbers and has the solution set as an equation over the real numbers${\ displaystyle x ^ {2} = 2}$${\ displaystyle \ lbrace {\ sqrt {2}}, - {\ sqrt {2}} \ rbrace}$
• the equation is unsolvable as an equation over the real numbers and has the solution set as an equation over the complex numbers${\ displaystyle x ^ {2} = - 2}$${\ displaystyle \ lbrace {\ sqrt {2}} i, - {\ sqrt {2}} i \ rbrace}$

In the case of determining equations, variables sometimes appear that are not sought but are assumed to be known. Such variables are called parameters . For example, the formula for solving the quadratic equation is

${\ displaystyle x ^ {2} + px + q \; = \; 0}$

for unknown unknowns and given parameters and${\ displaystyle x}$${\ displaystyle p}$${\ displaystyle q}$

${\ displaystyle x_ {1,2} \; = \; - {\ frac {p} {2}} \ pm {\ sqrt {{{\ frac {p ^ {2}} {4}} - q}}}$.

If you insert one of the two solutions into the equation, the equation is transformed into an identity, i.e. it becomes a true statement for any choice of and . For here the solutions are real, otherwise complex. ${\ displaystyle x_ {1}, x_ {2}}$${\ displaystyle p}$${\ displaystyle q}$${\ displaystyle 4q \ leq p ^ {2}}$

#### Equations of definition

Equations can also be used to define a new symbol. In this case, the symbol to be defined is written on the left, and the equal sign is often replaced by the definition sign (“: =”) or written over the equal sign “def”.

For example, which is derivative of a function at a position by ${\ displaystyle f}$${\ displaystyle x_ {0}}$

${\ displaystyle f '(x_ {0}): = \ lim _ {x \ to x_ {0}} {\ frac {f (x) -f (x_ {0})} {x-x_ {0}} }}$

Are defined. In contrast to identities, definitions are not statements; so they are neither true nor false, just more or less useful.

### Division on the right

#### Homogeneous equations

A defining equation of form

${\ displaystyle T (x) = 0}$

is called a homogeneous equation . If a function is , the solution is also called the zero of the function. Homogeneous equations play an important role in the solution structure of linear systems of equations and linear differential equations . If the right side of an equation is not equal to zero, the equation is said to be inhomogeneous. ${\ displaystyle T}$${\ displaystyle x}$

#### Fixed point equations

A defining equation of form

${\ displaystyle T (x) = x}$

is called the fixed point equation and its solution is called the fixed point of the equation. Fixed point theorems give more precise information about the solutions of such equations . ${\ displaystyle x}$

#### Eigenvalue problems

A defining equation of form

${\ displaystyle T (x) = \ lambda x}$

is called the eigenvalue problem, where the constant (the eigenvalue) and the unknown (the eigenvector) are sought together. Eigenvalue problems have diverse areas of application in linear algebra, for example in the analysis and decomposition of matrices , and in areas of application, for example structural mechanics and quantum mechanics . ${\ displaystyle \ lambda}$${\ displaystyle x \ neq 0}$

### Classification according to linearity

#### Linear equations

An equation is called linear if it is in the form

${\ displaystyle T \ left (x \ right) = a}$

can be brought, where the term is independent of and the term is linear in , so ${\ displaystyle a}$${\ displaystyle x}$${\ displaystyle T (x)}$${\ displaystyle x}$

${\ displaystyle T \ left (\ lambda x + \ mu y \ right) = \ lambda T \ left (x \ right) + \ mu T \ left (y \ right)}$

applies to coefficients . It makes sense to define the appropriate operations, so it is necessary that and are from a vector space and the solution is sought from the same or a different vector space . ${\ displaystyle \ lambda, \ mu}$${\ displaystyle T (x)}$${\ displaystyle a}$ ${\ displaystyle V}$${\ displaystyle x}$${\ displaystyle W}$

Linear equations are usually much easier to solve than nonlinear ones. The superposition principle applies to linear equations : The general solution of an inhomogeneous equation is the sum of a particulate solution of the inhomogeneous equation and the general solution of the associated homogeneous equation.

Because of the linearity there is at least one solution to a homogeneous equation. If a homogeneous equation has a unique solution, then a corresponding inhomogeneous equation also has at most one solution. A related but much more in-depth statement in functional analysis is Fredholm's alternative . ${\ displaystyle x = 0}$

#### Nonlinear equations

Nonlinear equations are often differentiated according to the type of nonlinearity. In school mathematics in particular , the following basic types of non-linear equations are dealt with.

##### Algebraic equations

If the equation term is a polynomial , one speaks of an algebraic equation. If the polynomial is at least degree two, the equation is called nonlinear. Examples are general quadratic equations of form

${\ displaystyle ax ^ {2} + bx + c = 0}$

or cubic equations of form

${\ displaystyle ax ^ {3} + bx ^ {2} + cx + d = 0}$.

