EQUATIONS. An equation may be defined to be an algebraical sentence stating the equality of two algebraical expressions, or of an algebraical expression to zero. From another point of view, it is the algebraical expression of the conditions which connect known and unknown quantities. Thus (1), xy = 24, and (2), + = 52, are two E. expressing the relations between the unknown quantities x and y and known quan tities. Generally, E. are formed from observations from which an object of inquiry may be inferred, but which do not directly touch the object. Thus, suppose we wish to ascertain the lengths of the sides of a rectangular board which we have no means of measuring, and that all the information we can get respecting it is, that it covers (say> 24 sq. ft., and that the square on its diagonal is (say) 52 sq. feet. From these facts, we can form E. from which we may determine the lengths of the sides. In the first place, we know that its area is equal to the product of its sides, and if we call these x and y, we have xy = 24, the first of the E. above given. Again, we know that the sum of the squares on the sides is equal to the square on the diagonal; hence, we have the second equation, + = 52. From these two E., we should be able to deter mine the values of x and y. The determination of these values is called the solution of the equations.
E. are of several kinds. -Simple E. are those which contain the unknown quantity in the first degree; thus, + 3 = 4, is a simple equation. Quadratic E. are those which contain the unknown quantity in the second degree: Sr — 36 = 0, is a quadratic equation. Cubic and biquadradic E involve the unknown quantity in the third and fourth powers respectively. For the higher E. there are no special names; they are said to be E. of the degree indicated by the highest power of the unknown which they contain. Simultaneous E. are those which involve two or more unknown quantities, and there must always be as many of them, in order to their determinate solution, as there are unknown quantities. The E. first mentioned—viz., xy = 24 — + = 52, are simultaneous equations. It may be mentioned, that in the course of solving such E., the principal difficulties encountered are always ultimately the same as in the solu tion of E. only one unknown quantity. For instance, in the E. just given, if we substitute in the second time value of y as given by the first, or 24 y = —x ' we have = 52, which may be solved as a quadratic equation. The general theory of E., then, is principally concerned with the solution of E. involving one unknown quantity only, for to this sort all others reduce themselves. Indeterminate E. are such as do not set forth sufficient relations between the unknown quantities for their absolute determination,. and which, accordingly admit.of various solutions. Thus,
xy = 24 is an indeterminate equation, which is Satisfied by the = 3, y = 8; or 6, y = 4; or x = 2, y = 12. We require some other relation, such as e + = 52, to enable us to fix on one of the sets of values, x and y, as those of x. For other kinds of E., see EXPONENT AND EXPONENTIAL, FUNCTIONS, and DIFFERENCE.
The object of all computation is the determination of numerical values for unknown quantities, by means of the relations which they bear to other quantities already known. The solution of E., accordingly, or, in other words, the evolution of the unknown quantities in them, is the chief business of algebra. But so difficult is this business, that, except in the simple cases where the unknown quantity rises to no higher than the second degree, all the resources of algebra are as yet inadequate to effect the solution of T. in general and definite terms. For E. of the second degree, or quadratic E., as they are called, there is a rigorous method of solution by a general formula; but as yet no such formula has been discovered for E. even of the third degree. It is true, that for E. of the third and fourth degrees general methods exist, which furnish formulas which express under a finite form the values of the roots. Sec CARD AN and CUBIC EQUATIONS. But all such formulas are found to involve imaginary expressions, which, except in particular cases, make the actual computations impracticable till the formulas are devel oped in infinite series, and the imaginary terms disappear by mutually destroying one another. What is called Cardan's formula, for instance (and all others are reducible to it), is in this predicament whenever the values of the unknown quantity are all real; and accordingly, in nearly all such cases, the values are not obtainable from the formulae directly, but from the infinite series of which they are the expression. But though such formula as Cardan's are useless for the purpose of numerical computation, the search for them has led to most of the truths which constitute the general theory of E., and through which their numerical solution may be said to have been at last rendered effective and general. This method of numerical solution is a purely arithmetical process, performed upon the numerical co-eificients of E., and it is universally applicable, whatever the degree of the equation may be. With this method are connected the names of Budan, Fourier, Homer, and Sturm. We cannot here enter into an account of it: the reader should consqlt on the subject Young's Theory and Solution, of Algebra ical Equations of the _Higher Orders; Peacock's Treatise on Algebra; and La Grange's work on Numerical Solutions.