# Mathematical equations

**Mathematical equations**

**Video: What is an Equation?**

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 that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variable. A conditional equation is true for only particular values of the variables.

An equation is written as two expressions, connected by a equals sign ("="). The expressions on the two sides of the equals sign are called the "left-hand side" and "right-hand side" of the equation.

The most common type of equation is an algebraic equation, in which the two sides are algebraic expressions. Each side of an algebraic equation will contain one or more terms. For example, the equation

has left-hand side , which has three terms, and right-hand side , consisting of just one term. The unknowns are *x* and *y* and the parameters are *A*, *B*, and *C*.

An equation is analogous to a scale into which weights are placed. When equal weights of something (grain for example) are placed 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 true.

In geometry, equations are used to describe geometric figures. As the equations that are considered, such as implicit equations or parametric equations, have infinitely many solutions, the objective is now different: instead of giving 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. When there is only one variable, polynomial equations have the form *P*(*x*) = 0, where *P* is a polynomial, and 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.

**Algebra**

**Polynomial equations**

The *solutions* –1 and 2 of the *polynomial equation* *x*^{2} – *x* + 2 = 0 are the points where ..................

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