33.

34.

35.

+

(ca) (a - b) ̄ (a - b) (b − c) *

b2 - ca

c2 - ab

(a + b) (a + c) (b + c) (b + a) * (c + a) (c + b) *

a2-bc

+

b2 + ca

+

c2 + ab

(a - b) (a−c) ́ ́ (b + c) (b − a) * (c − a) (c + b) '

bc

(a-b

са

(c− a) (a - b) (a − b) (b −c) ̄ (b − c) (c − a) *

a (a-b) (a-c) b (b−c) (b-a) c (c-a) (c-b)*

c-a (a - b) (b−c) (c − a)

c+ b

c2 +

69. Simplify() + + + 8).

70.

c2

x

Simplify()+(-3-3).

2

2

a-b

b2 a2 b2

71. Simplify (+-)+(-8-3-5).
(a b a b) (a2
÷

i+b2

a2

2x2

x + 3α

148. Any collection of Algebraical symbols is called an expression. When two expressions are connected by the sign of equality the whole is called an equation. The expressions thus connected are called sides of the equation, or members of the equation. The expression to the left of the sign of equality is called the first side, and the expression to the right the second side.

149. An identical equation is one in which the two sides are equal whatever numbers the letters stand for; for example, (x+b)(x —b) = x2 — ba

is an identical equation. An identical equation is called briefly an identity.

150. An equation of condition is one which is not true for every value of the letters, but only for a certain number of values; for example,

x+1=7

cannot be true unless x = 6. An equation of condition is called briefly an equation.

151. A letter to which a particular value or values must be given in order that the statement contained in an equation may be true is called an unknown quantity. Such particular value of the unknown quantity is said to satisfy the equation, and is called a root of the equation. To solve an equation is to find the particular value or values.

152. An equation involving one unknown quantity is said to be of as many dimensions as is denoted by the index of the highest power of the unknown quantity. Thus, if x denote the unknown quantity, the equation is said to be of one dimension when x occurs only in the first power; such an equation is also called a simple equation, or an equation of the first degree. If x2 occurs, and no power of a higher than x occurs, the equation is said to be of two dimensions; such an equation is also called a quadratic equation, or an equation of the second degree. If x3 occurs, and no power of x higher than 3 occurs, the equation is said to be of three dimensions; such an equation is also called a cubic equation, or an equation of the third degree. And so on.

It must be observed that these definitions suppose both members of the equation to be integral expressions so far as relates to x, and not to contain x under the radical sign.

153. We shall now indicate some operations which may be performed on an equation without destroying the equality which it expresses. It will be seen afterwards that these operations are useful when we have to solve equations.