17. A is 2 years older than B, and B 4 years older than C. What will represent the sum of their ages, C's being represented by x?

18. B has $4 more than A, and $5 less than C. What will represent the number of dollars B and C have, respectively, that which A has being x?

19. À begins trade with x dollars, and gains $100; B begins with twice as much money as A, and gains $150; C begins with as much as A and B both, and gains $70. What will represent the amount of their property, including stock and gain?

ART. 8. 1. If x represent the number of dollars A has, and B have $2 less, what will represent B's money? Ans. x

- 2.

2. If x represent the number of cows a farmer has, what will represent the number of his cows after he shall have sold 10?

3. If a shillings represent the price of a bushel of wheat, and rye be 4 s. a bushel cheaper, what will represent the price of a bushel of rye?

4. A drover, having 100 sheep, sold a number of them represented by x. What will represent the number he had remaining? Ans. 100

-X.

5. Of 150 acres of land, a part is cultivated, and the rest is woodland. If x represent the number of acres cultivated, what will represent the number of acres of woodland?

6. The sum of two numbers being 20, what will represent the second, if x represent the first?

7. A man has 50 coins, consisting of dollars and sovereigns. If x represent the number of dollars, what will represent that of the sovereigns?

8. A walked a number of miles, represented by x; B

walked twice as far wanting 4 miles. What will represent the distance B walked?

9. A man having a certain number, x, of dollars, doubled his money, and afterwards lost $12. What will represent the money he then had?

10. A draper, having 50 yards of cloth, sold a number of yards, represented by x. What will represent the number of yards remaining?

11. A grocer, having 500 lbs. of coffee, sold 3 bags, each containing z lbs. What will represent the number of pounds remaining?

12. A has x, and B, y dollars. If A give B $5, what will represent the number of dollars each has then?

Ans. A, x-5, and B, y +5. 13. If A have 3 x, and B, 2 y dollars, what will represent their money, after B shall have given A $10?

14. A farmer has 100 sheep in one pasture and 75 in another. What will represent the number in each flock, after he shall have taken 2x sheep from the larger and put them with the smaller?

15. A and B have each 50 cows. A sells a number, x, and B sells 3 times as many. What will represent the number each has left?

16. A poulterer has x turkeys and y geese. After selling 4 turkeys for 15 geese, what will represent the number of each he then has ?

17. If a woman be x years old, and her husband twice as old as she, at the time of marriage, what will represent their respective ages 5 years before marriage?

ART. 9. According to Ax. 5, if the same quantity be both added to, and subtracted from another, that other will not be changed in value. Thus, if 2 be added to 8,

the sum will be 10.

Now, if 2 be subtracted from 10,

the remainder will be 8, the same as at first.

If these operations are represented merely, the result will be expressed thus, 8+2-2, in which +2 and -2 destroy or cancel each other, and leave 8.

But if more is added to a quantity than is subtracted from it, that quantity is increased by the excess of what is added above what is subtracted. Thus, 12+7-6 is the same as 12+1, or 13, because 1 more is added to 12 than is subtracted from it.

On the other hand, if more is subtracted from a quantity than is added to it, that quantity is diminished by the excess of what is subtracted above what is added. Thus, 15—73, or 15+3-7, is the same as 15—4, or 11.

In like manner, x+5—3, is the same as x+2; and x-8+5, is the same as x- 3.

1. What is the same as x- 4+4? as x+6-6? as 5-5+x? as-7+7+2x?

2. What is the same as x+9-3? as x· 9+3? as 2x-7+3? as 3-7+2x? as 5x-5+7? as 5x+ '7—9?

3. What is the same as 4x-6+3? as 7x+10—4? as 12x-12+7?

The several parts of an algebraic quantity, connected together by the signs and

10+5 +3,72,

arate terms.

Thus,

are called terms. 10, 5x, and 3 are the sep

When a quantity contains many terms, all those consisting of numbers only may be reduced to one single term, and those which are alike with regard to the letters, to another. But it must be remembered, that when a term has no sign before it, it is supposed to have the sign +. Thus, x+2x+5 x'is the same as 8x; 4x+2+3x+7 is the same as 7x+9; 10x-2+3x- -5 is the same

as 13x-7. In the last example, 10 x and 3 x are added, whilst 2 and 5 are both subtracted, which is the same as subtracting their sum, 7

In a similar manner, 10x+7+3x— 5 is the same as 13x+2; 12x-8+4x+3 is 16x-5; and 12x +3-3x-8 is 9 x-5.

Again, 4x+10+3x-7+8x-3-4x6-3x -4 is 8x+2; 3x-7+4x+2-x+1+5x—3— 5 x is 6x-7; 12-3x+7+2x-3-5x is 166x; and 7+3x−4+2x−x + 2 is 5 + 4 x.

In the last four examples, we combine all the terms containing x, and preceded by the sign +, expressed or implied; then combine all those containing x, and preceded by the sign, and take the difference of the two sums, giving it the sign of the greater sum. With the numbers we proceed in the same way.

4. What is the same as x+2+3x-4+5x+7? as 4x-2+2x+7+3x-9+x+10? as 11x-6+ 3x+7-6x-4? as 7+3x-6+4x+10-13 x ? as 5x+7-10x+3-4x-2+x+5?

Let the answers to the following questions be reduced in the manner shown above.

5. A is x years old; B is twice as old as A, and three years more; and C's age is 2 years less than the sum of A's and B's. What will represent C's age?

A gains

Ans. x+2x+3-2, or 3x+1. 6. A has x dollars, and B has twice as many. $20, and B loses $5. of A's and B's money?

What will then represent the sum

Ans. x+20+2x-5, or 3x+15.

7. A has $50, and B $30. A loses x dollars, and B gains 4 times as much money as A loses. What will then represent the amount of their money?

Ans. 50-x+30 +4 x, or 80+3x.

SALEM FRATERNITY

PRELIMINARY EXERCISES.

15

Observe that, in the three preceding examples, the addition of quantities containing several terms is performed by writing them after each other, without altering the signs; after which the result is reduced. That the signs should not be changed, may easily be shown by figures. Thus, 12-5 and 17-3, when added, give 12-5+ 17 — 3, which is the same as 29—8 or 21; for 12—5 is 7, and 17-3 is 14, and the sum of 7 and 14 is 21.

8. An army consists of x officers, 6 more than 3 times as many cavalry, and 30 less than 10 times as many infantry as officers. What will represent the number of men in the army?

9. A is 10 years younger, and C 20-years older, than B. What will represent the sum of their ages, if x represent the number of years in B's age?

10. Four towns are in a straight line, and in the order of the letters A, B, C, and D. The distance from A to B is 20 miles more, and the distance from C to D is 30 miles less, than the distance from B to C. What will represent the whole distance from A to D, if x represent the number of miles from B to C?

11. Three men owed x guineas. A could pay the whole debt wanting 10 guineas, B could pay it wanting 20 guineas, and C could pay it and have 15 guineas left. What will represent the number of guineas they all had?

12. A man has lived x years in France, 5 years more than twice as long in England, and 15 years less than 3 times as long in America. These being his only places of residence, what will represent his age?

ART. 10. When a quantity containing several terms is to be multiplied, each term must be multiplied, and the signs remain unchanged, except under particular circumstances, which will be hereafter explained.