12. What integers have each the property that, if 9 be sub- ́ tracted from three times the integer, the remainder will be less than twice the integer, increased by 12? 13. A has three times as much money as B. If B gives A $10, then A will have more than seven times as much as B will have left. What are the possible amounts of money which A and B have? Identical Inequalities. 14. Many inequalities hold for all values of the literal numbers involved; as a2 + b2> a2. Such inequalities are analogous to identical equations. 15. Prove that if a is not equal to b, then a2 + b2 > 2 ab. We have (a - b)2 > 0, (1) since the square of any positive or negative number is positive, and therefore greater than 0. From (1), a2-2ab+b2 >0; whence a2+b2 >2 ab, by Art. 12 (i.). EXERCISES II. Prove the following inequalities, in which the literal numbers are all positive and unequal: 1. a2 + b2 + c2 > ab + ac + bc. 2. a2b2 + b2c2 + a2c2 > abc (a + b + c). 3. ab (a+b)+ be (b+c) + ac (a + c) > 6 abc. 4. If l2 + m2 + n2 = 1, and 2+ m‚2 + n‚2 = 1, then ll1 + mm2 + nn1 < 1. 5. a3+b3> a2b + ab2. 6. a+b> a3b + ab3. 7. (a + b) (b+c) (c + a) > 8 abc. 8. 3 (a2 + b2 + c2) > (a + b + c)2. CHAPTER XII. INDETERMINATE LINEAR EQUATIONS. 1. It was shown in Ch. X., Art. 1, that the linear equation in two unknown numbers x + y = 5 is satisfied by an indefinite number of sets of values of x and y. An Indeterminate Equation is an equation which, like the above, has an indefinite number of solutions. Evidently the number of solutions will be more limited if only positive integral values of the unknown numbers are admitted. In this chapter we shall consider a simple method of solving in positive integers linear indeterminate equations. 2. Ex. 1. Solve 4x + 7y = 94, in positive integers. Solving for 2, which has the smaller coefficient, we obtain That is, y must have such a value that 2-3 y shall be divisi 2-3y 4 termine integral values of y. But since the expression is to be an integer, any multiple of it will be an integer. We therefore multiply its numerator by the least number which will make the coefficient of y one more than a multiple of the Any integral value of m will give to x and y integral values. But since y is to be positive, m <1; and, since x is to be positive, m > -3. Therefore the only admissible values of m are 0, 1, -2. 3. An Indeterminate System is a system of equations which has an indefinite number of solutions. Thus, if the system x + y 2 = 9, 2x-y+72=33, be solved for x and y, we obtain x = 14-2z, y=3z-5. In these values of x and y we may assign any value to z and obtain corresponding values of x and y. 4. In solving a system of two linear equations in three unknown numbers, we first eliminate one of the unknown numbers, and apply to the resulting equation the preceding method. Pr. A party of 20 people, consisting of men, women, and children, pay a hotel bill of $67. Each man pays $5, each woman $4, and each child $1.50. How many of the company are men, how many women, and how many children? Let x stand for the number of men, y for the number of women, z for the number of children. Solving this equation, we obtain x=2-5m, y = 12 + 7 m, z=6-2 m. 9. 89x-144 y=1. 10. 14x-49 y=133. 11. 67x-43 y=5. 12. Divide 1000 into two parts so that one part shall be a multiple of 13, and the other a multiple of 53. 13. What positive integers when divided by 4 give a remainder 3, and when divided by 5 give a remainder 4? 14. A farmer received $16 for a number of turkeys and chickens. If he was paid $2 for each turkey and $.75 for each chicken, how many of each did he sell? 15. A gardener has fewer than 1000 trees. If he plants them in rows of 37 each, he will have 8 left; but if he plants them in a different number of rows of 43 each, he will have 11 left. How many trees has he? 16. A said to B: "If I had eight times as much money as I now have, and you had seven times as much money as you now have, and I were to give you $1, we should have equal amounts." How many dollars had each ? CHAPTER XIII. INVOLUTION 1. Involution is the process of raising a number to any These examples illustrate the following method of finding any required power of a given power: Multiply the exponent of the given power by the exponent of the required power; or, stated symbolically, Ex. 2 (xy) 10 (xy) (xy) (xy)... to 10 factors = ... mn =(xxx to 10 factors) (yyy... to 10 factors) These examples illustrate the following method of finding any required power of a product : Take the product of the factors, each raised to the required power; or, stated symbolically, (ab)"a"b"; (abc)" = a"b"c"; etc. |