In algebra, there is the concept of two types of equalities - identities and equations. Identities are those equalities that can be fulfilled for any meaning of the letters in them. Equations are also equalities, but they are feasible only with certain meanings of the letters that enter them.
Letters according to the conditions of the problem are usually unequal. This means that some of them can take any acceptable values, called coefficients (or parameters), while others - they are called unknown - take values that need to be found in the process of solving. As a rule, unknown quantities are denoted in the equations by letters that are last in the
Latin alphabet (xyz, etc.), or by the same letters, but with an index (x
1 , x
2 , etc.), and the known coefficients are first 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 at which the equation being solved turns into identity are called solutions of the equations. The equation can be considered solved if all its solutions are found or it is proved that it does not have such. The task of "solving the equation" is often encountered in practice and means that you need to find the root of the equation.
Definition : the roots of an equation are those values of unknowns from the region of admissible for which the equation being solved turns into identity.
The algorithm for solving absolutely all equations is the same, and its meaning is to bring this expression to a simpler form using mathematical transformations.
Equations that have the same roots are called equivalent in 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 have no roots at all).
If the root of the equation is simultaneously the root of another, simpler equation obtained from the original by transformations, then the latter is called a consequence of the previous equation.
If one of their two equations is a consequence of the other, then they are considered equivalent. They are also called equivalent. The above example illustrates this.
Solving even the simplest equations in practice often causes difficulties. As a result of the solution, you can get one root of the equation, two or more, even an infinite number - it depends on the type of equations. There are those that have no roots, they are called unsolvable.
Examples:
1) 15x -20 = 10; x = 2. This is the only root of the equation.
2) 7x - y = 0. The equation has an infinite number of roots, since each variable can have countless values.
3) x 2 = - 16. A number raised to the second power always gives a positive result, so 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 checked by substituting the found roots instead of letters and solving the resulting example. If the identity is respected, the decision is correct.