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remains after having diminished 36 by 3 x that is to be subtracted from 5; so that the difference 5 x 36 ought to be augmented by 3x in order to form the quantity that should remain after having taken from 5 x the number denoted by 36 → 3x. This quantity will then be

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x = 64 = 8.

There have been then 8 successful throws of the net and 4 unsuccessful ones.

Indeed 8 throws at 5 cents a throw give 4 throws at 3 cents a throw give

difference

as required by the conditions of the question.

40 cents,

12

28

To render the solution general, let a represent the sum given by the father to the son for each successful throw of the net, and b the sum returned by the son for each unsuccessful one, and c the total number of throws, and d the sum received on the whole by the son. If x be put equal to the number of successful throws, C- x will express the number of unsuccessful ones; each throw of the former kind being worth to the son a sum a, x throws would be worth a X x or a x, and the unsuccessful throws would be worth to the father the sum b multiplied by the number c-x. The reasoning by which we have found the parts of the product of 3 by 12-x, applies equally to the general case. If we neglect in the first place in forming the product be of b by the whole of c, the sum b will be repeated a times too much, and consequently the true product will be bc-bx.

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In order to subtract this product from the sum a x, it is necessary to observe, as in the numerical example, that if we subtract the whole of the quantity bc we take the quantity be too much, by which the former ought to have been first diminished, and that consequently the true remainder is not merely a x-bc, but a x -bc + bx.

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As this sum is equal to d, we have the equation

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As this general formula indicates what operations are to be performed upon the numbers a, b, c, d, in order to obtain the unknown quantity x, we may reduce it to a rule or carefully write instead of the letters a, b, c, d, the numbers given. This last process is called substituting the values of the given quantities, or putting the formula into numbers. Applying here those of the foregoing example, we have

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Methods for performing, as far as is possible, the Operations indicated upon Quantities that are represented by Letters.

16. FROM the preceding question it is evident, that in certain cases a multiplication indicated upon the sum or difference of several quantities is made to consist of several partial multiplications; and in art. 11, we have exactly the reverse, by resolving the quantity axbx + cx, which represents the result of several multiplications, followed by additions and subtractions, into the two factors a b + c and x, which indicate only a single multiplication preceded by addition and subtraction. The reasoning pursued in these two circumstances, will suggest rules. for performing, upon quantities represented by letters, operations which are called algebraic multiplication and division, from the analogy which they have with the corresponding operations of arithmetic.

We have also by the same analogy two algebraic operations, which bear the names of addition and subtraction, in which the object is to unite several algebraic expressions in one, or to take one expression from another. But these operations, like the preceding, differ from those of arithmetic in this, that their results are, for the most part, only indications of the operations

to be performed; they present only a transformation of the operations originally indicated into others, which produce the same effect. All that is done, is either to simplify the expressions, or to give them a proper form for exhibiting the conditions that are to be fulfilled.

In order to explain these operations, we give the name of simple quantities to those which consist only of one term, as + 2 α, 3 ab, &c. binomials to those which consist of two, as a + b, α b, 5 a 2 x, &c. trinomials to those which consist of three terms, quadrinomials to those which consist of four terms, and polynomials to those which consist of more than four terms. It may be observed also, that we call polynomials compound quantities.

Of the Addition of Algebraic Quantities.

17. THE addition of simple quantities is performed by writing them one after the other, with the sign + between them; thus, a added to b is expressed by a + b. But when it is proposed to add together several algebraic expressions, we aim at the same time to simplify the result by reducing it to as small a number of terms as possible by uniting several of the terms in one. This is done in articles 2 and 5, by reducing the quantity x + x to 2x, and the quantity x+x+x to 3 x. It can take place only with respect to quantities expressed by the same letters, and which are for this reason called similar quantities. A literal quantity that is repeated any number of times is regarded as a unit, it is thus, that the quantities 2 a and 3 a considered as two and three units of a particular kind, form when added 5 a or 5 units of the same kind. Also 4 a b and 5 a b make 9 a b.

In this case, the addition is performed with respect to the figures which precede the literal quantity, and which show how many times it is repeated. These figures are called coefficients. The coefficient then is the multiplier of the quantity before which it is placed, and it must be recollected, that when there is none expressed, unity is understood; for 1 a is the same as ɑ.

18. When it is proposed to unite any quantities whatever, as 4 a 5b and 2 c + 3 d,

the sum total ought evidently to be composed of all the parts joined together; we must write then

4a5b2c+ 3 d.

4a5b and 2 c - 3 d.

If we have on the contrary

The sign

must be retained in the sum, to mark as subtractive the quantity 3 d, which, as it is to be taken from 2 c, must necèssarily diminish by so much the sum formed by uniting 2 c with the first of the quantities proposed; we have then,

4a + 5b + 2 c. 3 d.

From these two examples it is evident, that in algebra the addition of polynomials is performed by writing in order, one after the other, the quantities to be added with their proper signs, it being observed that the terms which have no signs before them are considered as having the sign +.

The above operation is, properly speaking, only an indication. by which the union of two compound quantities is made to consist in the addition and subtraction of a certain number of simple quantities; but, if the quantities to be added contained similar terms, these terms might be united by performing the operation upon their coefficients.

Let there be, for example, the quantities

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2 a 3 c + 4d,
76+ c

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- e ;

the sum indicated would be, according to the rule just given, 2c+2a 3 c4d + 76+ c —

4a+9b

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But the terms 4a, + 2a, being formed of similar quantities, may be united in one sum equal to 6 a.

Also the terms + 96, +76 give + 16 b.

The terms - 2 c and 3 c, being both subtractive, produce on the whole, the same effect as the subtraction of a quantity equal to their sum, that is to say, as the subtraction of 5c; and as by virtue of the term + c, we have another part c to be added, there will remain therefore to be subtracted only 4 c.

The sum of the expressions proposed then, will be reduced to 6a16b- 4 c + 4 d e.

The last operation exhibited above, by which all similar terms are united in one, whatever signs they have, is called reduction. It is performed by taking the sum of similar quantities having the sign +, that of similar quantities having the sign-, and subtracting the less of the two sums from the greater, and giving to the remainder the sign of the greater.

It is to be remarked, that reduction is applicable to all algebraic operations.

The following examples of addition, with their answers, are intended as an exercise for the learner.

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Answer, 7m+3n-14p+17r+3a+9n-11m+2r+5p-4m+8n +11n2bm-r+ s.

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9p+18r+ 3 a
9p+18r+ 3 a

2b+s, 2b+s,

by beginning with the term having the sign +.

2. To add the quantities

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11bc4ad-8ac + 5 c d +8 ac+7bc2ad +4 m n 2cd-3ab5ac +an+9an-2bc-2ad+5cd.

By reducing this quantity it becomes

16 b c + 5 ac + 12 c d + 4 m n — 3 ab + 10 an.

Of the Subtraction of Algebraic Quantities.

20. THE subtraction of single quantities, according to established usage, is represented by placing the sign between the quantity to be subtracted, and that from which it is to be taken; b subtracted from a is written a

b.

When the quantities are similar, the subtraction is performed directly by means of the coefficients.

If 3 a be subtracted from 5 a, we have for a remainder 2 a. With regard to the subtraction of polynomials, it is necessary to distinguish two cases.

1. If the terms of the quantity to be subtracted have each the sign, we must clearly give to each the sign, since it is required to deduct successively all the parts of the quantity to be subtracted.

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