RESOLUTION OF FRACTIONS INTO SERIES.

109

Thus, in the infinite series, 1—ax+a2x2—a3x3+a*x*, &c., aby term may be found, by multiplying the preceding term by -ax.

Any proper algebraic fraction, whose denominator is a polynomial, can, by division, be resolved into an infinite series; for, the numerator is a dividend, and the denominator a divisor, so related to each other, that the process of division never can terminate, and the quotient will, therefore, be an infinite series. After a few of the terms of the quotient are found, the law of the series will, in general, be easily seen, so that the succeeding terms may be found without continuing the division.

nite series, 1+x+x2+x23+x2+, &c.

In a similar manner, let each of the following fractions be resolved into an infinite series, by division.

=1−x+x2—x3+x1—, &c., to infinity.

1

2.

1+x

a-x

REVIEW-144. What is an infinite series? What is the law of a series? Give an example. Why can any proper algebraic fraction, whose denom inator is a polynemial, be resolved into an infinite series, ky division?

CHAPTER IV.

EQUATIONS OF THE FIRST DEGREE.

DEFINITIONS AND ELEMENTARY PRINCIPLES.

ART. 145.-The most useful part of Algebra, is that which relates to the solution of problems. This is performed by means of equations.

An equation is an Algebraic expression, stating the equality between two quantities.

Thus, x—3—4, is an equation, stating, that if 3 be subtracted from x, the remainder will be equal to 4.

ART. 146.-Every question is composed of two parts, separated from each other by the sign of equality. The quantity on the left of the sign of equality, is called the first member, or side of the equation. The quantity on the right, is called the second member, or side. The members or quantities are each composed of one or more terms.

ART. 147.-There are generally two classes of quantities in an equation, the known and the unknown. The known quantities are represented either by numbers, or the first letters of the alphabet, as a, b, c, &c.; and the unknown quantities by the last letters of the alphabet, as x, y, z, &c.

ART. 148.-Equations are divided into degrees, called first, second, third, and so on. The degree of an equation, depends on the highest power of the unknown quantity which it contains.

An equation which contains no power of the unknown quantity higher than the first, is called an equation of the first degree.

Thus, 2x+5=9, and ax+b=c, are equations of the first degree. Equations of the first degree are usually called Simple Equations.

An equation in which the highest power of the unknown quantity is of the second degree, that is, a square, is called an equation of the second degree, or a quadratic equation.

REVIEW.-145. What is an equation? Give an example. 146. Of how many parts is every equation composed? How are they separated? What is the quantity on the left of the sign of equality called? On the right? Of what is each member composed? 147. How many classes of quantities are there in an equation? How are the known quantities represented? How are the unknown quantities represented? 148. How are equations divided? On what does the degree of an equation depend? What is an equation of the first degree? Give an example. What are equations of the first degree usually called? What is an equation of the second degree? Give an exam ple. What are equations of the second degree usually called.

Thus, 4x2—7—29, and ax2+bx=c, are equations of the second degree.

In a similar manner, we have equations of the third degree, fourth degree, &c.; the degree of the equation being always the same as the highest power of the unknown quantity which it contains.

When any equation contains more than one unknown quantity, its degree is equal to the greatest sum of the exponents of the unknown quantities in any of its terms.

Thus, xy+ax+by=c, is an equation of the second degree.

x3y+x2+cx=a, is an equation of the third degree.

ART. 149.-An identical equation, is one in which the two members are identical; or, one in which one of the members is the result of the operations indicated in the other.

Thus, 2x-1=2x-1, 5x+3x=8x, and (x+2)(x-2)=x2-4, are identical equations.

Equations are also distinguished as numerical and literal. A numerical equation is one in which all the known quantities are expressed by numbers.

Thus, x2+2x=3x+7, is a numerical equation.

A literal equation is one in which the known quantities are represented by letters, or by letters and numbers.

Thus, ax—b=cx+d, and ax2+bx=2x-5, are literal equations. ART. 150.-Every equation is to be regarded as the statement, in algebraic language, of a particular question.

Thus, x—3—4, may be regarded as the statement of the following question: To find a number, from which, if 3 be subtracted, the remainder will be equal to 4.

If we add 3 to each member, we shall have x-3+3=4+3, or x=7.

