7. The sum of two numbers which are formed by the same two digits in reverse order is ğ of their difference; and the difference of the squares of the numbers is 3960. Determine the numbers. 8. The hypotenuse of a right triangle is 20, and the area of the triangle is 96. Determine the sides. NOTE. The square on the hypotenuse sides; and the area of a right triangle = sum of the squares on the product of sides. 9. Two boys run in opposite directions round a rectangular field the area of which is an acre; they start from one corner and meet 13 yards from the opposite corner; and the rate of one is of the rate of the other. Determine the dimensions of the field. 10. A, in running a race with B, to a post and back, met him 10 yards from the post. To make it a dead heat, B must have increased his rate from this point 414 yards per minute; and if, without changing his pace, he had turned back on meeting A, he would have come 4 seconds after him. How far was it to the post? 11. The fore wheel of a carriage turns in a mile 132 times more than the hind wheel; but if the circumferences were each increased by 2 feet, it would turn only 88 times more. Find the circumference of each. 12. A person has $6500, which he divides into two parts and loans at different rates of interest, so that the two parts produce equal returns. If the first part had been loaned at the second rate of interest, it would have produced $180; and if the second part had been loaned at the first rate of interest, it would have duced $245. Find the rates of interest. pro CHAPTER XVI. SIMPLE INDETERMINATE EQUATIONS. 242. If a single equation be given which contains two unknown quantities, and no other condition be imposed, the number of its solutions is unlimited; for, if any value be assigned to one of the unknown quantities, a corresponding value may be found for the other. Such an equation is said to be indeterminate. 243. The values of the unknown quantities in an indeterminate equation are dependent upon each other; so that, though they are unlimited in number, they are confined to a particular range. This range may be still further limited by requiring these values to satisfy some given condition; as, for instance, that they shall be positive integers. 244. The method of solving an indeterminate equation in positive integers is as follows: (1) Solve 3x+4y=22, in positive integers. Since the values of x and y are to be integral, x + y − 7 will be 1-2 will be integral, though written in the integral, and hence, form of a fraction. Let 3 1—1=m, an integer; Substitute this value of y in the original equation, 3x+412m m = 22, ..x = 6 + 4m. The equation y=1-3m shows that m in respect to y may be 0, or have any negative value, but cannot have a positive value. = 6 + 4m shows that m in respect to x may be 0, The equation x = but cannot have a negative value greater than 1. (2) Solve 5x-14y= 11, in positive integers. To avoid this difficulty, it is necessary in some way to make the coefficient of y equal to unity. Since 1+4 is integral, any multiple of 1+ 4y 5 is integral. Multiply, then, by such a number as will make the coefficient of y greater by 1 than some multiple of the denominator. In this case, multiply by 4. Then Since x = f (11 + 14y), from the original equation, .. x = 14m - 9. Here it is obvious that m may have any positive value, and The required multiplier will always be less than the denominator; and for this reason it is best to divide the original equation by the smaller of the two coefficients, in order to have the multiplier as small as possible. 245. The necessity for a multiplier may often be obviated by a little ingenuity. Thus, 246. It will be seen from (1) and (2) that when only positive integers are required, the number of solutions will be limited or unlimited according as the sign connecting x and y is positive or negative. (3) Find the least number that when divided by 14 and 5 will give remainders 1 and 3 respectively. (4) Solve 5x+6y=30, so that x may be a multiple of y, and both positive. 1 – 8 y. (5) Solve 14x+22y=71, in positive integers. x= 5 −y + 14 a form which shows that there can be no integral solution. There can be no integral solution of ax ± by c if a and b have a common factor not common also to c; for, if d be a factor of a and also of b, but not of c, the equation may be written, |