Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/576

Rh 538 ALGEBRA [SIMPLE EQUATIONS. Ex. 5. Find the vulgar fraction which is equivalent to the recurring decimal. 3142 Let Z--3142, then 1 Ox =3-142 10,000^ = 3142-142 .-.subtracting 9990.r = 3139 o!39 ~9&amp;gt;90 Ex. C. A sum of money doubles itself in fifteen years at a rate a little below 5 per cent. A noble Scotch family have retained in their possession gold coins of the value of 500 since the days of Mary Stuart (300 years) ; what have they lost by not allowing the money to accumulate at the above rate ] Every pound would have amounted to 2 20 ;. . 500 (2 20 -1) is the loss. It amounts to upwards of 524,000,000. Ex. 7. The sum of the mixed series (a+n-ll)}r n l-r SECT. VI. RESOLUTION OF EQUATIONS INVOLVING ONE UNKNOWN QUANTITY. limple 53. The primary object of algebraic investigation is to Equations, discover certain unknown quantities, by comparing them with other quantities which are given, or supposed to be known. The relation between the known and unknown quantities is either that of equality, or else such as may be reduced to equality; and a proposition which affirms that certain combinations of quantities are equal to one another is called an equation. Such are the following : x^x _24 2 + 3 x 2x + oy = xy. The first of these equations expresses the relation between an unknown quantity x and certain known numbers; and the second expresses the relation which the two indefinite quantities x and y have to each other. The conditions of a problem may be such as to require several equations and symbols of unknown quantities for their complete expression. These, however, by rules here after to be explained, may be reduced to one equation, involving only one unknown quantity and its powers, be sides the known quantities; and the method of expressing that quantity by means of the known quantities consti tutes the theory of equations, one of the most important as well as most intricate branches of algebraic analysis. An equation is said to be resolved when the unknown quantity is made to stand alone on one side, and only known quantities on the other side ; and the value of the un known quantity is called a root of the equation. The general definition of a root of an equation is, that it is a numerical quantity (i.e., some combination of numbers) which, when written in place of the unknown quantity, renders the equation a numerical ideality; thus 1 is the root of the equation x = l, 1 and -1 are both roots of the equation x~ } 1, fJ ) t J and -1 are all roots of the equation x^ l. 54. Equations containing only one unknown quantity and its powers, are divided into different orders, according to the highest power of that quantity contained in any one of its terms. The equation, however, is supposed to be reduced to such a form that the unknown quantity is found only in the numerators of the terms, and that the exponents of its powers are expressed by positive integers. If an equation contains only tlie first power of the unknown quantity, it is called a simple equation, or an equation of the first order. Such is ax + b = c, where x denotes an unknown, and a, l&amp;gt;, c, known quantities. If the equation contains the second power of the un known quantity, it is said to be of the second degree, or is called a quadratic equation; such is &amp;lt;ix 2 + 3x = l2, and in general ax 2 + bx = c. If it contains the third power of the unknown quantity, it is of the third degree, or is a cubic equation ; such are x 3 + Ix&quot; + 4# = 10, and ax^ + bx 2 + ex = d ; and so on with respect to equations of the higher orders. A simple equation is sometimes said to be linear, or of one dimension. In like manner, quadratic equations are said to be of two dimensions, and cubic equations of three dimensions. When in the course of an algebraic investigation wo arrive at an equation involving only one unknown quantity, that quantity will often be so entangled in the different terms as to render several previous reductions necessary before the equation can be expressed under its character istic form, so as to be resolved by the rules which belong to that form. These reductions depend upon the operations which have been explained in the former part of this treatise, and the application of a few self-evident principles, namely, that if equal quantities be added to or subtracted from equal quantities, the sums or remainders will be equal; if equal quantities be multiplied or divided by the same quantity, the products or quotients will be equal; and, lastly, if equal quantities be raised to the same power, or have the same root extracted out of each, the results will still be equal. From these considerations are derived the following rules, which apply alike to equations of all orders, and are alone sufficient for the resolution of simple equations. 55. Eule 1. Any quantity may be transposed from one side of an equation to the other, by changing its sign. Thus, Then Or Again, Then Or if 3x- 10-2 if ax + b = ex - dx + e f ax ex -I- dx e b , (a c + d}x = e - b . The reason of this rule is evident, for the transposing of a quantity from one side of an equation to the other is nothing more than adding the same quantity to each side of the equation, if the sign of the quantity transposed was - ; or subtracting it, if the sign was + . From this rule we may infer, that if any quantity be found on each side of the equation with the same sign, it may be left out of both. Also, that the signs of all the terms of an equation may be changed into the contrary, without affecting the truth of the equation. Thus, if Then And if a x = b d, Then x - a = d - b . 5G. Rule 2. If the unknown quantity in an equation be multiplied by any quantity, that quantity may be taken away, by dividing all the other terms of the cquatiou by it. If 3vC = 24, f)A Then a; = ^ = 8. If ax = b-c, rrn. 6 C 6 C Then x = = a a a