Equation solving

In mathematics, equation solving refers to finding what values (numbers, functions, sets, etc.) fulfill a condition stated as an equality (an equation). Usually, this condition involves expressions with variables (or unknowns), which are to be substituted by values in order for the equality to hold. More precisely, an equation involves some free variables.

Overview

In one general case, we have a situation such as

f(x₁,...,x_n) = c,

where c is a constant, which has a set of solutions S of the form

{(a₁,...,a_n) ∈ Tⁿ|f(a₀,...,a_n) = c}

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

For example, an expression such as

3x + 2y = 21z

can be solved by first modifying the equation in some way as to preserve the equality, such as subtracting 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 = 20/3, y = 11, z = 2. In fact, this particular set of solutions describes a plane in three dimensions, which passes through the point (20/3, 11, 2).

Solution sets

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

f(x₀,...,x_n) = c

becomes true for a given c.

For example, let us examine the classic one-variable case, given a function

consider the equation

f(x) = −1

The solution set is {}, in that no positive real number solves this equation. However note that in attempting to find solutions for this equation, if we modify the function's definition – more specifically, the function's domain, we can find solutions to this equation. So, if we were instead to define

g(x) = −1

has a solution set {i, −i}, where i is the imaginary unit. This equation has exactly two solutions.

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

Elementary algebra

Equations involving linear functions of x, such as

can be solved using the methods of elementary algebra.

Polynomial equations

Polynomial equations with degrees 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 numerical methods (see below) or special functions such as Bring radicals.

Inverse functions

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

h(x) = c, c constant

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

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

h⁻¹(h(x)) = h(h⁻¹(x)) = x.

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

h(x) = c, c constant

we obtain

h⁻¹(h(x)) = h⁻¹(c)

x = h⁻¹(c)

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

Examples 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).

Numerical methods

With more complicated equations, 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 within some applications can be entirely sufficient to solve some problem.

Taylor series

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

Trial and error

If the solution set of an equation is restricted to a finite set (equations in modular arithmetic, for example), or can be limited to a finite number of possibilities (as is the case with some Diophantine equations) then the solution set can be found by trial and error by testing each of the possible values.

Solving other equations

It is important to note that we can create even more complex equations, involving differential operators, matrices, and so on. The underlying principle of solving equations by finding a value which satisfies the equation is maintained, but with vastly differing methodologies used to find them.