The root of the equation is familiarization information

In algebra there is a concept of two kinds of equations - identities and equations. Identities are such equalities that are feasible for any values of the letters entering them. Equations are also equalities, but they are doable only for certain values of the letters entering into them. The letters by the condition of the problem are usually unequal. This means that some of them can take any admissible values, called coefficients (or parameters), while others - they are called unknowns - take values that must be found in the solution process. As a rule, unknown quantities are denoted by the letters in the equations, the last ones in the Latin alphabet (xyz, etc.), or by the same letters, but with the index (x ₁ , x ₂ , etc.), and the known coefficients are the first ones Letters of the same alphabet.

By the number of unknowns, equations with one, two, and several unknowns are distinguished. Thus, all the values of the unknowns for which the solved equation is transformed into an identity are called solutions of the equations. The equation can be considered solved in the event that all its solutions are found or it is proved that it does not. The task of "solving an equation" in practice occurs frequently and means that we need to find the root of the equation.

Definition : the roots of the equation are those values of unknowns from the domain of admissible for which the solved equation is transformed into an identity.

The algorithm for solving all equations is the same, and its meaning is that with the help of mathematical transformations, this expression leads to a simpler form.
Equations that have the same roots are called equivalent in an algebra.

The simplest example: 7x-49 = 0, the root of the equation x = 7;
X-7 = 0, similarly, the root x = 7, therefore, the equations are equivalent. (In special cases, equivalent equations may not have roots at all).

If the root of the equation is simultaneously the root of another, a simpler equation obtained from the original by means of transformations, then the latter is called the consequence of the previous equation.

If their two equations are one consequence of the other, then they are considered equivalent. They are also called equivalent. The example above illustrates this.

Solving even the simplest equations in practice often causes difficulties. As a result of the solution one can get one root of the equation, two or more, even an infinite number - it depends on the kind of equations. There are also those who do not have roots, they are called insoluble.

Examples:
1) 15x-20 = 10; X = 2. This is the only root of the equation.
2) 7x - y = 0. The equation has an infinite set of roots, since each variable can have an infinite number of values.
3) x ² = - 16. The number raised to the second power always gives a positive result, therefore it is impossible to find the root of the equation. This is one of the unsolvable equations discussed above.

The correctness of the solution is verified by substituting the found roots for the letters and solving the resulting example. If the identity is observed, the decision is correct.