Elements of Algebra

CHAP. II. OF ALGEBRA. 189 whether we take the logarithms of those quantities, as we have already done in the preceding section. 572. The equations which are most easily resolved, are those in which the unknown quantity does not exceed the first power, after the terms of the equation have been properly arranged; and these are called simple equations, or equations of the first degree. But if, after having reduced an equation, we find in it the square, or the second power, of the unknown quantity, it is called an equation of the second degree, which is more difficult to resolve. Equations of the third degree are those which contain the cube of the unknown quantity, and so on. We shall treat of all these in the present section. CHAP. II. Of the Resolution of Simple Equations, or Equations of the First Degree. 573. When the number sought, or the unknown quantity, is represented by the letter $x$, and the equation we have obtained is such, that one side contains only that $x$, and the other simply a known number, as, for example, $x = 25$, the value of $x$ is already known. We must always endeavour, therefore, to arrive at such a form, however complicated the equation may be when first obtained: and, in the course of this section, the rules shall be given, and explained, which serve to facilitate these reductions. 574. Let us begin with the simplest cases, and suppose, first, that we have arrived at the equation $x + 9 = 16$. Here we see immediately that $x = 7$: and, in general, if we have found $x + a = b$, where $a$ and $b$ express any known numbers, we have only to subtract $a$ from both sides, to obtain the equation $x = b - a$, which indicates the value of $x$. 575. If we have the equation $x - a = b$, we must add $a$ to both sides, and shall obtain the value of $x = b + a$. We must proceed in the same manner, if the equation have this form, $x - a = a^3 + 1$: for we shall immediately find $x = a^3 + a + 1$. In the equation $x - 8a = 20 - 6a$, we find $x = 20 - 6a + 8a$, or $x = 20 + 2a$.