Any of the three foregoing rules may be applied to the solution of

4. Given ar+by=c, and dx+ey=f, it is required to find x and y.

To exterminate three unknown quantities, or to reduce three simple equations, containing them, to a single one.

Rule.-1. Let x, y, and z, be the three unknown quantities which are to be exterminated.

2. From each of the three given equations, find the value of x. 3. Compare the first value of with the second, and there wil

arise an equation involving only y and z.

4. Compare, in like manner, the first value of x with the third, and there will arise another equation involving only y and x.

5. From these two equations, find the values of y and z, according to the former rules, and x, y, and z, will be exterminated as was required.

Note.-In nearly the same manner, any number of unknown quantities may De exterminated; but practice will often suggest much shorter methods for performing the operation.

1. Given

Examples.

x + y + z=zy
x+2y+3x=62
•r+y+4x=10J

it is required to find x, y, and z.

2. Given

( F x + jy ++x=62)

}

3x + y + z 47 it is required to find x, y, and z. y + zx=38)

( { x +

First, the given equations, cleared of fractions, become

And, if the second of these equations be subtracted from double the first, and three times the third from five times the second, we shall have

And again, if the second of these be subtracted from three times the first, it will give

3. Given x+y+x=53, x+2y+3x-105, and x+3y+4x=134, it is required to find x, y, and z.

Ans. x=24, y=6, and x=23.

4. Given x+y=a, x+x=b, and y+xc, it is required to find r, y, and z.

Miscellaneous Questions, producing Simple Equations.

1. To find two numbers, such that their sum shall be 40, and their difference 16.

Ans. 12 least number, and 28 = greater number.

2. A prize of 1000l. is to be divided between two persons, whose shares therein are in the proportion of 7 to 9; required the share of each. Ans. 4371. 10s. = first share, and 5621. 10s, = second share.

3. What number is that whose one-third part exceeds its one-fourth part by 16? Ans. 192.

4. Divide 1000l. between A, B, and C, so that A shall have 721. more than B, and C. 100l. more than A.

Ans. 2521. B.'s share; 3241. A.'s share; and 4241. C.'s share.

5. What fraction is that to the numerator, of which, if 1 be added, the value will be one-third; but if 1 be added to the denominator, its value will be one-fourth?

6. A labourer agreed to serve for 40 days, upon condition that for every day he worked he should receive 201. but for every day he played, or was absent, he should forfeit 8d. Now, at the end of his engagement, he had to receive 11. 11s. 8d. it is required to find how many days he worked, and how many he was idle.

Ans. He worked 25 days, and he was idle 15.

7. If the paving of a square, at 2s. a yard, costs as much as the inclosing of it at 5s. a yard; required the side of the square. Ans. 10 yards.

8. A market-woman bought a certain number of eggs, at 2 a penny, and as many at 3 a penny, and sold them all again at the rate of 5 for 2d. and by so doing lost 4d. what number of eggs had she?

Ans. 120.

9. If one agent A, alone, can produce an effect e, in the time a, and another agent B, alone, in the time b; in what time will they both produce the same effect?

10. If A, alone, can do a piece of work in ten days, and B in thir teen; set them both about it, in what time will it be finished?

11. What two numbers are those whose difference is 7, and sum 33? Ans. 13 and 20.

12. Two travellers set out at the same time from London and York, the distance being 150 miles; one of them goes 8 miles a day, and the other 7; in what time will they meet? Ans. in 10 days.

13. A post is in the mud, in the water, and 10 feet above the water; what is its whole length? Ans. 24 feet. 14. In a mixture of cider and wine, one-half of the whole plus 25 gallons was wine, and one-third part minus 5 gallons was cider; how many gallons were there of each?

Ans. 85 of wine, and 35 of cider. fish weighs 9lb. its head weighs as much as its and its body weighs as much as its head and its

15. The tail of a tail and half its body, tail; what is the whole weight of the fish ›

Ans. 72lb. 16. A bill of 120l. was paid in guineas and moidores, and the number of pieces of both denominations that were used amounted to 100; how many were there of each? Ans. 50 of each.