There are general solution formulas for polynomial equations up to degree four .

##### Fractional equations

If an equation contains a fraction term in which the unknown occurs at least in the denominator , one speaks of a fraction equation, for example

${\ displaystyle {\ frac {x + 2} {x ^ {2} +3}} = {\ frac {2} {x + 1}}}$.

By multiplying by the main denominator, in the example , fractional equations can be reduced to algebraic equations. Such a multiplication is usually not an equivalence conversion and a case distinction must be made, in the example the fraction equation is not included in the definition range . ${\ displaystyle (x ^ {2} +3) (x + 1)}$${\ displaystyle x = -1}$

##### Root equations

In the case of root equations, the unknown is at least once under a root , for example

${\ displaystyle {\ sqrt {x}} = 1-x}$

Root equations are special power equations with an exponent . Root equations can be solved by a root is isolated and then the equation with the root exponent (in the example ) potentiates is. This process is repeated until all roots are eliminated. Increasing to the power of an even-numbered exponent does not represent an equivalence conversion and therefore in these cases a corresponding case distinction must be made when determining the solution. In the example, squaring leads to the quadratic equation , the negative solution of which is not in the definition range of the output equation . ${\ displaystyle {\ tfrac {1} {n}}}$${\ displaystyle n}$${\ displaystyle n = 2}$${\ displaystyle x = (1-x) ^ {2}}$

##### Exponential equations

In exponential equations , the unknown appears at least once in the exponent , for example:

${\ displaystyle 2 ^ {3x + 2} = 4 ^ {x + 1}}$

Exponential equations can be solved by taking logarithms . Conversely, logarithmic  equations - i.e. equations in which the unknown occurs as a number (argument of a logarithm function) - can be solved by exponentiation .

##### Trigonometric equations

If the unknowns appear as an argument of at least one angle function , one speaks of a trigonometric equation, for example

${\ displaystyle \ sin (x) = \ cos (x)}$

The solutions to trigonometric equations are generally repeated periodically , unless the solution set is limited to a certain interval , for example . Alternatively, the solutions can be parameterized by an integer variable . For example, the solutions to the above equation are given as ${\ displaystyle [0.2 \ pi)}$${\ displaystyle k}$

${\ displaystyle x = {\ frac {\ pi} {4}} + \ pi k}$   with   .${\ displaystyle k \ in \ mathbb {Z}}$

### Classification according to unknown unknowns

#### Algebraic equations

In order to distinguish equations in which a real number or a real vector is searched for from equations in which, for example, a function is searched, the term algebraic equation is sometimes used, but this term is not restricted to polynomials . However, this way of speaking is controversial.

#### Diophantine equations

If one looks for integer solutions of a scalar equation with integer coefficients, one speaks of a Diophantine equation. An example of a cubic Diophantine equation is

${\ displaystyle 2x ^ {3} -x ^ {2} -8x = -4}$,

of the integers that satisfy the equation, here the numbers . ${\ displaystyle x \ in \ mathbb {Z}}$${\ displaystyle x = \ pm 2}$

#### Difference equations

If the unknown is a consequence , one speaks of a difference equation. A well-known example of a second order linear difference equation is

${\ displaystyle x_ {n} -x_ {n-1} -x_ {n-2} = 0}$,

whose solution for starting values and the Fibonacci sequence is. ${\ displaystyle x_ {0} = 0}$${\ displaystyle x_ {1} = 1}$ ${\ displaystyle 1,2,3,5,8,13, \ ldots}$

#### Functional equations

If the unknown of the equation is a function that occurs without derivatives, one speaks of a functional equation. An example of a functional equation is

${\ displaystyle f (x + y) = f (x) f (y)}$,

whose solutions are precisely the exponential functions . ${\ displaystyle f (x) = a ^ {x}}$

#### Differential equations

If a function is sought in the equation that occurs with derivatives, one speaks of a differential equation. Differential equations are very common when modeling scientific problems. The highest occurring derivative is called the order of the differential equation. One differentiates:

${\ displaystyle f '(x) + xf (x) = 0}$
• partial differential equations in which partial derivatives occur according to several variables, for example the linear transport equation of the first order
${\ displaystyle {\ frac {\ partial f (x, t)} {\ partial t}} + {\ frac {\ partial f (x, t)} {\ partial x}} = 0}$
{\ displaystyle {\ begin {aligned} {\ ddot {x}} _ {1} & = 2x_ {1} \ lambda \\ {\ ddot {x}} _ {2} & = 2x_ {2} \ lambda - 1 \\ 0 & = x_ {1} ^ {2} + x_ {2} ^ {2} -1 \ end {aligned}}}
${\ displaystyle {\ rm {d}} S_ {t} = rS_ {t} {\ rm {d}} t + \ sigma S_ {t} {\ rm {d}} W_ {t}}$

#### Integral equations

If the function you are looking for occurs in an integral, one speaks of an integral equation. An example of a linear integral equation of the 1st kind is

${\ displaystyle \ int _ {0} ^ {x} (xt) f (t) ~ \ mathrm {d} t = x ^ {3}}$.