An equation is said to be verified, when the value of the unknown quantity being substituted for it, the two members are rendered equal to each other.

Thus, in the equation x-3=4, if 7, the value of x, be substituted instead of it, we have 7-3-4, or, 4=4.

To solve an equation, is to find the value of the unknown quantity; or, to find a number, which being substituted for the unknown quantity, will render the two members identical.

REVIEW.-148. When an equation contains more than one unknown quantity, to what is its degree equal? Give an example. 149. What is an identical equation? Give examples. What is a numerical equation? Give an example. What is a literal equation? Give an example. 150. How is every equation to be regarded? Give an example. When is an equation said to be verified? What do you understand, by solving an equation?

ART. 151.-The value of the unknown quantity in any equation, is called the root of that equation.

SIMPLE EQUATIONS, CONTAINING BUT ONE UNKNOWN QUANTITY.

ART. 152.-The operations that we employ, to find the value of the unknown quantity in any equation, are founded on this evident principle: If we perform exactly the same operation on two equal quantities, the results will be equal. This principle, or axiom, may be otherwise stated, as follows:

1. If, to two equal quantities, the same quantity be added, the sums will be equal.

2. If, from two equal quantities, the same quantity be subtracted, the remainders will be equal.

3. If two equal quantities be multiplied by the same quantity, the products will be equal.

4. If two equal quantities be divided by the same quantity, the quotients will be equal.

5. If two equal quantities be raised to the same power, the resuls will be equal.

6. If the same root of two equal quantities be extracted, the results will be equal.

REMARK.-An axiom is a self-evident truth. The preceding axioms are the foundation of a large portion of the reasoning in mathematics.

ART. 153. There are two operations of frequent use in the solution of equations. These are, first, to clear an equation of fractions; and, second, to transpose the terms, in order to find the value of the unknown quantity. These are named in the order in which they are generally used, in the solution of an equation; we shall, however, first consider the subject of

TRANSPOSITION.

Suppose we have the equation 2x-3=x+5.

Since, by the preceding principle, the equality will not be affected, by adding the same quantity to both members; or, by subtracting the same quantity from both members; if we add 3 to each member, we have 2x-3+3=x+5+3.

If we subtract x from each member, we have

2x-x-3+3=x-x+5+3.

REVIEW. 151. What is the root of an equation? 152. Upon what principle are the operations founded, that are used in solving an equation? What are the axioms which this principle embraces? 153. What two operations are frequently used, in the solution of equations?

But, -3+3 cancel each other; so, also, do x-x; omitting these, we have 2x-x=5+3.

Now, the result is the same as if we had removed the terms -3 and +x, to the opposite members of the equation, and, at the same time, changed their signs.

Again, take the equation ax+b=c—dx.

If we subtract b from each side, and add dx to each side, we bave ax+dx=c-b.

But, this result is also the same as if we had removed the terms +b and dx to the opposite members of the equation, and, at the same time, changed their signs. Hence,

Any quantity may be transposed from one side of an equation to the other, if, at the same time, its sign be changed.

TO CLEAR AN EQUATION OF FRACTIONS.

ART. 154.-1. Let it be required to clear the following equation of fractions.

X x

+ 23

5.

Since the first term is divided by 2, if we multiply it by 2, the divisor will be removed; but if we multiply the first term by 2, we must multiply all the other terms by 2, in order to preserve the equality of the members. Multiplying both sides by 2, we have

2x
x+ =10.
3

Again, since the second term is divided by 3, if we multiply it by 3, the divisor will be removed; but, if we multiply the second term by 3, we must multiply all the terms by 3, in order to preserve the equality of the members. Multiplying both sides by 3, we have 3x+2x=30.

Instead of multiplying first by 2, and then by 3, it is plain that we might have multiplied at once, by 2×3, that is, by the product of the denominators.

2. Again, let it be required to clear the following equation of fractions.

X X

+ =d. ab bc

Since the first term is divided by ab, if we multiply it by ab, the divisor will be removed; but, if we multiply the first term by ab, we must multiply all the other terms by ab, in order to preserve the equality of the members.

REVIEW.-154. How may a quantity be transposed from one member of an equation to the other? Explain the principle of transposition by an example.