17. After paying away one-fourth and had 66 guineas left; what had I at first?

one-fifth of my money, I Ans. 120 guineas.

18. At an election, 375 persons voted, and the candidate chosen had a majority of 91; how many voted for each candidate ?

Ans. 233 for one, and 142 for the other.

19. A.'s age is double of B.'s, and B.'s is triple of C.'s, and the sum of all their ages is 140; what is the age of each?

Ans. A.'s=84, B.'s=42, and C.'s=14.

20. What number is that from which, if 5 be subtracted, twothirds of the remainder will be 40?

Ans. 65.

21. Two persons, A and B, lay out equal sums of money in trade; A gains 1261. and B loses 871. and A.'s money is now double of B.'s; what did each lay out?

Ans. 300l.

22. Out of a cask of wine, which had leaked away one-third, 21 gallons were drawn; and then, being gauged, it appeared to be half full; how much did it hold? Ans. 126 gallons.

23. A gentleman bought a chaise, horse, and harness, for 601. the horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness; how much did he pay for each? Ans. 131. 6s. 8d. for the horse, 61. 13s. 4d. for the harness, and 401. for the chaise.

24. A person was desirous of giving some beggars 3d. a-piece, but found that he had not money enough in his pocket by 8d. he therefore gave them 2d. each, and had then 3d remaining; how many beggars Ans. 11.

25. Two persons, A and B, have both the same income; A saves one-fifth of his yearly, but B, by spending 501. per annum more than A, at the end of four years finds himself 1001. in debt; what is their income? Ans. 1251.

26. A footman agreed to serve for 81. a year and a livery, but was turned away at the end of seven months, receiving only 21. 13s. 4d. and his livery; what was its value?

Ans. 41. 16s.

27. A hare is fifty leaps before a greyhound, and takes four leaps to the greyhound's three; but two of the greyhound's leaps are as much as three of the hare's; how many leaps must the greyhound take to catch the hare? Ans. 300.

28. The hour and minute hand of a clock are exactly together at 12 o'clock; at what time are they next together?

Ans. 1 ho. 55 min.

29. A gamester lost one-fourth of his money at play, and then won three shillings; after which he lost one-third of what he then had, and then won two shillings; lastly, he lost one-seventh of what he then had; and, this done, found he had only 12s. remaining; what had he at first? Ans. 20s.

30. A man and his wife usually drank a cask of beer in twelve days; but when the man was from home, it lasted his wife thirty days; how many days would the man alone be in drinking it?

Ans. 20 days.

31. A merchant has two kinds of tea, one worth 9s. 6d. per pound, the other 13s. 6d. How many pounds of each must he take to form a chest of 104lbs. which shall be worth 561.?

Ans. 33 at 13s. 6d. and 71 at 9s. 6d.

32. Divide the number 90 into 4 such parts, that if the first be increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2; the sum, difference, product. and quotient, shall be all equal to each other.

Ans. The parts are 18, 22, 10, and 40, respectively.

33. If A and B together perform a piece of work in 8 days; A and C together in 9 days; and B and C in 10 days; how many days will it take each person to perform the same work?

Ans. A 143 days, B 17, and C 23 T.

34. How much foreign brandy at 8s. per gallon, and British spirits at 3s. per gallon, must be mixed, that in selling a compound of 65 gallons, at 9s. per gallon, the distiller may clear 30 per cent. ?

Ans. 51 gallons of brandy, and 14 of spirits.

35. If three agents, A, B, and C, produce the effects a, b, c, m the times e, f, g, respectively; in what time would they jointly produce the effect d?

a ს
Ans. a+ (+/+) time.

e

36. A gentleman has two horses, and a saddle worth 50l.; now if the saddle be put on the back of the first horse, it will make his value double that of the second; but if it be p. on the back of the second, i will make his value triple that of the first; what is the value of each Ans. One 30l. and the other 401.

norse?