Using Equations To Solve Problems

Using Equations To Solve Problems-73
This may be due to a lack of mathematical knowledge; some problems were only solved after centuries of effort.

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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 , which simplifies this to a quadratic equation in z).

In Diophantine equations the solutions are required to be integers.

Equations involving linear or simple rational functions of a single real-valued unknown, say x, such as can be solved using the methods of elementary algebra.

Smaller systems of linear equations can be solved likewise by methods of elementary algebra.

However, in some areas of mathematics the convention is to reserve some variables as known and others as unknown.

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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.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: Its solution set is , the empty set, since 2 is not the square of an integer, so no integer solves this equation.A solution of an equation is often also called a root of the equation, particularly but not only for algebraic or numerical equations.A problem of solving an equation may be numeric or symbolic.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. 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".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. Note that the distinction between known variables and unknown variables is made in the statement of the problem, rather than the equation.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. In general, given a class of equations, there may be no known systematic method (algorithm) that is guaranteed to work.

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