Solution.

Let us first establish, by the aid of the algebraic symbols, the connexion which exists between the given and unknown numbers of the question.

If the least 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 make

and

= the less number,

x+19= the greater,

2x+19= the sum.

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

fore we have

2x + 1967.

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

2x = 67 - 19, or performing the subtraction, 2x = 48.

Hence, x is equal to half of 48, that is,

General Solution of this Problem.

The sum of two numbers is a, and their difference is b. What

2. Reduce the polynomial 7abc2 - abc2 - 7abc2 - 8abc2 + 6abc2

REMARK. It should be observed that the reduction affects only the co-efficients, and not the exponents.

The reduction of similar terms is an operation peculiar to algebra. Such reductions are constantly made in Algebraic Addition, Subtraction, Multiplication, and Division.

30. In the operations of algebra, there are two kinds of quantities which must be distinguished from each other, viz.

1st, Those whose values are known or given, and which are called known quantities; and

2dly, Those whose values are unknown, which are called unknown quantities.

The known quantities are represented by the first letters of the alphabet, a, b, c, d, &c.; and the unknown, by the final letters, x, y, z, &c.

31. A problem is a question proposed which requires a solution. It is said to be solved when the values of the quantities sought are discovered or found.

A theorem is a general truth, which is proved by a course of reasoning called a demonstration.

32. The following question will tend to show the utility of the algebraic analysis.

1

Question.

The sum of two numbers is 67, and their difference 19; what are the two numbers?

Solution.

Let us first establish, by the aid of the algebraic symbols, the connexion which exists between the given and unknown numbers of the question.

If the least 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.

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

fore we have

2x + 1967.

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

2x = 67 19, or performing the subtraction, 2x = 48.

Hence, x is equal to half of 48, that is,

General Solution of this Problem.

The sum of two numbers is a, and their difference is b. What

are the two numbers?

Let

=

the less number;

And by adding b to each side of the equality, we obtain the greater number,

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 questions of the same nature, differing only in the numerical values of the given quantities. Hence,

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

The principles enunciated by the formulas above, are these: The greater of any 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.

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

the greater number

and the less

237 99 237 +99
= +
2 2

=

2 237 99 237-99 2 2 2 and these are the true numbers; for,

=

=

168 + 69 = 237 which is the given sum,

and 168 69 = 99 which is the given difference. From the preceding explanations, we see that Algebra is a language composed of a series of symbols, by the aid of which, we can abridge and generalize the operations required in the solution of problems, and the reasonings pursued in the demonstration of theorems.

CHAPTER II.

OF ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION.

ADDITION.

33. ADDITION, in algebra, consists in finding the simplest equivalent expression for several algebraic quantities. Such equivalent expression is called their sum.

34. Let it be required to add together the monomials,

The result of the addition is

56

2c

3a +56 + 2c

an expression which cannot be reduced to a more simple form.

35. As a course of reasoning similar to the above would apply to all algebraic expressions, we deduce, for the addition of algebraic quantities, the following general

RULE.

I. Write down the quantities to be added, with their respective signs, so that the similar terms shall fall under each other.

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