4 x + 4. And = the price of x eggs at 3 for a cent. 3 These added together make what the eggs cost. The whole number is 2 x ; these at 5 for two cents come to 4 cents. 5 , + 2 3 5 32. A cistern has two fountains to fill it ; the first will fill it alone in 7 hours, and the second in 5 hours. In what time will the cistern be filled, if both run together ? Let x = the number of hours required to fill it. The first would fill of it in an hour, and the second would fill of it in an hour. Both together then would fill + in an hour ; and in * hours both would fill + 7 5 was to be filled in x hours. Therefore, = 1 cistern. 5 33. A gentleman, having a piece of work to do, hired two men and a boy to do it; one man could do it alone in 5 days, the other could do it alone in 8 days, and the boy could do it alone in 10 days. How long would it take the three together to do it ? 34. A cistern, into which the water runs by two cocks, А and B, will be filled by them both running together in 12 hours ; and by the cock A alone in 20 hours. In what time will it be filled by the cock B alone ? Let x = the time in which В will fill it alone. Both will fill in of it in an hour, A alone zo of it, and B will fill 's of it in an hour, &c. 35. A man and his wife usually drank out a vessel of beer in 12 days : but when the man was from home it would usually last the wife alone 30 days. In how many days would the man alone drink it out? + 7 36. The hold of a ship contained 442 gallons of water. This was emptied out by two buckets, the greater of which holding twice as much as the other, was emptied twice in three minutes, but the less three times in two minutes; and the whole time of emptying was 12 minutes. Required the size of each. The greater was emptied 3 times in the 12 minutes, &c. 37. Two persons, A and B, have the same income. A saves of his ; but B, by spending £80 a year more than A, at the end of 4 years finds himself £220 in debt. What did each receive and expend annually? 38. After paying of my money, and į of the remainder, I had 72 guineas left. How much had I at first ? 39. A bill of £120 was paid in guineas and moidores, the guineas at 21s., and the moidores at 27s. each ; the number of pieces of both sorts was just 100. How many were there of each > 40. It is required to divide the number 26 into three such parts, that if the first be multiplied by 2, the second by 3, and third by 4, the products shall all be equal. 2 Let x = the first part. The second part must be and 3' 2 x 4 2 41. It is required to divide the number 54 into three such parts, that of the first, of the second, and of the third, may be all equal to each other. Let 2x = the first part. Then 3x = the second part, &c. 42. A person has two horses and a saddle, which of itself is worth £25. Now if the saddle be put upon the back of the first horse, it will make his value double that of the second; but if it be put upon the back of the second, it will make his value triple that of the first. What is the value of each horse ? 43. A man has two horses and a chaise, which is worth $183. Now if the first horse be harnessed to the chaise, the horse and chaise together will be worth once and two sevenths the value of the other ; but the other horse being harnessed, the horse and chaise together will be worth once and five х the third part 2 ore eighths the value of the first. Required the value of each horse. Equations with two Unknown Quantities. VIII. Many examples involve two or more unknown quantities. In fact, many of the examples already given involve several unknown quantities, but they were such, that they could all be derived from one When it is necessary to use two unknown quantities in the solution, the question must always contain two conditions, from which two equations may be derived. When this is not the case the question cannot be solved. 1. A boy bought 2 apples and 3 oranges for 13 cents ; he afterwards bought, at the same rate, 3 apples and 5 oranges for 21 cents. How much were the apples and oranges apiece? Let x = the price of an orange, and y= the price of an apple. 1. 3x + 2 y = 13, 2. 5x + 3y = 21. Multiply the first equation by 3, and the second by 2, 3. 9 x + 6y= 39 4. 10x + 6y 42. Subtract the first from the second, because the y's being alike in each, the difference between the numbers 39 and 42 must depend upon the x's. 5. X = 3 cents, the price of an orange. Putting this value of x into the first equation, 6. 9 + 2 y = 13 7. y= 2 cents, the price of an apple. Proof. 2 apples at 2 cents each come to 4 cents, and 3 oranges at 3 cents come to 9 cents. 9 +4= 13. So 3 apples and 5 oranges come to 21 cents. Note. In this example I observed, that the coefficient of y in the first equation is 2, and in the second, the coefficient of is 3. I multiplied the whole of the first equation by 3, and the whole of the second by 2 ; this formed two new equations in which the coefficients of y are alike. If the first equation had been multiplied by 5 and the second by 3, the coefficients of x would have been alike, and x instead of y would have been y made to disappear by subtraction, and the same result would have been finally obtained. It is evident, that the coefficients of either of the unknown quantities may always be rendered alike in the two equations, by multiplying the first equation by the coefficient which the quantity that you wish to make disappear has in the second equation ; and the second equation by the coefficient which the same quantity has in the first equation. They may be rendered alike more easily, when they have a common multiple less than their product. 2. A person has two horses, and a saddle which of itself is worth £10; if the first horse be saddled, he will be worth as much as the other, but if the second horse be saddled, he will be worth as much as the first. What is the value of each horse ? A question similar to this has already been solved with one unknown quantity, but it will be more easily solved by using two of them. Let x = the value of the first horse, and y = the value of the second horse. 1. By the conditions, = x + 10 7 бу · y = = 10 = 70 8 x 5 Multiply the 3d by 7, and the 4th by 5, to free them from denominators; 5. 7 x + 6y 6. 5 50 - 35 x + 30 y = 350 8. 48 — 30 y = 300 Add together 7th and 8th, 9. 48x-35 x + 30 y-30 y = 350 + 300 10. Uniting terms, 13 x 11 = 650 X = 50 6 y = 420 Putting 50 the value of x, into the 5th, 12 бу — 350 = 70 13 14. y 70 Ans. The first is worth £50, and the second £70. Note. In this example the 30 y in the 7th equation had the sign +, and in the 8th the sign before it, hence it was necessary to add the two equations together in order to make the y disappear, or as it is sometimes called, to eliminate y. 3. A market-woman sells to one person, 3 quinces and 4 melons for 25 cents, and to another, 4 quinces and 2 melons, at the same rate, for 20 cents. How much are the quinces and melons apiece ? 4. In the market I find I can buy 5 bushels of barley and 6 bushels of oats for 27s., and of the same grain 4 bushels of barley and 3 bushels of oats for 19s. What is the price of each per bushel ! 5. My shoemaker sends me a bill of $12 for 1 pair of boots and 3 pair of shoes. Some months afterwards he sends me a bill of $20 for 3 pair of boots and 1 pair of shoes. What are the boots and shoes a pair ? 6. Three yards of broadcloth and 4 yards of taffeta cost 575., and at the same rate 5 yards of broadcloth and 2 yards of taffeta cost 81s. What is the price of a yard of each ? 7. A man employs 4 men and 8 boys to labour one day, and pays them 40s. ; the next day he hires, at the same wages, 7 men and 6 boys, and pays them 50s. What are the daily wages of each: 8. A vintner sold at one time 20 dozen of port wine and 30 doz. of sherry, and for the whole received £120 ; and at another time, sold 30 doz. of port and 25 doz. of sherry at the same prices as before, and for the whole received £140. What was the price of a dozen of each sort of wine ? 9. A gentleman has two horses and one chaise. The first horse is worth $180. If the first horse be harnessed to the chaise, they will together be worth twice as much as the second horse ; but if the second be harnessed, the horse and chaise will be worth twice and one half the value of the first. What is the value of the second horse, and of the chaise ? |