In the same manner, if the sum were $360; then a would stand for $360, and the son's share would be of 360; that is, $120. And in the same manner we may make a represent any sum; and still the son's share of it would be of it. Hence, this substitution of a letter for a number, is called generalizing the operation.

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We see that this result has given us a general rule for dividing any sum between two, so that one of them shall have twice as much as the other. The rule is, The least share shall be one-third of the sum; and the greatest share, twothirds of it.

§142. In the same manner, we may generalize all the questions in the First Section of Equations. That is, with each question we may find a rule by which any other question like it, may be answered with fewer figures than the Algebraic operation required. But, as such sums are not apt to occur, there will be no practical use of finding rules for them. Notwithstanding, as the exercise may be instructive, the teacher may require his pupils to go through them if he sees fit.

Example 2. What number is that, which, with 5 added to it, will be equal to 40?

This is the first problem in section 2, which we will generalize; using a for 40, and b for 5.

We see that the answer is found by subtracting the 5

from the 40. Thus, 40-5=35.

The third question is similar to it; and in that, a represents 23, and b represents 9. The answer is 23—9=14.

Example 3. Divide $17 between two persons, so that one may have $4 more than the other. [Prob. 4. Sect. 2.] Represent 17 by a, and 4 by b.

The answer is found by subtracting the difference or 4, from the whole sum, and then dividing by 2.

And this is the rule for all similar sums.

The 5th question is similar to it; and in that, a represents 55, and b represents the difference or 7. It is found by the rule just shown.

Perform the 7th and 10th by the same rule.

$143. As this rule is of some importance, it will be well to remember it. If, from a number to be divided into two parts, we subtract the difference of those parts, half the remainder will be equal to the smaller part.

Example 4. In the same questions, let us take a for the greatest share. Then x-b = the less.

Forming the equation,

Transposing and uniting,

Dividing by 2,

x+x-b=a

2x=a+b

x=a+b

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§144. Here we have another rule. If, to a number to be divided into two parts, we add the difference between those parts, half the sum will be equal to the greater part.

Find the greater part in questions 4, 5, 7, and 10, by this rule.

§145. The mere letters in the answer of an algebraical operation are called a formula. They are not called a rule, until they are turned into common language.

Example 5. The learner may now generalize question 6; and by the formula that he obtains, he may find the answers in questions 8, 9, and 11. The two differences will be b and c.

We have said that in Algebra the arithmetical operations on numbers are only represented by different methods of combining the signs that stand for those quantities. And now, although we have shown in our progress thus far, what some of those methods are, it may be well to review them a little.

ADDITION AND SUBTRACTION OF ALGEBRAICAL QUANTITIES.

§146. One algebraical quantity is added to another by writing one quantity after the other, taking care to put the sign plus+between them. Thus, a+f-c is added to d-e+b, so as to make d-e+b+a+f-c. Or, as it is easier to read the letters in their alphabetical order, their sum may be written a+b-c+de+f.

§147. One algebraical quantity is subtracted from another, by changing the sign or signs of the quantity which is to be subtracted, and then writing that quantity after the other. Thus, a+h-y is subtracted from b-x+c, by first

making it -a-h+y, and then writing the whole quantity, b-x+c-a-h+y; or, b—a+ch+y—x.

§148. When quantities that have been made by addition. or subtraction, have like terms in them, they may be reduced to smaller expressions, by uniting the like terms. This is done by putting into one part, all those which have the sign, and into another part, all those which have the sign ; then subtract the least result from the greatest, and give to the remainder the sign that belonged to the greatest result.

§149. In uniting the terms of compound numbers, we consider the literal part of the term as a unit; thus, 2a and 3a, are regarded as 2 units and 3 units of a particular kind, which when put together, make 5 units of that kind. Now, we have seen, §139, that the co-efficient of a quantity may also be literal; as in ba, ca, &c. In such cases, the whole term ba or ca becomes the unit, each of a different kind; and of course are not like quantities and cannot be united.

§150. But if there are several similar units of this kind, they may be united by the general rule. Thus, ba-ca +ba+ca+ba+ca, can be united into, 3ba+ca. ax-bx +ax+2bx-3ax+bx, are equal to, -ax+2bx; or 2bx

-ax.

§151. Again, we have seen, §65, that several quantities are sometimes united by a vinculum. In such cases all that is embraced by the vinculum, is regarded as a unit of that kind; and may have a co-efficient. Thus, in the expres sions, 3xa-b+x, and 5(x+ax-y), a-b+x is a quantity taken 3 times, and x+ax-y is a quantity taken 5 times. Like quantities of this kind can be united; thus, 2(ay—bx+x)+5(ay—bx+x)=7(ay—bx+x).

§152. In uniting terms, great care must be taken that the literal part be entirely alike. Thus, 2bx+3cx, cannot be united. Neither can 3y-2ay; nor, 6(a+bx)+2(ax +bx); nor, 3. ay-by + 2.ay-by, nor, 4(ax-bx)— 2(ax+bx); neither in any other case where there is the least difference in any part but the leading co-efficient.

EXAMPLES.

Unite the following quantities.

1. 3ax-2y+4x-5y+ax-3y. Ans. 8ax-10y. 2. 3x+ay-2x — ay+4x+3ay — 2x+4ay.

Ans. 3x+7ay.

3. 4ax-y+3ay-2-2ax+ay-7y+8+2ay+y. Ans. 2ax+6ay — 7y+6.

4. axay3 3ay+5ax-2ay+7ay-4ax- 8ay3.

Ans. 2ax+2ay-9ay.

Ans. 16(a-y).

5. 3(a — y)+4(a− y)+2(a− y) +7(a—y).

6. —4(a+b)+3(a+b)−2(a+b)+7(a+b.)

Ans. 4. a+b.

7. 2(ab+x)+3(ax+b) — 4(x − y) — 2(ab+x)

Ans. 3(ax+b)- 4(x — y). 8. 7y—4(a + b) + by + 2y + 2(a + b)+(a+b)+y - 3(a + b.) Ans. 16y-4(a+b).

9. x2+ax-ab+ab-x3+xy+ax+xy-4ab+x2+ x3-x+xy+xy+ax. Ans. 2x2+3ax-4ab+4xy-x.

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10. x+12—ax+y (48―x — ax+3y).