Now, the denominators being the same, that fraction will be the greatest which has the greater numerator. But the two numerators, ab+am, and ab+bm, have a common part ab; and the part bm of the second is greater than the part am of the first, since b>a. Hence the second fraction is greater than the first.

If the given fraction is improper, or a>b, it is plain that the numerator of the second fraction will be less than that of the first, since bm would be less than am.

CHAPTER II.

Of Equations of the First Degree.

79. An Equation is the expression of two equal quantities with the sign of equality placed between them. Thus, x=a+b is an equation, in which x is equal to the sum of a and b.

=.

80. By the definition, every equation is composed of two parts, separated from each other by the sign The part on the left of the sign, is called the first member, and the part on the right, is called the second member; and each member may be composed of one or

more terms.

81. Every equation may be regarded as the enunciation, in algebraic language, of a particular problem. Thus, the equation x+x=30, is the algebraic enunciation of the following problem;

To find a number which, being added to itself, shall give a sum equal to 30.

Were it required to solve this problem we should first express it in algebraic language, which would give the equation

Hence we see that the solution of a problem by algebra, consists of two distinct parts.

1st. To express algebraically the relation between the known and unknown quantities.

2d. To find a value for the unknown quantity, in terms of those which are known, which substituted in its place in the given equation will satisfy the equation; that is, render the first member equal to the second.

This latter part is called the solution of the equation.

82. An equation is said to be verified, when such a value is substituted for the unknown quantity as will prove the two members of the equation to be equal to each other.

83. Equations are divided into different classes. Those which contain only the first power of the unknown quantity, are called equations of the first degree. Thus,

ax + b = cx+d

is an equation of the 1st. degree. 2x2-3x=5 -2x2 is an equation of the 2d. degree. 4x3-5x2=3x+11 is an equation of the 3d. degree.

In general, the degree of an equation is denoted by the greatest of the exponents with which the unknown quantity is affected.

84. Equations are also distinguished as numerical equations and literal equations. The first are those which contain numbers only, with the exception of the unknown quantity, which is always denoted by a letter. Thus, 4x-3=2x+5, 3x2-x=8, are numerical equations. They are the algebraical translation of problems, in which the known quantities are particular numbers.

The equations ax-b=cx+d, ax2+bx=c, are literal equations, in which the given quantities of the problem are represented by letters.

19

85. It frequently occurs in algebra, that the algebraic sign + or which is written, is not the true sign of the term before which it is placed. Thus, if it were required to subtract -b from a, we should write

a-(-b)=a+b.

Here the true sign of the second term of the binomial is plus, although its algebraic sign, which is written in the first member of the equation, is -. This minus sign, operating upon the sign of b, which is also negative, produces a plus sign for b in the result. The sign which results, after combining the algebraic sign with the sign of the quantity, is called the essential sign of the term, and is often different from the algebraic sign.

By considering the nature of an equation, we perceive that it must possess the three following properties.

1st. The two members are composed of quantities of the same kind. 2d. The two members are equal to each other.

3d. The essential sign of the two members must be the same.

Equations of the First Degree involving but one unknown quantity.

86. An axiom is a self-evident proposition. We may here state the following.

1. If equal quantities be added to both members of an equation, the equality of the members will not be destroyed.

2. If equal quantities be subtracted from both members of an equation, the equality will not be destroyed.

3. If both members of an equation be multiplied by the same number, the equality will not be destroyed.

4. If both members of an equation be divided by the same number, the equality will not be destroyed.

87. The transformation of an equation consists in changing its form without affecting the equality of its members.

The following transformations are of continued use in the resolution of equations.

First Transformation.

88. When some of the terms of an equation are fractional, to reduce the equation to one in which the terms shall be entire.

First, reduce all the fractions to the same denominator, by the known rule; the equation becomes

and since we can multiply both members by the same number without destroying the equality, we will multiply them by 72, which is the same as suppressing the denominator 72, in the fractional terms, and multiplying the entire term by 72; the equation then becomes

89. The last equation could have been found in another manner by employing the least common multiple of the denominators.

The common multiple of two or more numbers is any number which they will divide without a remainder; and the least common multiple, is the least number which they will so divide. The least common multiple will be the product of all the numbers, when, in comparing either with the others, we find no common factors. But when there are common factors, the least common multiple will be the product of all the numbers divided by the product of the common factors.

The least common multiple, when the numbers are small, can generally be found by inspection. Thus, 24 is the least common multiple of 4, 6, and 8, and 12 is the least common multiple of 3, 4 and 6.

Take the last equation

2x 3 Ꮖ
-x+
3 4

11.

6

We see that 12 is the least common multiple of the denominators, and if we multiply all the terms of the equation by 12, and divide by the denominators, we obtain

8x-9x+2x=132.

the same equation as before found.

90. Hence, to make the denominators disappear from an equation, we have the following

RULE

I. Form the least common multiple of all the denominators.

II. Multiply each of the entire terms by this multiple, and each of the fractional terms by the quotient of this multiple divided by the denominator of the term thus multiplied, and omit the denominators of the fractional terms.

the least common multiple of the denominators is ab2; hence clear

ing the fractions, we obtain

a1bx-2a2bc2x+-4a2b2=4b3c2x—5a® +2a2b3c2 — 3a3b3.

Second Transformation,

91. When the two members of an equation are entire polynomials, to transpose certain terms from one member to the other.

Take for example the equation

If, in the first place we subtract 2x from both members, the equality will not be de

stroyed, and we have

Whence we see that the term 2x, which was additive in the second member becomes subtractive in the first.