V. INDETERMINATE EQUATIONS OF THE FIRST DEGREE 1. It was shown in § 189 that a linear equation involving two unknown quantities, such as 3x+2y=7, has an infinite number of solutions. Similarly any system of equations is indeterminate, if the number of unknown quantities is greater than the number of equations. By introducing the condition that the roots shall be positive integers, the number of solutions can frequently be limited. 2. If the equation ax + by = c (1) is satisfied by the values ar1, and y=r2, then it is also satisfied by the values: 3. If one set of integers is known that satisfies an indeterminate equation involving two unknown quantities, all solutions can be found by the preceding paragraph. = Thus, the equation 3x + 2y 34 is satisfied by the values x = y = 2. Hence it is satisfied by any values of the form: = 10, Dividing both members by the smaller coefficient, i.e. 7, Since x and y, and hence x+2y, are to be integers, the fractional parts of the two members must either be equal, or differ by an integral number. Assume the first of these cases, if it produces an integral y : Hence if m denotes any integer, the general solutions are (§ 3) : To obtain positive values for x and y, m must be zero. Equating the fractional parts of the two members produces a fractional y, hence let Or 14 y 23 10 = +n, where n is an integer. 23 14 y = 10 + 23 n. This is another indeterminate equation, but a simpler one than (1). In complex cases, this equation may be treated by the regular method, while in simpler ones we find by trial a value of n which produces an integral y. Evidently, if n = 2, 10+ 23 n is divisible by 14. Or, if To make x positive, m≤3; to make y positive, m m≥0. By assigning to m any positive integral value, we obtain an unlimited number of solutions : m = 0, 1, 2, 3, x=13, 30, 47, 64, ..., 29. Divide 142 into two parts such that one part is a multiple of 9, the other a multiple of 14. 30. Divide 1591 into two parts such that one part is a multiple of 23, the other of 34. 31. Find two fractions whose denominators are respectively 7 and 3, and whose sum equals 1. 32. In what manner can $15 be paid in five-dollar bills and two-dollar bills? 33. A farmer sold a number of horses and cows for $447, receiving $112 for each horse and $37 for each cow. How many did he sell of each? 34. A grocer bought a number of pounds of tea and coffee for $5.10, paying for tea 40¢ per pound, and for coffee 19¢ per pound. How many pounds did he buy of each? 35. A farmer sold a number of cows, sheep, and pigs for $140, receiving $31 for each cow, $11 for each sheep, and $9 for each pig. How many did he buy of each, if the total number of animals was 10? VI. VARIATION 1. Direct variation. When the ratio of two variables is constant, each variable is said to vary directly as the other. Thus, x varies directly as y (or briefly, a varies as y), if =m, a constant. x х y E.g. The weight of a quantity of water varies as its volume. The distance traversed by a man walking at a uniform rate varies as the time during which he walks, etc. |