Equation solving

In mathematics, equation solving is the problem of finding what values (numbers, functions, sets...) 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.

In one general case, we have a situation such as

f(x₀,...,x_n)=c, c constant

which has a set of solutions S in the form

{(a₀,...,a_n)∈Tⁿ|f(a₀,...,a_n)=c}

with T the domain of the function. Note that the set of solutions can be empty (there are no solutions), singleton (there is exactly 1 solution), finite (there are only n number of solutions), or infinite (there are always 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 both sides by 21z to obtain

3x+2y-21z=0

Now, it occurs that in solving this equation, that there is not just one solution to this equation, but a 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 describe a plane in three dimensions, which passes through the point (20/3, 11, 2).

Contents

1 Solution sets 2 Methods of solution 2.1 Inverse functions 2.2 Numerical methods 2.2.1 Taylor series 3 Solving other equations

Solution sets

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

f(x₀,...,x_n)=c

becomes true.

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

<math>f : \mathbb{R}^+ \rightarrow \mathbb{R} ; x \longmapsto x^2<math>

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

<math>g : \mathbb{C} \rightarrow \mathbb{C} ; x \longmapsto x^2<math>

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=b with a,b 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

In simple cases, it is rather easy to solve an equation provided certain conditions are met. However, in more complicated cases, exact symbolic forms for solutions are often difficult to obtain or cumbersome to manipulate with, and an approximate numerical solution may be in fact sufficient for use.

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.

If 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.

Numerical methods

With more complicated equations, simple methods to solve equations can fail. In certain circumstances, a root-finding algorithm 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.

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.de:Lösen von Gleichungen nl:oplossen van vergelijkingen