In algebra there is a concept of two types of equalities - identities and equations. Identities are such equalities that are feasible for any values of the letters included in them. Equations are also equalities, but they are feasible only for certain values of the letters included in them.
Letters are usually unequal in terms of the task. This means that some of them can take on any allowed values, called coefficients (or parameters), while others - they are called unknowns - take on values that need to be found in the solution process. As a rule, unknown quantities are denoted in equations by letters, the last ones in the Latin alphabet (x.y.z, etc.), or by the same letters, but with an index (x1, x 2, etc.), and the known coefficients are given by the first letters of the same alphabet.
Based on the number of unknowns, equations with one, two and several unknowns are distinguished. Thus, all values of the unknowns for which the equation being solved turns into an identity are called solutions of the equations. An equation can be considered solved if all its solutions are found or it is proved that it does not have any. The task "solve the equation" in practice is common and means that you need to find the root of the equation.
Definition: the roots of an equation are those values of the unknowns from the range of admissible values at which the equation being solved becomes an identity.
The algorithm for solving absolutely all equations is the same, and its meaning is to reduce 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 not have roots at all.)
If the root of an equation is also the root of another, simpler equation obtained from the original one by transformations, then the latter is called a consequence of the previous equation.
If one of the two equations is a 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 is often difficult. 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 also those that have no roots, they are called undecidable.
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 countlessnumber of values.
3) x2=- 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 mentioned above.
The correctness of the solution is checked by substituting the found roots instead of letters and solving the resulting example. If the identity holds, the solution is correct.
