16. Divide (a3 +5 - y)3 by (a3 + 5 — y)7.` 17. Divide a b3 + 2 a3 b + 2 a2 b2 + a1 by a b2 + a3 + a2 b.

Before we begin to divide compound quantities, we should arrange the terms of the divisor and dividend according to the powers of their letters, as this will greatly facilitate the work. The highest power of a letter should come first, and the lower powers should succeed in order. The first term of the divisor and the first term of the dividend should contain the same letter.

To arrange this question, we place the letter of the divisor which is of the highest power, first, and the other terms in order, thus; a3 + a2 b+ a b2. Now, as the first term of the divisor is a, the first term of the dividend should also contain a, and the whole should be arranged thus ; a1 +2 a3 b + 2 a2 b2 + a b3.

20. Divide 39x2+4x-80 by x +5..

21. Divide 6o — 16 c3 by b2

2 c2.

22. Divide a2 x — b2 x + 8 x − a2 y3 + b2 y3

—

23. Divide ao — a1 x — a2 x3 + 2 x1 by aa — x3.

CHAPTER VIII.

EQUATIONS OF THE FIRST DEGREE.

SECTION 1.

Introduction.

WHEN two equal quantities, differently expressed, are compared together by means of the sign = between them, such an expression is called an equation. Thus, 8+4=18 6 is an equation; for the sums are equal, though expressed in different numbers. So, too, if x + 5 and a- 7 represent equal quantities, we have the equation x + 5 = a -7.

It is by means of equations that most of the investigations of Algebra are carried on; and the preceding chapters may be regarded as merely preparatory to this part of the science.

An equation of the first degree contains only the first power of the unknown quantity, as x. When some

igher power of the unknown quantity, as x2, or x3, nters into the equation, it is said to be of the second r third degree.

The terms on the left of the sign, taken together, are called the first member of the equation; those on the right, the second member. Thus, in the equation

5, the first member is x +6, and a 5

Any changes, which convenience requires, may be made in the members of an equation, provided the same be made in both, so that their equality is preserved; as may be seen in the following examples: Given the equation 8+ 4 = 18 — 6. Add 10 to each member; 8410186 10. Subtract 12 from each member;

Multiply each term by 2;

16820-24 36 12 20-24.

Divide every term by 4;

4+2+56=9—3—5— 6.

Although the members of the given equation, 8+4 186, are changed in form and value by each successive operation, it will be seen that their equality is preserved throughout. Whence we infer that,

The same quantity may be added to both members of an equation;

The same quantity may be subtracted from both members of an equation;

All the terms of an equation may be multiplied by the same quantity; and

All the terms of an equation may be divided by the same quantity; without affecting, in either of these cases, the equality of the two members.

The application and use of these and other changes in the terms and members of equations, will be ex

plained hereafter, as it is found necessary to introduce them.

When a question is proposed, to be solved by Algebra, the first step is, to express its conditions in the form of an equation; or, in common language, to put the question into an equation.

The next step is, to find the value of the unknown quantity from the terms with which it is associated; which process is called the reducing or resolving of an equation.

The rules for reducing equations are few and simple; and they will be given in the subsequent sections. But no particular mode of putting questions

to an equation, can be prescribed, as the process must vary with the conditions of every question. The following general directions may be of some service:

When a question is proposed, before its solution is attempted, get a clear and distinct understanding of its nature and design.

Let the thing required, the answer to the question, be represented by some letter, as x or y.

Regard the letter used as the answer of the question, and perform the same operations on it as would be necessary to prove the real answer to be correct.

The result thus obtained will be one member of the equation; and the other member will be found in the corresponding conditions of the question.

To illustrate these general directions by a single example: Let us suppose that a man gave 175 dollars for his watch, chain and seal; that the chain cost twice as much as the seal; and the watch twice as much as the chain. What was the price of each?

This question has three required quantities, or answers, namely, the prices of the watch, the chain, and the seal, either of which may be assumed and represented by a letter, as x; and, either being known, the others can readily be found.

First, let us suppose the watch cost x dollars; then the chain cost dollars, or half as much as the watch; and the seal cost dollars, or half as much as the chain; and, according to the answer assumed, they

all cost ++ dollars. But they actually cost,

2

4

by the question, 175 dollars; we have, therefore, this equation :

Again, let us suppose the chain cost x dollars; then the watch cost twice as much, or 2 x dollars; and

the seal cost half as much as the chain, or dollars. According to this supposition, the cost of the whole was 2 x + x + — dollars; and we have this equation:

Finally, let us suppose the seal cost x dollars; then the chain cost 2 x dollars; and the watch cost twice as much as the chain, or 4 x dollars. By this supposition, the price of the whole was 4x + 2 x + x dollars; and we have the following equation:

4 x + 2 x + x = 175.

Either of these equations will give one answer to