Hence, 5 is a root of the equation, because if substituted for x, it will render the two members equal. Again, let x = — 7. We have

(− 7)2 + (−7 × 2) = 35,

49- - 14 = 35,

35 = 35.

Hence, is also a root of the given equation.

141. A Numerical Equation is one in which all the known quantities are expressed by figures, as, 323—x2+2x = 17. 142. A Literal Equation is one in which some or all of the known quantities are expressed by letters; as ax2—3bx=5d. 143. An Equation of Condition is one which must exist between certain known or arbitrary quantities, in order that certain other equations may be true. Thus, the two equations, x + c = 5α,

x— c = a,

cannot both be true at the same time, unless

c = 2a;

that is, the last equation expresses the condition which will render the other two equations true; it is therefore called an equation of condition.

144. An Identical Equation is one in which the two members are the same algebraic expression, or are reducible to the same. Thus,

are identical equations.

a2 3x a2-3x,

x2

x2- a2 = (x + a) (x − a),

145. Equations are said to be of different degrees or dimen

sions.

The Degree of an equation is denoted by the greatest number of unknown factors occurring in any term. Hence,

1. If an equation involves but one unknown quantity, its degree is denoted by the highest exponent of this quantity in any term. 2. If an equation involves more than one unknown quantity, its degree is denoted by the greatest sum which the exponents of the unknown quantities give in any term.

Thus, for example :

x + ax = b

ax + y = c2 x2+4x=8

x2 + xy = a2b

ax3 + bx2 + cx = 2α1b

x3 + 3xy + y3 = ab5

}

are equations of the first degree;

} are equations of the second degree ;

are equations of the third degree.

146. A Simple Equation is an equation of the first degree.

147. A Quadratic Equation is an equation of the second degree.

148. A Cubic Equation is an equation of the third degree.

TRANSFORMATION OF EQUATIONS.

149. The Transformation of an equation is the process of changing its form without destroying the equality of its members.

From the nature of an equation, it is evident that all the operations to which it can be subjected without destroying the equality, are embraced in the axioms (39); they may be stated as follows:

1. The same or equal quantities may be added to both members (Ax. 1).

2. The same or equal quantities may be subtracted from both members (Ax. 2).

3. Both members may be multiplied by the same or equal quantities (Ax. 3).

4. Both members may be divided by the same or equal quantities (Ax. 4).

5. Both members may be raised, by involution, to the same power (Ax. 8).

6. Both members may be reduced, by evolution, to the same root (Ax. 9).

CASE L

150. To transpose the terms of an equation.

Transposition is the process of changing a term from one member of an equation to the other, without destroying the equality.

To exhibit the law of transposition, let us consider the three following examples:

If we subtract a from both members of this equation, the result will be

x=b-a;

and we perceive that the term, +a, has been removed from the first member, and appears as — a in the second member.

2. Let

If we add a to both members of this equation, the result will be

x = b+a;

and we perceive that the term, -a, has been removed from the first member, and appears as + a in the second.

Subtracting a from both members of the equation, we have

If we now multiply both members of this result by 1, we

and by comparing this last result with the given equation, we observe that a has been removed from the first to the second member, but the signs of both the other terms of the equation have been changed.

Hence, for changing the sign or place of any term of an equation we have the following

RULE.-I. Any term may be transposed from one member of an equation to the other by changing its sign (1, 2).

II. Any term may be transposed without changing its sign, provided the signs of all the other terms be changed (3).

III. The sign of any term may be changed without transposition, by changing the signs of all the terms simultaneously (3).

EXAMPLES FOR PRACTICE.

In the following equations, transpose the unknown terms to the first member, and the known terms to the second (I):

In the following, transpose the unknown terms to the first member, and the known to the second (II)

151. To clear an equation of fractions.

We have seen (135, 2), that if a fraction be multiplied by any multiple of its denominator, the product will be entire; consequently, if several fractions be multiplied by a common multiple of their denominators, all the products will be entire.

Multiplying every term by 30, which is the least common multiple of the denominators, we have

9x4x360,

in which all the terms are entire.

Multiplying every term by a2b2, observing that the product obtained from the second fraction is to be subtracted, we have ax + ac = bx + bc.

b2x

Hence the following

-

RULE.-Multiply all the terms of the equation by the least common multiple of the denominators, observing that when a fraction has the minus sign before it, the signs of the terms derived from its numerator must be changed.

NOTES.—1. The pupil should observe that in multiplying any fraction it will be most convenient to divide the multiplier by the denominator and multiply the numerator by the quotient.

2. It will be obvious, also, that the equation will be cleared of fractions, by multiplying by the several denominators, successively.