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 8 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 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.

Let x = the first part. The second part must be

2 x

3'

and

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
Then

2x the first part.

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

eighths the value of the first. Required the value of each horse.

tities.

Equations with two Unknown Quantities.

VIII. Many examples involve two or more unknown quanIn 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?

1.

2.

3

Let x = the price of an orange,

and y the price of an apple.

=

3x+2y=13,

5x + 3y = 21.

Multiply the first equation by 3, and the second by 2,

4.

9x6y39

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.

7.

9+2y=13

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.

y

Note. In this example I observed, that the coefficient of 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 a would have been alike, and x instead of y would have been

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 whicl. 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.

Multiply the 3d by 7, and the 4th by 5, to free them from denominators;

5.

6.

―7x+6y=70
8 x

5 Y

= 50

Multiply the 5th by 5 and the 6th by 6, in order to make the coefficients of y alike in the two;

6 y=420
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 18s. 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.

the boots and shoes a pair?

What are

6. Three yards of broadcloth and 4 yards of taffeta cost 57s., 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 ?

10. Two men, driving their sheep to market, A says to B, give me one of your sheep and I shall have as many as you; B says to A, give me one of your sheep and I shall have twice as many as you. How many had each?

Let

And

x = the number A had,
y = the number B had.

If B gives Ä one, their numbers will be

x+1 and y-1.

If A gives B one, their numbers will be

11. If A gives B $5 of his money, B will have twice as inuch as A has left; but if B gives A $5 of his money, A will have three times as much as B has left. How much has each?

12. A man bought a quantity of rye and wheat for £6, the rye at 4s. and the wheat at 5s. per bushel. He afterwards sold of his rye and of his wheat at the same rate for £2. 17s. How many bushels were there of each?

13. A man bought a cask of wine, and another of gin for $210; the wine at $1.50 a gallon, and the gin at $0.50 a gallon. He afterwards sold of his wine, and of his gin for $150, which was 15 more than it cost him. How many gallons were there in each cask?

14. A countryman, driving a flock of gecse and turkeys to market, in order to distinguish his own from any he might meet with on the road, pulled three feathers out of the tail of each turkey, and one out of the tail of each goose, and found that the number of turkeys' feathers exceeded twice those of the geese by 15. Having bought 10 geese and sold 15 turkeys by the way, he was surprised to find that the number of geese exceeded the number of turkeys in the proportion of 7 to 3. Required the number of each at first.

3. Freeing the 2d from fractions, 3y+30=7x-105 Instead of the method employed above for eliminating one of the unknown quantities, we may find the value of one of them in one equation, as if the other were known; and then