FIRST LESSONS IN ALGEBRA.

CHAPTER I.

INTRODUCTION.

SECTION I.

Letters and Coefficients.

1. A BOY bought a peach and a melon for 12 cents, and the melon cost three times as much as the peach. What was the price of each?

Let the letter x represent the number of cents the boy gave for the peach; then, as he gave one a for the peach, whatever the value of r may be, and as the melon cost three times as much, he must have given three x's for that; and, of course, he gave one x and three x's, that is, four x's for both. But, by the question, he gave 12 cents for both; therefore, the four 's must be equal in value to the 12 cents. But, if four x's are equal to 12 cents, one must be equal to one fourth part of 12 cents, or 3 cents, which was the price of the peach; and if one x be

equal to 3 cents, three x's must be equal to three times 3 cents, or 9 cents, the price of the melon.

It will be observed, that, in this operation, the answer of the question, the thing unknown, is assumed and represented by the letter x, which is therefore called the unknown quantity. Any other letter, mark or character, may be used with equal propriety, provided always that its value be indefinite. This

will be evident if the word share or part be substituted for the letter x, in the above operation. It is usual, however, to represent unknown quantities by the last letters of the alphabet, as x, y, z.

It is sometimes necessary to express quantities, whose values either are, or are supposed to be, determined, by letters. These are called known quantities, and are usually represented by the first letters of the alphabet, as a, b, c.

2. John is four times as old as James, and the sum of their ages is 20 years. What is the age of each? Let x represent the age of James; then, as John is four times as old, four x's will represent his age; and their joint ages must be one x and four x's, that. is, five x's. But the sum of their ages is 20 years, by the question; therefore, five x's must be equal to 20 years, and one x to one fifth part of 20, namely, 4 years, which is the age of James; and, if one x be 4 years, four x's must be four times 4, or 16 years, the age of John.

Instead of writing one x, three x's, four x's, five x's, &c., as in these examples, we use the expressions x, 3 r, 4 x, 5 x, &c. The numbers placed before the

letters, as 3, 4, 5, are called their coefficients. When no number is placed before a letter, as x, its coefficient is always understood to be 1.

In the last example, the algebraic expression for John's age was 4 x, and the value of x was found to be four years. To find the age of John in years, this value of x was multiplied by 4, its coefficient. Any quantity is always supposed to be multiplied by its coefficient. Thus, if the value of x be 6, 3 x will be 3 times 6, or 18; and if the value of x be 10, then 3 x will be 3 times 10, or 30; and 7 x will be 70, 9 x will be 90, and so on.

It is often convenient, in algebraic calculations, to use a letter for a coefficient, instead of a number, as m x, where m is regarded as the coefficient of x; thus, if m be 3, and x be 5, m x will be 3 times 5, or 15.

3. A leaves Boston, and walks three miles an hour, and B leaves Newburyport, at the same time, and walks 5 miles an hour. In how many hours will they meet, the places being 32 miles apart?

In this question, the thing required is, in how many hours A and B will meet; that is, how many hours they will travel. Let it be assumed that they will meet in a hours. Then if A walk 3 miles in 1 hour, in x hours he will walk x times 3 miles, that is, x 3 or 3 x miles; and if B walk 5 miles in 1 hour, in x hours he will walk x 5 or 5 x miles; and they will both walk 3 x and 5x, or 8 x miles, which is the whole distance. But the distance given in the question is 32 miles; therefore, 8 x miles must be the same as 32 miles; or, to use a general expression, 8 x is equal in

value to 32. And if 8x be equal to 32, x must be one eighth part of 32, or 4. They will meet in 4 hours.

The expressions x 3 and 3 x, used in this operation, mean the same thing; for any two or more quantities are supposed to be multiplied together, when they are not separated; and, of course, it is of no consequence which is placed first. Thus, if the value of x be 4, as in the question, 3 x is 3 times 4, or 12; and a 3 is 4 times 3, or 12. But it is more convenient to place the number before the letter, which is always done.

4. A farmer sold a calf, a sheep and a cow, for 36 dollars; for the sheep he received twice as much as for the calf, and for the cow three times as much as.for both the calf and the sheep. What was the price of each?

Let x represent the price of the calf; then 2x will be the price of the sheep; x and 2x, or 3x, will be the price of the two, and three times 3 x, or 9 x, will be the price of the cow. The three animals were, therefore, sold for x, and 2 x, and 9 x, that is, for 12 x. But, according to the question, they were sold for 36 dollars; 12 x must, therefore, be equal to 36 dollars, and the value of x must be one twelfth part of 36, namely, 3 dollars, which is the price of the calf: if the value of x be 3 dollars, 2 x is twice 3, or 6 dollars, the price of the sheep; and 9x is 9 times 3, or 27 dollars, the price of the cow.

5. A gentleman gave a purse, containing a certain sum of money, to his three children, to be divided among them in such a manner, that Mary should have

twice as much as Ellen, and John as much as both his sisters. What was the share of each?

As the sum contained in the purse is not named, we will call it a.

Let x denote Ellen's share; then Mary's share is twice as much, or 2x; and John's, x and 2x, that is, 3x; and the sum of their shares is x and 2 x and 3x, or 6x, which must be equal to a, the sum to be divided, whatever the value of a may be. And if 6x is equal to a, x is equal to one sixth part of a, which is Ellen's share.

If the purse contained 18 dollars, Ellen's share was 3, Mary's 6, and John's 9 dollars.

If the purse contained 24 dollars, Ellen's share was 4, Mary's 8, and John's 12 dollars.

In this manner the share of each may be determined, whatever be the sum indicated by a.

This section will serve to give the learner a general notion of the nature and use of Algebra, and the manner in which it is applied to the performing of questions.

SECTION II.

Algebraic Signs.

Besides letters, certain other signs are used in Algebra, some of which are also used in Arithmetic, though less frequently. It is by means of these and other arbitrary signs, that calculations in Algebra are per