3d. SYMMETRICAL EQUATIONS. When the equations are of this description, they may be solved by taking advantage of multiple forms, and of the necessary relations existing between the sum, difference, and product of two quantities. Proceeding now as in the third example, we have x2+2xy+y=169, x2-2xy+y=25, x—y=±5, x= 9 or 4, y = 4 or 9 In this example, the auxiliary letters were used to avoid fractional exponents in the operation. This practice, however, is not a necessity, but only a convenience. The auxiliary letters should be made to represent the lowest powers of the unknown quantities. Substituting these values in the given equations, and factoring, P' + or, Substituting these values in (1) and (2), we have (4) � (5) = 13.81. 3 9 If we take the minus sign in the second member of equation (4), Whence, by combining (1) and (6) as in the 3d example, x = 5 or 3, y = 3 or 5. 300. For examples of more than two unknown quantities, no additional illustrations are necessary. a final equation in the quadratic form methods that apply to the preceding. The few cases which lead to are to be treated by the same And skill in the management of this whole class of examples, must depend less upon precept than upon practice. 301. As uxiliary to the solution of certain questions, particularly in geometrical progression, we give the following PROBLEM.-Given x+y=s and xy = p, to find the values of x3 +y3, x3+y3, xʻ+yʻ, and x'+y°, expressed in terms of s and p. The following example will illustrate the use of these formulas. (0) = sp2; If we take xy = p'-162p-2072, p-162p+6561 = 4489, p-8167, xy=p=148 or 14. 148, the values of x and y will be imaginary. Taking xy 14, with the equation x+y= 9, we readily obtain x = 7 or 2, y = 2 or 7. |