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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
= y + 10
y = 10
8 x 2.
6 у 3. By transposition,
8 x 4.
5 Multiply the 3d by 7, and the 4th by 5, to free them from denominators; 5.
-73 +6y= 70 6.
= 50 Multiply the 5th by 5 and the 6th by 6, in order to make the coefficients of y alike in the two; .
35 x + 30y= 350
48 x - 30 y = 300 Add together 7th and 8th, 9.
48x-35 x + 30 y — 30 y=350 + 300 10. Uniting terms,
13 x 650 11.
x = 50
y = 70
Putting 50, the value of x, into the 5th, 12
6y- 350 = 70 13.
6 y = 420 14.
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 27., and of the same grain 4 bushels of barley and 3 bushels of oats for 18s. What is the price of each
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 ?
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 x = the number A had,
x + 1 and y-1.
-1 and y +1, &c. 11. If A gives B $5 of his money, B will have twice as much as A has left; but if B gives À $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. 175. 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 geese 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
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.
Let x = the number of turkeys,
and y= the number of geese. 1.
3x = 2y + 15
- 105 2.
y + 10 =
3 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
this value may be substituted in the other, and an equation will be obtained, containing only one unknown quantity, which may be solved the usual way. 4. Divide the first by 3,
2y + 15
3 5. Multiply the 4th by 7,
14y + 105
3y + 30 =
3 7. Multiply by 3, 9y+90 = 14 y + 105 - 315 8. Transposing & uniting, 300 = 5 y
y's The value of x may be found by substituting 60 for y in the 4th,
120 + 15 9.
= 45. 3
Ans. 45 turkeys, and 60 geese. Let the learner go back and solve, in this manner, the preceding examples in this Art. Sometimes one method is preferable and sometimes the other.
15. A person expends $1 in apples and pears, buying his apples at 3 for a cent, and his pears at 2 cents apiece; afterwards he accommodates his neighbour with f of his apples and 4 of his pears for 30 cents. How many of each did he buy ?
Let x = the number of apples.
&c. 16. A market-woman bought , eggs, some at the rate of 2 for a cent, and some at the rate of 3 for two cents, to the amount of 65 cents ; she afterwards sold them all for 120 cents, and thereby gained one half cent on each egg. How many of each kind did she buy?
17. It is required to find two numbers such, that if of the first be added to the second, the sum will be 30, and if of the second be added to the first, the sum will be 30.
18. It is required to find two numbers such, that of the first and of the second added together will make 12, and if the first be divided by 2 and the second be multiplied by 3, ; of their sum will be 26.
19. Two persons, A and B, talking of their ages, says A to B, 8 years ago I was three times as old as you were, and 4 years hence I shall be only twice as old as you. Required their present ages.
20. There is a certain fishing rod, consisting of two parts, the
upper of which is to the lower as 5 to 7; and 9 times the upper part, together with 13 tires the lower part, is equal to 11 times the whole rod and 8 feet over. Required the length of the two parts.
21. A vintner has two kinds of wine, one at 5s. a gallon, and the other at 12s. of which he wishes to make a mixture of 20 gallons, that shall be worth 8s. a gallon. How many gallons of each sort must he use ?
22. A vintner has 2 casks of wine, from each of which he draws 8 gallons; and finds that the number of gallons remaining in the less, is to that in the greater as 2 to 5. He then puts 1 gallon of water into the less, and 5 gallons into the greater, and then the quantities are in the proportion of 5 to 13. What quantity did each contain at first ?
23. A farmer, after selling 13 sheep and 5 cows, found that the number of sheep he had remaining, was to that of his cows in the proportion of 4 to 3. After three years he found that he had 57 more sheep, and 10 more cows than he had at first; and that the proportions were then as 3 to 1. What number of each had he at first ?
24. When wheat was 8 shillings a bushel, and rye 5 shillings, a man wished to fill his sack with a mixture of wheat and rye, for the money he had in his purse. If he bought 15 bushels of wheat, and laid out the rest of his money in rye, he would want 3 bushels to fill his sack; but if he bought 15 bushels of rye, and then filled his sack with wheat, he would have 15 shillings left. How much of each-must he purchase in order to lay out his money and fill his sacks ?
25. A grocer had 2 casks of wine, the smaller at 7s. per gallon, the larger at 10s. The whole was worth $112. When