## Chains of equations

If there are several equal signs in one line, one speaks of an equation chain . In an equation chain, all expressions separated by equal signs should have the same value. Each of these expressions must be considered separately. For example, is the equation chain

${\ displaystyle 17 + 3 = 20/2 = 10 + 7 = 17}$

wrong, because broken down into individual equations it leads to wrong statements. For example, it is true

${\ displaystyle 17 + 3 = 40/2 = 10 + 10 = 20}$.

Chains of equations can be meaningfully interpreted , especially because of the transitivity of the equality relation. Chains of equations often appear together with inequalities in estimates , for example for${\ displaystyle n \ geq 3}$

${\ displaystyle 2n ^ {2} = n ^ {2} + n ^ {2} \ geq n ^ {2} + 3n> n ^ {2} + 2n + 1 = (n + 1) ^ {2}}$.

## Systems of equations

Often several equations that must be fulfilled at the same time are considered and several unknowns are searched for at the same time.

### Systems of linear equations

A system of equations - that is, a set of equations - is called a system of linear equations if all equations are linear. For example is

{\ displaystyle {\ begin {aligned} x + y + z & = 5 \\ 2x-z & = 13 \ end {aligned}}}

a system of linear equations consisting of two equations and three unknowns and . If both the equations and the unknowns are combined into tuples , an equation system can also be understood as a single equation for an unknown vector . In linear algebra, for example, a system of equations is written as a vector equation ${\ displaystyle x, y}$${\ displaystyle z}$

${\ displaystyle A \ cdot {\ vec {x}} = {\ vec {b}}}$

with a matrix , the unknown vector and the right hand side , where is the matrix-vector product . In the example above are ${\ displaystyle A}$${\ displaystyle {\ vec {x}}}$${\ displaystyle {\ vec {b}}}$${\ displaystyle (\ cdot)}$

${\ displaystyle A = {\ begin {pmatrix} 1 & 1 & 1 \\ 2 & 0 & -1 \ end {pmatrix}}}$,     and   .${\ displaystyle {\ vec {x}} = {\ begin {pmatrix} x \\ y \\ z \ end {pmatrix}}}$${\ displaystyle {\ vec {b}} = {\ begin {pmatrix} 5 \\ 13 \ end {pmatrix}}}$

### Nonlinear systems of equations

Systems of equations whose equations are not all linear are called nonlinear systems of equations. For example is

${\ displaystyle \ left \ {{\ begin {array} {rcl} 3x ^ {2} + 2xy & = & 1 \\\ sin (x) \ cdot \ ln (y) & = & e ^ {x} \ end {array }} \ right.}$

a nonlinear system of equations with the unknowns and . There are no generally applicable solution strategies for such systems of equations. Often one only has the possibility to determine approximate solutions with the help of numerical methods. A powerful approximation method is, for example, the Newton method . ${\ displaystyle x}$${\ displaystyle y}$

A rule of thumb states that the same number of equations as there are unknowns are required for a system of equations to be uniquely solvable. However, this is actually only a rule of thumb, it is valid to a certain extent for real equations with real unknowns because of the main theorem on implicit functions .

## Solving equations

### Analytical solution

As far as possible, one tries to find the exact solution of a determining equation. The most important tools here are equivalence transformations , through which one equation is gradually transformed into other equivalent equations (which have the same set of solutions) until an equation is obtained whose solution can be easily determined.

### Numerical solution

Many equations, especially those from scientific applications, cannot be solved analytically. In this case one tries to calculate an approximate numerical solution on the computer. Such procedures are dealt with in numerical mathematics . Many nonlinear equations can be solved approximately by approximating the nonlinearities occurring in the equation linearly and then solving the resulting linear problems (for example using Newton's method ). For other problem classes, for example when solving equations in infinite-dimensional spaces, the solution is sought in suitably chosen finite-dimensional subspaces (for example in the Galerkin method ).

### Qualitative analysis

Even if an equation cannot be solved analytically, it is still often possible to make mathematical statements about the solution. In particular, we are interested in the question of whether a solution exists at all, whether it is unique, and whether it depends continuously on the parameters of the equation. If this is the case, one speaks of a correctly posed problem . A qualitative analysis is also or especially important for the numerical solution of an equation so that it is ensured that the numerical solution actually provides an approximate solution of the equation.