Examples of putting Questions into Equations. 15. A person buys some pieces of cloth, at equal prices, for $60. Had he got 3 pieces more for the same sum, each piece would have cost him $ 1 less. How many pieces did he buy? Ans. If x = the number of pieces bought the required equation is 16. Two drapers A and B cut, each of them, a certain number of yards from a piece of cloth; A however 3 yards less than B, and jointly receive for them $35. "At my own price," said A to B, "I should have received $ 24 for your cloth." "I must admit," answered the other, "that, at my low price, I should have received for your cloth no more than $121." How many yards did each sell? Solution. Let x = the number of yards sold by A ; then x+3= the number sold by B. Now since A would have sold x + 3 yards for $ 24, 24 A's price per yard = ; and since B would have sold x yards for $ 124, the sum for which A sells x yards the sum for which B sells x + 3 yards and the required equation is Examples of putting Questions into Equations. 17. Two travellers, A and B, set out at the same time from two different places, C and D; A, from C to D; and B, from D to C. When they met, it appeared that A had already gone 30 miles more than B; and, according to the rate at which they are travelling, A calculates that he can reach the place D in 4 days, and that B can arrive at the place C in 9 days. What is the distance between C and D? Ans. If, when they meet, 18. Some merchants jointly form a certain capital, in such a way that each contributes 10 times as many dollars as they are in number; they trade with this capital, and gain as many dollars per cent. as exceed their number by 8. Their profit amounts to $288. How many were there of them? Ans. If x = equation is the number of merchants, the required To x2 (x + 8) = 288. 19. Part of the property of a merchant is invested at such a rate of compound interest, that it doubles in a number of years equal to twice the rate per cent. What is the rate of interest? Ans. If x = the rate per cent., the required equation is (100+) 2* = 2. Degree of an Equation. SECTION II. Reduction and Classification of Equations. 104. The portions of an equation, which are separated by the sign =, are called its members; the one at the left of the sign being called its first member, and the other its second member. 105. Equations are divided into classes according to the form in which the unknown quantities are contained in them. But before deciding to which class an equation belongs, it should be freed from fractions, from negative exponents, and from the radical signs which affect its unknown quantities; its members should, if possible, be reduced to a series of monomials, and the polynomials thus obtained should be reduced to their simplest forms. 106. When the equation is thus reduced, it is said to be of the same degree as the number of dimensions of the unknown quantities in that term which contains the greater number of dimensions of the unknown quantities. Thus, x and y being the unknown quantities, the equations ax + b = c, 10 x + y = 3, are of the first degree; x2+3x+1 = 5, x y = 11, are of the second degree, &c. Transcendental Equations; Roots of Equations. 107. But when an equation does not admit of being reduced to a series of monomials, or, when being so reduced, it contains terms in which the unknown quantities or their powers enter otherwise than as factors, it is said to be transcendental; and the consideration of such equations belongs to the higher branches of mathematics. 108. An equation is said to be solved, when the values of its unknown quantities are obtained; and these values are called the roots of the equation. 109. The reduction and solution of all equations depends upon the self-evident proposition, that Both members of an equation may be increased, diminished, multiplied, or divided by the same quantity, without destroying the equality. 110. Corollary. If all the terms of an equation have a common factor, this factor may be suppressed. 111. EXAMPLES. 1. If the factor common to the terms of the equation α2 x5 + 3 α3 x2 a2 x2 is suppressed, what is the resulting equation? Ans. 23+3a = 1. 2. If the factor common to the terms of the equation ax + 3 ax + 1 x = ax − 1 is suppressed, what is the resulting equation? To free an Equation from Fractions. 112. Problem. To free an equation from fractions. Solution. Reduce, by arts. 67 and 68, all the terms of the equation to fractions having a common denominator, and suppress the common denominator, prefixing to the numerators the signs of their respective fractions. Demonstration. For suppressing the denominator of a fraction is the same as multiplying the fraction by its denominator; and, consequently, both the members of this equation are, by the preceding process, multiplied by the common denominator. 113. Corollary. It must be strictly observed that, when the denominator of a fraction is removed, the sign, which precedes the fraction, affects all the terms of the numerator. If therefore this sign is negative, all the signs of the numerator are to be reversed. Solution. This equation, when its terms are reduced to bd x + b d x bd x |