CHAPTER IV. ON SIMPLE EQUATIONS, INVOLVING TWO OR MORE UNKNOWN QUANTITIES. 202. It has been observed (Art. 184), that an equation was the translation into algebraic language of two equivalent phrases comprised in the enunciation of a question; but this question may comprehend in it a greater number, and if they are well distinguished two by two, and independent of one another, they furnish a certain number of equations. Thus, for example, let us propose to find two numbers, such that double the first added to the second, gives 24, and that five times the first, plus three times the second, make 65. We find here two phrases, which express the same thing in different terms; 1st, the double of an unknown number, plus another unknown number, then the equivalent 24; 2d, five times the first unknown number, plus three times the "second, then the equivalent 65. The translation is easy, and it gives these two de terminate equations 2x+y=24; 5x+3y=65. When two or more equations, involving as many unknown quantities, are independent of one another, they are called determinate. But if for the second of these two conditions we had substituted this: and such that six times the first number, plus three times the second, make 72; these two phrases ex press nothing more than the first two, since that we have only tripled two equal results; we should have but one translation, and consequently a single equation. It can therefore happen that we may have less equations than unknown quantities, and then the question is said to be indeterminate; because the number of conditions would be insufficient for the determination of the unknown quantities, as we shall see clearly illustrated in the following section. §I. ELIMINATION OF UNKNOWN QUANTITIES FROM ANY NUMBER OF SIMPLE EQUATIONS. 203. Elimination is the method of exterminating all the unknown quantities, except one, from two, three, or more given equations, in order to reduce them to a single, or final equation, which shall contain only the remaining unknown, and certain known quantities. 204. In order to simplify the calculations, by avoiding fractions, we shall here make use of literal equations, which will modify the process of elimination: And also, to avoid the inconvenience arising from the multitude of letters which must be employed in order to represent the given quantities, when the number of equations involving as many unknown quantities surpasses two, we shall represent by the same letter all the coefficients of the same unknown quantity; but we shall affect them with one or more accents, in order to distinguish them, according to the number of equations. |