dividing (2) by (3), member by member, we have, x? — xy + y2 = 21 ; . squaring (3), we have, Q2 + 2xy + y2 = 81 ; . (4) . (5) subtracting (4) from (5), we have, 3xy = 60; or, xy = 20; combining (6) and (3), we have, Raising both members of (1) to the 4th power, we have, x4 · 4x8y + 6x?y? — 4xyz + y4 = 16 subtracting (3) from (2), and factoring, (3) multiplying (1) by 2, squaring, multiplying by xy, and subtracting from (4), we have, 2.c?y2 = 256 or, æy2 + 8xy = 128. 16xy ; Taking the first value xy = 8, and combining with (1), we find, 2 = 4, 1. Find two numbers, such that their product, added to their sum, shall be 47, and their sum, taken from the sum of their squares, shall leave 62. Let x and y denote the numbers; then, from the conditions of the problem, (1) (2) (2C + y) + xy 47 x2 + y2 (x + y) = 62 ; multiplying equation (1) by 2, we have, 2xy + 2(x + y) = 94; . adding (2) and (3), we have, 2c2 + 2xy + y2 + (oC + y) = 156 ; . or, (2C + y)2 + (x + y) = 156 ; solving (4), with respect to x + y, and taking the first value of 2C + y, we have, 12 ; . 2 + y = 1 + V156 + 1 substituting in (1), we have, xy = 47 – 12 = 35; combining (5) and (6), x = 5, and y = 7: the numbers required. (6) . 2. The sum of two numbers is 7, and the sum of their cubes is 91. What are the numbers ? Ans. 3 and 4. 3. Required two numbers, whose product is equal to the square of two thirds the first, and the difference of whose squares is greater, by 1, than the square of twice the second. Ans. 9 and 4. 4. Find two numbers, whose sum, multiplied by the greater, is 209, and whose difference, multiplied by the less, is 24. Ans. 11 and 8. 5. Find two numbers, such that the sum of their squares is equal to 181, and the difference of their squares equal to 19. Ans. 9 and 10. III, INEQUALITIES. Definitions and Explanations. C. 152. An inequality is an algebraic expression indicating that one quantity is greater or less than another. Thus, a> b + c, and a<b-c are inequalities; the former indicates that a is greater than b + c, and the latter that a is less than 5 - The two parts connected by the sign of inequality are called members; that on the left of the sign is called the first member, and that on the right of the sign is called the second member. Of two unequal quantities, that is algebraically the greater which is nearer to to, thus Two inequalities subsist in the same sense when the greater quantity is either in the first member of both, or in the second member of both; they subsist in a contrary sense when the greater quantity is in the first member of one and in the second member of the other. Thus, the inequalities, 5 > 3 and 7>2, subsist in the same sense; whilst the inequalities, 57 and 3 > 1 subsist in a contrary sense. Transformations. 153. The following transformations of inequalities follow from the preceding definitions and explanations: 1°. If the same quantity is added to, or subtracted from, both members of an inequality, the sense of the inequality will not be changed. Thus, if we have the inequality 13 >12, we also have the inequalities 13 + 2 > 12 + 2, and 13 – 3 > 12 — 3. Hence, we may transpose a term from one member to the other by changing its sign. 2°. If both members of an inequality are either multiplied or divided by a positive quantity, the sense of the inequality will not be changed. Thus, if we have the inequality 12 > 6, we also have the inequalities 12 6 12 X 3 X 6 X 3 and > 4 4 This principle enables us to clear an inequality of fractions by the rule for clearing an equation of fractions. 3°. If we change the signs of either member of an inequality, we must change the sense of the inequality. Thus, if 3 > 2, we have, – 3 < 2. Solution of Inequalities. 154. The solution of an inequality is the operation of finding an inequality in which the unknown quantity shall form one member; the other member is then a limiting value of that quantity. |