EQUATION (Lat. crquatio, from c•quarc, to equalize. from wrytins, equal). In algebra, an equality which exists only for particular values of certain letters representing the unknown quantities is called an equation. These partieu• lar values are called the roots of the equation, and the determination of these roots is called the solution of the equation. Thus 2x + 3 = 9 is an equation, because the equality is true only for a pa rticular value of the unknown quantity x, viz., for x = 3. The expression 2 + 5 =7 ex. presses an equality, but it is not an equation as the word is technically used in mathematics. Expressions like (a + x) 2 = + 2ax e are true for all values of the letters and are called identities to distinguish them from equations. If an algebraic function f (x) equals zero, and is arranged according to the descending, integral, positive powers of x. and in its relation to 0 ex pressed as an equation, it has the form f (.r) = a,at + = Such an equation is called complete equation of the nth degree with one unknown quantity; e.g. + 0 is a complete equation, while + 0 is au incomplete equation, both of the second degree. The letters a,, stand for known quantities, and in the theory of equations, so called, they stand for real quantities. They are all coefficients of powers of x, except the absolute term, a„, which might, however, be considered the coefficient of e. In case a,. a,. are all expressed as numbers, the equation is said to be wumerical ; otherwise it is known as literal.
Equations may be classified as to the number of their unknown quantities. Those already mentioned involve a single unknown, but (iv+ :// = 0 and xy = 1 involve two unknowns. There is no theoretical limit to the number of unknown quantities. Equations may also be classified as to degree, this being determined by the value of n in the complete equation already given. Thus, 5+ +al + az + a , 4 x = 0 +al 5S + + 0 are equations, respectively, of the first degree (linear equation), of the second degree ratic equation), of the third degree (cubic equa tion), and of the fourth degree (quartic or biquadratic equation).
If two or more equations are satisfied by the same value of the unknown quantities they are said to be simultaneous, as in the case of x' y = 7, x 11, where x = 2, y = 3; but x' y = 7 and 3x' +3y = 9 are not simulta neous; they are inconsistent, there being no val ues of x and y that will satisfy both: and + y = 7 and 3y = 21 are said to be identical, each being derivable from the other. In case sufficient relations are not given to en able the roots of an equation to be determined, exactly or approximately, the equation is said to be indeterminate; e.g. in the equation 211 = 10, any of the following pairs of values satisfies the equation: (0, 5), (I. 4.5), (2, 4). (3. 3.5), (10, 0), (11, —0.5), In general, n linear each containing n + 1 or more unknown quantities, are indeterminate.
Thus 2x + 3y 4. z = 10, 2x 4- 2y 4- = S. give rise to the simple equation x+ y= 2. which is indeterminate. Equations may also he clas sified as rational, irrational, integral. or fraction al, according as the two members, when like terms are united, are composed of expressions which are rational, irrational (or partly so), tegral, or fractional for partly so), respectively, with respect to tI, unknowu quantities; e.g.: =0 is a rational integral equation, 6 = 0 is an irrational integral equation, = 0 is a rational fractional equation, 5 is an irrational fractional equation.
Algebra is chiefly concerned with the solution of equations, and definite methods have been de vised for determining the roots of algebraic equa tions of the first, second, third, and fourth de grees. Equations of the first degree are solved by applying the common axioms: If equals are added to equals, the results are equal; if equals are subtracted from equals, the results are equal; and the corresponding ones of multiplica tion and division. Equations of the second de gree may be solved by reducing the quadratic function to the product of two linear factors, thus making the solution of the quadratic equa tion depend upon that of two linear equations. Thus px q = 0 reduces to (x + (it+ = 0, whence — = O.