equal to x; for, things equal to the same thing are equal to each other.

2.

If

8x+5y=26 And 4x+3y=14

What are the values of x and y?

3. A man has two horses, and a chaise which is worth 180 dollars. If the first horse be harnessed to the chaise, they will, together, be equal to twice the value of the second horse. If the second horse be harnessed to the chaise, they will, together, be equal to three times the value of the first horse. What is the value of the two horses?

4. A says to B, if you were 5 years younger, and I, 10 years older, I should be three times as old as you; but, if I were 8 years older, and you 1 year younger, I should be just twice as old as you. How old are A and B ?

5. A and B have each a sum of money. If B give A one dollar, B will have as much as A; but, if A give B one dollar, B will have twice as much as A. How much money has A, and how much has B?

6. A man bought two lots of ground, one at 10 and the other at 15 dollars per acre, for 270 dollars; he then sold one at 12 dollars, and the other at 20 dollars per acre, and gained 66 dollars. How many acres of each kind were there?

7. A man bought two kinds of wine for 10 dollars, one at 1 dollar per gallon, and the other at 50 cents. He then mixed the two together, and sold the mixture at 1 dollar per gallon, and by that means he gained 5 dollars. How many gallons of each kind did he buy?

SECTION XXVI.

Simple Equations with Two Letters.

Two equations containing x and y may be reduced by mu tiplying one or both equations, so as to get the same number of x's or y's in both equations, and then subtracting one equation from the other.

1. If And

(1) 4x+3y=302

2x+y=145
14 (4)

Multiply the second of these equations by 3, and we have the two following equations; namely,

4x+3y=302
6x+3y=42 S

(B)

Subtract the first of these last two equations (B) from the last of the two, and we have

2x=12

x= 6 &c.

We might have multiplied the second of equations (A) by 2, and subtracted it from the first, and we should have obtained immediately y=2.

(1)

Thus

4x+3y=30
4x+2y=28

(A)

3y =

3

What are the values of x and y?

Add the two equations together, and we have

8x 32
x = 4
y 3

Here we added the equations together in order to eliminate because the y's in the two equations have different signs.

5. A man bought 4 turkeys and 3 dozen chickens for $2.36.. He afterwards bought 2 turkeys and 4 dozen chickens, for $1.98. What was the price of a turkey, and of a dozen of chickens?

6. A pole is partly under water. The part under water is 22 feet less than four times the part out of water; and three times the part under water is equal to twice the length of the pole. What is the length of the pole?

7. A man has 20 gallons of three different kinds of wine. The first is worth one dollar per gallon; the second two dol

lars, and the third three dollars per gallon. The whole co$43; and what the first and second cost is 5 dollars less than what the third kind cost. How many gallons were there of

each kind?

Let the number of gallons of the first kind,
y = the number of gallons of the second kind,
-y the number of gallons of the third kind.

20-x

(1) x+2y+60-3x-3y=43

x+2y+560-3x-3y
-2xy=— 17

8. Divide the number 30 into three parts, such, that twice the first, three times the second, and four times the third, may make 86; and that the third added to four times the first, may be equal to six times the second, wanting 4. What are the parts?

9. Three men commenced trade on 200 dollars. What the first and second put in, is equal to three times what the third put in; and the difference between what the first and second put in, is equal to of what the third put in. What did each put in?

10. Four men, A, B, C, D, commenced trade together. B had twice as much as A; and the sum of A's and B's money was 20 more than the sum of C's and D's shares. B's share is equal to twice the difference between C and D ; and the sum of all these shares, makes 100 dollars. What was each man's share?

11. What fraction is that, to the numerator of which, if 1 be added, the fraction will be ; but, if 1 be added to the denominator, the fraction will be

?

12. What fraction is that, to the numerator of which, if 2 be added, the fraction will be; but, if 2 be added to the denominator, the fraction will be ;

1, 2, 3, 4, 5, 6, 7, 8, 9, are called digits. A number expressed by two digits, as 86,= 10 times the left hand digit plus, the right hand one.

13. There is a number consisting of two digits. If it be

divided by the sum of its digits, the quotient will be 6; and, if 9 be subtracted from the number, the result will be a number expressed by the digits inverted. What is the number?

x= the digit on the left hand,
y =the digit on the right hand.
the number.

Let

And

Then

10x+y

10x+y

And

x + y

- 6

And 10x+y-9-10y+x, &c.

14. There is a number consisting of two digits, which, divided by the sum of its digits, gives 4; and, if 27 be added to the number, the result will be a number expressed by the digits inverted. What is the number?

SECTION XXVII.

Equations of the Second Degree with Two Letters.

If we wish to express the result of one letter multiplied by another, as x multiplied by y, we write it thus: x Xy, or x.y, or xy.

If xxy, or x.y, or xy, be divided by y, the quotient will be x. Why? xy may be written y x. For x multiplied by y, is the same as y multiplied by x. If we wish to express two, three, or four times xy, we can write it 2xy, зxy 4xy, &c.

1. If x= 2, and y = 3; 2. If x = 4, and y = 6; 3. If x = 8, and y =

what would x y be equal to? what would x y be equal to? 3; what would 6 x y be equal to ?

4. Which is the greater, six times x multiplied by y, or six times y multiplied by x?

5. If 7 x y be divided by y; what will the quotient be?

6. The product of two numbers is equal to three times their sum; and, if the sum be divided by the less of the two numbers, the quotient will be 4.

Let
And

(1)

(2)

What are the numbers?
x = the greater,
y the less.

xy=3x+3y x + y

4

y

From the second we get, by clearing it of fractions,

x+y=4y
x=3y

Put 3 y in place of x in the first, and it becomes

7. The product of two numbers is equal to twelve times their difference; and the sum of the two numbers divided by the less, will give 3 for the quotient. What are the numbers? 8. A man had two flocks of geese. Twice the difference of the numbers of geese in the two flocks, was equal to their He sold each flock for as many cents per goose as there were geese in the other flock, and found that he had received as much money for the whole, as his neighbor, who had sold as many geese at 30 cents a-piece. How many geese were there in each flock?

sum.

9. There are two numbers which have a certain product. If the first were 2 less, and the second 3 more, the product would be the same; and, if the first were 4 less, and the second 8 more, the product would be the same. What are the

numbers?

10. A certain number of men have to pay a certain sum of money. If there were one man more, they would have to pay 2 shillings a-piece less; and if there were one man less, they would have to pay 4 shillings a-piece more, How many

men were there, and how many shillings had each to pay?

11. Twice the product of two numbers is equal to eight times the less number; and five times their difference is equal to their sum. What are the numbers?

12. The quotient of one number by another is equal to 25; and the product of the two is equal to 100. What are the

numbers?

13. Two men were 48 miles apart and set out with a design to meet; when they met, each had traveled as many hours as the other traveled miles per hour; and the sum of the numbers of miles they traveled per hour, was equal to five times the difference. How many hours, and at what rate did they travel?