Elements of Algebra

If the less of the two numbers were known, the greater could be found by adding to it the difference 19; or in other words, the less number, plus 19, is equal to the greater.

If, then, we denote the less number by x,

and

x+19 will denote the greater,

2x+19 will denote the sum.

But from the enunciation, this sum is to be equal to 67. There fore,

2x1967.

Now, if 2x augmented by 19, is equal to 67, 2x alone is equal to 67 minus 19, or

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The less number being 24, the greater is

x + 19 = 24+ 1943.

And, indeed, we have

43 + 24 = 67, and 43 — 24 = 19.

GENERAL SOLUTION.

The sum of two numbers is a, and their difference is b What are the two numbers?

Let

Then will

x denote the less number;

x+b denote the greater number.

Now, from the conditions of the problem,

x+x+b, or 2x + b

will be equal to the sum of the two numbers: hence,

2x + b = a.

Now, if 2x+b is equal to a, 2x alone must be equai te

If the value of x be increased by b, we shall have the greater number: that is,

That is, the greater of two numbers is equal to half their sum increased by half their difference; and the less is equal to half their sum diminished by half their difference.

As the form of these results is independent of any particular values attributed to the letters a and b, the expressions are called formulas, and may be regarded as comprehending the solution of all problems of the same kind, differing only in the numerical values of the given quantities. Hence,

A formula is the algebraic expression of a general rule, or principle.

To apply these formulas to the case in which the sum is 237 and difference 99, we have

and these are the true numbers; for,

16869237 which is the given sum,

CHAPTER IL

ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION.

ADDITION.

31. ADDITION, in algebra, is the operation of finding the simplest equivalent expression for the aggregate of two or more alge. braic quantities. Such equivalent expression is called their sum.

32. If the quantities to be added are dissimilar, no reductions can be made among the terms. We then write them one after the other, each with its proper sign, and the resulting polynomial will be the simplest expression for the sum. For example, let it be required to add together the monomials

3a, 56 and 2c;

we connect them by the sign of addition,

3a +56 +2c,

a result which cannot be reduced to a simpler form.

33. If some of the quantities to be added have similar terms, we connect the quantities by the sign of addition as before, and then reduce the resulting polynomial to its simplest form, by the rule already given. This reduction will, in general, be more readily accomplished if we write down the quantities to be added, so that similar terms shall fall in the same column. Thus ;

Let it be required to find the sum of the quantities,

Their sum, after reducing (Art. 29), is

3a2 4ab

2a23ab + b2

2ab 5a2 5ab

562

462

34. As operations similar to the above apply to all algebraic expressions, we deduce, for the addition of algebraic quantities, the following general

RULE.

1. Write down the quantities to be added, with their respective signs, so that the similar terms shall fall in the same column.

II. Reduce the similar terms, and annex to the results those terms which cannot be reduced, giving to each term its respective sign.

EXAMPLES.

1. Add together the polynomials,

3a22b2-4ab, 5a2 — b2 +2ab and 3ab 3c2-262.

The term 3a2 being similar to 5a2 we write Sa2 for the result of the reduction of these two terms, at the same time slightly crossing them as in the terms of the example.

382-4ab 242

5×2+2ab

282

+ 3ab
Sa2 + ab 562

3c2

Passing then to the term - 4ab, which is similar to the two terms2ab and +3ab, the three reduce to ab, which is placed after Sa2, and the terms crossed like the first term. Passing then to the terms involving b2, we find their sum to be -562, after which we write 3c2.

The marks are drawn across the terms, that none of them may be overlooked and ornitted.