Equations: common information

Equality. Identity. Equation (unknowns, roots
of an equation, solving). Equivalent equations.

Equality - two expressions (numerical or literal ones), jointed by sign " = ".
Identity - a valid numerical equality or a literal equality, valid at any numerical values of letters, contained in it.

E x a m p l e s :   1)  A numerical equality  4 · 7 + 2 = 30 is an identity.

2)  A literal equality  ( a + b )( a – b ) = a 2 – b 2 is an identity,
because it is valid at all values of letters, contained in it.

Equation – a literal equality, which is valid ( i.e. it becomes an identity ) only at some values of letters, contained in it. These letters are called unknowns and the values, at which an equality is valid – roots of an equation. Procedure of finding all roots of an equation is called solving. To solve an equation means to find all its roots. Substitution of each root into an equation instead of unknown converts it into a valid numerical equality ( identity ). Two or some equations are called equivalent equations , if they have the same roots .

E x a m p l e :   Equations  5 x – 25 = 0  and  2 x – 7 = 3  are equivalent, because
they have the same root: x = 5 .