15.) From 24x+3yz-12 abc-6mnp+5 Take 16 abc-6mn p+23x + y z +12—xy.

21.), From 3 a x + 5 ay-2 a b take 2 a x + 5 ay

and the reasoning which follows them, we derive the following general RULE for Subtraction in Algebra : Change all the signs in the quantity to be subtracted, each + to and each to +; and unite the terms

that are similar, as in Addition.

The subtraction is, in fact, performed, when the signs of the terms to be subtracted are changed. The object of the remaining part of the operation, is, to reduce the number of terms, by uniting or cancelling such as are similar, that the answer may be presented in its simplest form.

The expression used in this question implies, that the whole quantity included in the parenthetical marks (), namely, a+b-c, is to be subtracted from x; of course, all the signs must be changed.

4. What is the value of a b-(- cx + d — 16)? ANS. abc x

5.) From 6x + 3y Take 5 x 4 y

d + 16.

It is recommended to the student, in performing these examples, actually to change the signs; at least, until he becomes perfectly familiar with the operation. The last two examples, thus prepared, will stand

CHAPTER IV.

MULTIPLICATION.

SECTION 1.

Simple Quantities.

1. What will 5 oranges come to, at a cents apiece? ANS. 5 a cents.

It is evident that 5 oranges must cost five times as much as one orange; therefore, if one orange cost a cents, whatever the value of a may be, 5 oranges will cost 5 a cents. Let a 2 cents; then 5 a = 5 X 2 or 10 cents.

MULTIPLICATION is merely a short way of performing ADDITION, when the quantities to be added happen to be equal. When these quantities are unequal, their united value or amount can be found only by adding them all together. And it is evident that, when they are equal, their amount can be found in the same way.

2+2+2+2+2=10, or 2×5=10;

where, in the first column, the same results are ob

tained by addition, as are obtained, in the second, by multiplication.

So, too, in literal quantities; if one orange cost a cents, two oranges will cost a + a = 2 a; three, a + a + a = 3 a; four, a + a + a + a = 4 a ; and five, a + a + a + a + a 5 a cents, as in the question.

To multiply a literal quantity by a number, therefore, all that is required is, to make the number the coefficient of the quantity; for the coefficient shows how many times the value of the quantity is to be taken.

In Algebra, as in Arithmetic, the number to be multiplied is called the multiplicand; the number by which we multiply, the multiplier; and the result of the operation, the product. The multiplier and multiplicand, when spoken of together, are called factors. 2. If a yard of cloth is worth a dollars, what is the value of 6 yards? ANS. $6 a.

It is recommended to the student to prove his answers to the questions, by substituting such numbers for the letters given, as he pleases.

3. If there be a apples in a bushel, how many are there in 9 bushels? ANS. 9 x apples. 4. If the interest of a given sum of money be y dollars per annum, what will the interest be for 8 years?

5. How many bushels of corn are there in a field of 12 acres, which produces a bushels to the acre? 6. How much will 10 yards of cloth come to, at c dollars per yard?

7. A man gave away a dollars every day; how much did he give away in a week?

8. What are a man's expenses for a year, who expends a dollars a day?

9. If there be x yards of cloth in one piece of linen, how many yards are there in 15 pieces?

10. How many rods of wall can a man build in three weeks, if he build a rods a day?

11. How many panes of glass are required for 14 windows, each window having c panes ?

12. What is the value of a yards of calico, at c cents a yard? ANS. a c cents.

It is evident that the price, c, must be multiplied by the quantity, a; which may be expressed thus, a × c, or thus, without the sign, a c. Let a 7, and c = 12; then a ×c, or a c = 7 × 12, or 84.

If the price had been c d cents, we should have multiplied that quantity by a, and the answer would have been a c d.

Or, if the quantity had been a b yards, and the price c d cents, we should still have multiplied the one by the other, and the answer would have been a b c d.

Hence, to multiply one simple literal quantity by another, we write all the letters in both quantities together.

13. If x men can do a piece of work in y days, how long will it take one man to do it? ANs. x y days. 14. Multiply a x by y.

15. Multiply b c by m n.
16. Multiply a b x by y z.
17. Multiply a m n by x y z.

ANS. a xy.