Equation solving - Wikipedia, the free encyclopedia. An example of using Newton- Raphson method to solve the equation f(x)=0. The Newton- Raphson method is a procedure to find a numerical solution. By using known values of the coefficients and evaluating, the numerical solution for the quadratic equation with those coefficients is found. 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). When seeking a solution, one or more free variables are designated as unknowns. A solution is an assignment of expressions to the unknown variables that makes the equality in the equation true. In other words, a solution is an expression or a collection of expressions (one for each unknown) such that, when substituted for the unknowns, the equation becomes an identity. A problem of solving an equation may be numeric or symbolic. Solving an equation numerically means that only numbers represented explicitly as numerals (not as an expression involving variables), are admitted as solutions. Solving an equation symbolically means that expressions that may contain known variables or possibly also variables not in the original equation are admitted as solutions. For example, the equation x + y = 2x . It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x . Or x and y can both be treated as unknowns, and then there are many solutions to the equation. Instantiating a symbolic solution with specific numbers always gives a numerical solution; for example, a = 0 gives (x, y) = (1, 0) (that is, x = 1andy = 0) and a = 1 gives (x, y) = (2, 1). Note that the distinction between known variables and unknown variables is made in the statement of the problem, rather than the equation. However, in some areas of mathematics the convention is to reserve some variables as known and others as unknown. When writing polynomials, the coefficients are usually taken to be known and the indeterminates to be unknown, but depending on the problem, all variables may assume either role.
Depending on the problem, the task may be to find any solution (finding a single solution is enough) or all solutions. The set of all solutions is called the solution set. In the example above, the solution (x, y) = (a + 1, a) is also a parametrization of the solution set with the parameter being a. 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 . Its solutions are the members of the inverse image. Note that the set of solutions can be the empty set (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. 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. Using the squaring function on the integers, that is, the function . However note that in attempting to find solutions for this equation, if we modify the function's definition . So, if we were instead to define that the domain of . 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 . The variety in types of equations is large, and so are the corresponding methods. Only a few specific types are mentioned below. In general, given a class of equations, there may be no known 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 1. 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. This MATLAB function starts at x0 and tries to solve the equations fun(x) = 0, an array of zeros. Do you know a program that can solve equatation by given variable? Program for solving mathematical equations. Export transparent-background equations from Latex. Common choices of dom are Reals, Integers, and Complexes. 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. 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 . 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. For solving larger systems, algorithms are used that are based on linear algebra. Polynomial equations. 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. In some cases 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. 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 identityh(h. For example, the projection. For example, the equationtan. Often, root- finding algorithms like the Newton. A particular class of problem that can be considered to belong here is integration, and the analytic methods for solving this kind of problems are now called symbolic integration.
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