1

$250. If the first be harnessed, the horse and chaise will be worth twice as much as the second horse; but if the second be harnessed, they will be worth three times as much as the first horse. What is the value of each horse?

19. A merchant bought two lots of flour, for $576; the first lot for $5, and the second for $6, per barrel. He then sold of the first lot and ≠ of the second for $353, by which he gained $11. How many barrels were there in each lot?

20. What fraction is that, whose numerator being doubled, and denominator increased by 7, the value becomes; but the denominator being doubled, and the numerator increased by 7, the value becomes 1?

21. A teacher, being asked the dimensions of his school-room, answered, that if it were 5 feet broader and 3 feet longer, the floor would contain 422 feet more; but if it were 3 feet broader and 5 feet longer, the floor would contain 400 feet more. What were the dimensions of the room?

Let x = the length of the room,
and y = the breadth.

Now, to find the area, we must multiply the length by the breadth. Consequently, xy = the area of the floor.

Then (x + 5) × (y + 3) = x y + 400,
and (x+3) × (y + 5) = xy + 422.

22. If 6 feet were added to each side of a hall, the breadth would be to the length as 6 to 7; but if 6 feet were taken from each of the sides, they would be to

each other as 4 to 5. Required the dimensions of the hall. 56.98.

23. A farmer has 86 bushels of wheat at 4s. 6d. a bushel, with which he wishes to mix rye at 3s. 6d. a bushel, and barley at 3s. a bushel, so as to make 136 bushels, that shall be worth 4s. a bushel. How much rye and barley must he take?

24. A merchant put 13 crates and 33 bales of goods into a warehouse, which was all that it would hold. After he had removed 5 crates and 9 bales, he found that the house was two thirds full. How many crates or bales would it take to fill it?

25. If you multiply the greater of two numbers by 3 and the less by 4, the difference of their products is 48; but if you divide the greater by 4 and the less by 3, the sum of their quotients will be 14. What are the numbers?

26. A gentleman, having a quantity of gold and silver coins, finds that 24 pieces of gold and 40 pieces of silver, will pay a certain debt; of which 5 pieces of gold and 15 pieces of silver, will pay + part. How many pieces of gold, and how many of silver, will pay the whole debt?

27. In an election, the two candidates received 1384 votes; but if the successful candidate had received but half his number of votes, and three times as many as he received had been given for the other, the whole number of votes would have been 2102. How many votes were given for each?

28. The length of a certain garden, which contains 128 square rods, is twice as great as its width; and if the garden were 4 rods longer, it would contain an acre. Required its length and width.

ELIMINATION BY ADDITION AND SUBTRACTION.

29. A man bought 3 bushels of wheat and 5 bushels of rye for 38 shillings; he afterwards bought 6 bushels of wheat and 3 bushels of rye for 48 shillings. What did he give a bushel for each?

Let x = the price of a bushel of wheat,
and y = the price of a bushel of rye.

A. Then 3 x + 5y = 38, by the first purchase, в. and 6 x + 3y = 48, by the second purchase. If we multiply all the terms of equation A by 2, [See Sec. I. of this Chap.] we shall have

c. 6 x + 10 y = 76.

Now, if we subtract equation B from equation c, the remainder will be a new equation, containing only one unknown quantity, whose value may be found as before. Thus,

c. 6 x + 10 y = 76

в. 6 х + 3 у = 48

D.

*

7 y = 28, by subtraction.

y = 4, the price of the rye.

By substituting this value of y, in either of the

above equations, we shall obtain the value of x.

x = 6, the price of the wheat.

The student will now understand why equation a is multiplied by 2. Having determined to eliminate x, I wish to make the coefficients of the terms containing a, in the equations A and B, equal, that they might cancel each other when subtracted. It matters not which of the unknown quantities is eliminated in this way.

30. A boy bought 7 oranges and 5 lemons for 55 cents; and afterwards let one of his companions have 4 oranges and 3 lemons for 32 cents, which was their cost. What was the price of each?

Let x = the price of an orange, and y = the price of a lemon. x+5y B. and 4 x + 3y = 32,

A. Then 7

=

55,2 by the conditions of the

question.

1

We will first eliminate y. As we cannot multiply either equation by any number, which will make the coefficients of the terms containing y alike, we must multiply both equations by such numbers as will produce this result.

c. 21 x + 15 y = 165, by multiplying equation A by 3. D. 20 x + 15 y = 160, by multiplying equation в by 5.

в. (4×5)+3у = 32, by substitution.

3y = 32-20 = 12; and y = 4.

ANS. Oranges, 5; lemons, 4 cents. 31. A gentleman paid for 6 pair of boots and 8 pair of shoes $52; he afterwards paid for 3 pair of boots and 7 pair of shoes $32. How much were the boots and shoes a pair?

Let x = the price of the boots,

and y = the price of the shoes.

A. Then 6 x + 8y = 52, by the conditions of the в. and 3 х + 7 y = 32, 5

question.

c. 3 x + 4y = 26, by dividing equation a by 2.

E. * 3y = 6, by subtracting equation c

from в.

y = 2, the price of a pair of shoes.

c. 3x + (4x2) = 26, by substitution.

and x = 6, the price of a pair of boots.

The equation A is divided by 2, to make the coefficient of x, the quantity to be eliminated, equal to the coefficient of the same quantity in equation B.

32. If twice A's money be subtracted from 3 times B's, the remainder is $38; but if twice B's money be subtracted from 3 times A's, the remainder is $83. How much has each?

If we multiply equation A by 3, and equation B by 2, the coefficients of the terms containing will be alike.