Equation solving

In mathematics, to solve an equation is to find what values (numbers, functions, sets, etc.) fulfill a condition stated in the form of an equation (two expressions related by equality). These expressions contain one or more unknowns, which are free variables for which values are sought that cause the condition to be fulfilled. To be precise, what is sought are often not necessarily actual values, but, more in general, mathematical expressions. A solution of the equation is an assignment of expressions to the unknowns that satisfies the equation; in other words, expressions such that, when they are substituted for the unknowns, the equation becomes a tautology (a provably true statement).

For example, the equation x + y = 2x – 1 is solved for the unknown x by the solution x = y + 1, since substituting y + 1 for x in the equation results in (y + 1) + y = 2(y + 1) – 1, a true statement. It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1. Or x and y can both be treated as unknowns, and then there are many solutions to the equation, some of which are (x, y) = (1, 0) – that is, x = 1 and y = 0 – and (x, y) = (2, 1), and, in general, (x, y) = (a + 1, a) for all possible values of a.

Depending on the problem, the task may be to find one solution – any solution will do – or all solutions. The set of all solutions is called the solution set. It is also possible that the task is to find a solution, among possibly many, that is best in some respect. Problems of that nature are called optimization problems; solving an optimization problem is generally not referred to as "equation solving".

A wording such as "an equation in x and y", or "solve for x and y", implies that the unknowns are as indicated: in these cases x and y.

Overview

In one general case, we have a situation such as

ƒ (x₁,...,x_n) = c,

where c is a constant, whose solutions are the members of the inverse image

ƒ ⁻¹[c] = {(a₁,...,a_n) ∈ T₁×···×T_n | ƒ (a₁,...,a_n) = c},

where T₁×···×T_n is the domain of the function. Note that the set of solutions can be empty (there are no solutions), a singleton (there is exactly one solution), finite, or infinite (there are infinitely many solutions).

For example, an equation such as

3x + 2y = 21z

with unknowns x, y and z, can be solved by first modifying the equation in some way while keeping it equivalent, such as subtracting 21z from both sides of the equation to obtain

3x + 2y − 21z = 0

In this particular case there is not just one solution to this equation, but an infinite set of solutions, which can be written

{(x, y, z) | 3x + 2y − 21z = 0}.

One particular solution is x = 0, y = 0, z = 0. Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. In fact, this particular set of solutions describes a plane in three-dimensional space, which passes through the three points with these coordinates.

Solution sets

If the solution set is empty, then there are no x_i such that the equation

ƒ (x₁,...,x_n) = c,

in which c is a given constant, becomes true.

For example, let us examine a classic one-variable case. Using the squaring function on the integers, that is, the function ƒ whose domain are the integers (the whole numbers) defined by:

ƒ (x) = x²,

consider the equation

ƒ (x) = 2.

Its solution set is {}, the empty set, since 2 is not the square of an integer, so no integer solves this equation. However note that in attempting to find solutions for this equation, if we modify the function's definition – more specifically, the function's domain, we can find solutions to this equation. So, if we were instead to define that the domain of ƒ consists of the real numbers, the equation above has two solutions, and its solution set is

{√2, −√2}.

We have already seen that certain solutions sets can describe surfaces. For example, in studying elementary mathematics, one knows that the solution set of an equation in the form ax + by = c with a, b, and c real-valued constants, with a and b not both equal to zero, forms a line in the vector space R². However, it may not always be easy to graphically depict solutions sets – for example, the solution set to an equation in the form ax + by + cz + dw = k (with a, b, c, d, and k real-valued constants) is a hyperplane.

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; a comprehensive treatment is not possible.

In general, given a class of equations, there may be no 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 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 systems, 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 equations), the solution set can be found by brute force, that is, by testing each of the possible values. 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 x, such as

$3x + 4 = 5x + 2 \, , \quad \frac{4x + 9}{3x + 4} = 2 \, ,$

can be solved using the methods of elementary algebra.

Systems of linear equations

Smaller systems of linear equations can be solved likewise by methods of elementary algebra. For solving large systems numerically, algorithms are used that are based on linear algebra.

Polynomial equations

Main article: Solving polynomial equations

See also: Systems 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 radicals, although some specific cases may be solvable algebraically, for example

4x⁵ − x³ − 3 = 0

(by using the Rational root theorem), and

x⁶ − 5x³ + 6 = 0,

(by using the substitution x = z^1/3, which simplifies this to a quadratic equation in z).

Diophantine equations

In Diophantine equations the solutions are required to be integers. In some case 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-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

$2x^5-5x^4-x^3-7x^2+2x+3=0\,$

has as rational solutions x = −1/2 and x = 3, and so, viewed as a Diophantine equation, it has the unique solution x = 3.

In general, however, Diophantine equations belong to the hardest equations to solve.

Inverse functions

See also: Inverse problem

In the simple case of a function of one variable, say, h(x), we can solve an equation of the form

h(x) = c, c constant

by considering what is known as the inverse function of h.

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

h⁻¹(h(x)) = h(h⁻¹(x)) = x.

Now, if we apply the inverse function to both sides of

h(x) = c, where c is a constant value in B,

we obtain

h⁻¹(h(x)) = h⁻¹(c)

x = h⁻¹(c)

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

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

h(h⁻¹(x)) = x

holds. For example, the projection π₁ : R² → R defined by π₁(x, y) = x has no post-inverse, but it has a pre-inverse π₁⁻¹ defined by π₁⁻¹(x) = (x, 0). Indeed, the equation

π₁(x, y) = c

is solved by

(x, y) = π₁⁻¹(c) = (c, 0).

Examples of inverse functions include the nth root (inverse of xⁿ); the logarithm (inverse of a^x); the inverse trigonometric functions; and Lambert's W function (inverse of xe^x).

Factorization

If the left-hand side expression P of an equation P = 0 can be factored as P = QR, the solution set of the original solution consists of the union of the solution sets of the two equations Q = 0 and R = 0. For example, the equation

$\tan x + \cot x = 2\,$

can be rewritten as

$\tan x + \cot x - 2 = 0\,,$

which can be factored, using the identity tan x cot x = 1 (provided that the components are defined), as

$(\tan x - 1)(\cot x - 1) = 0\,.$

The two equations tan x = 1 and cot x = 1 have identical solution sets

$x = \tfrac{\pi}{4} + k\pi, k = \cdots, -2, -1, 0, 1, 2, \ldots \,,$

which therefore is the solution set of the original equation.

Numerical methods

With more complicated equations in real or complex numbers, simple methods to solve equations can fail. Often, root-finding algorithms 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.

Taylor series

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

Matrix equations

Equations involving matrices and vectors of real numbers 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 equations, both numerically and analytically. A particular class of problem that can be considered to belong here is integration, and a method for finding analytical solutions for the indefinite case is the Risch algorithm – unfortunately too complicated for use with pencil and paper.

See also

• Simultaneous equations
• Equating coefficients
• Solving the geodesic equations