Equations

In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. Variables are also called unknowns and the values of the unknowns which satisfy the equality are called solutions of the equation. There are two kinds of equations: identity equations and conditional equations. An identity equation is true for all values of the variable. A conditional equation is true for only particular values of the variables.

Each side of an equation is called a member of the equation.  Each member will contain one or more terms. The equation{\displaystyle Ax^{2}+Bx+C=y} has two members. {\displaystyle Ax^{2}+Bx+C} and y. The left member has three terms and the right member one term. The variables are x and y and the parameters are AB, and C.

An equation is analogous to a scale into which weights are placed. When equal weights of something (grain for example) are place into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, an equal amount of grain must be removed from the other pan to keep the scale in balance. Likewise, to keep an equation in balance, the same operations of addition, subtraction, multiplication and division must be performed on both sides of an equation for it to remain an equality.

In geometry, equations are used to describe geometric figures. As equations that are considered, such as implicit equations or parametric equations have infinitely many solutions, the objective is now different: instead of given the solutions explicitly or counting them, which is impossible, one uses equations for studying properties of figures. This is the starting idea of algebraic geometry, an important area of mathematics.

Algebra studies two main families of equations: polynomial equations and, among them the special case of linear equations. Polynomial equations have the form P(x) = 0, where P is a polynomial. Linear equations have the form ax + b = 0, where a and b are parameters. To solve equations from either family, one uses algorithmic or geometric techniques, that originate from linear algebra or mathematical analysis. Algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from number theory. These equations are difficult in general; one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions.

Differential equations are equations that involve one or more functions and their derivatives. They are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in areas such as physics, chemistry, biology, and economics.

The “=” symbol, which appears in every equation, was invented in 1557 by Robert Recorde, who considered that nothing could be more equal than parallel straight lines with the same length.

Source: Wikipedia and trinhmanhdo

By Trinh Manh Do

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