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3x+ax-4=20 is a simple equation.

A quadratic Equation, or an equation of the second degree, is one in which the highest power of the unknown quantity is its second power or square.

2x2+3ax+5=30 is a quadratic equation.

A cubic Equation, or an equation of the third degree, is one in which the highest power of the unknown quantity is its third power, or cube; and so on.

Equations are also distinguished as

Numerical, Literal, and Identical Equations.

(111.) A numerical Equation is one in which all the known quantities are expressed by numbers.

2x+5x=25-3 is a numerical equation.

A literal Equation is one in which some or all of the known quan tities are represented by letters.

2x+ax=25-36 is a literal equation,

in which a and b are supposed to represent quantities whose values are known.

An identical Equation is one in which the two members are the same, or become the same by performing the operations which are indicated in them.

Thus 3x-3ab=3(x-ab) is an identical equation.

Transformation of Equations.

(112.) The transformation of an Equation consists in changing its form, without destroying the equality of the two members,-for the purpose of finding the value of the unknown quantity, or of discovering some general truth or principle.

These transformations depend, for the most part, on the following

Axioms.

(113.) An Axiom is a truth which is self-evident,—neither admitting nor requiring any demonstration; such as,

1. Things which are equal to the same thing, are equal to each other.

2. If equals be added to equals, the sums will be equal.

3. If equals be taken from equals, the remainders will be equal. 4. If equals be multiplied by equals, the products will be equal. 5. If equals be divided by equals, the quotients will be equal. 6 Any like powers or roots of equal quantities, are equal

SOLUTION OF SIMPLE EQUATIONS CONTAINING BUT ONE

UNKNOWN QUANTITY.

(114.) The value of the unknown quantity is found by making its symbol stand alone on one side of the Equation, so as to be equal to known quantities on the other side.

In order to this, the following transformations may be necessary, or at least may be expedient.

1. Clearing the Equation of Fractions.

2. The Transposition and Addition of Terms.

3. Changing the Signs of all the Terms in the Equation.

4. Dividing the Equation by the Coefficient of the Unknown quantity.

We shall apply each of these transformations to the solution of the same Equation.

Clearing an Equation of Fractions.

(115.) An Equation is cleared of fractions by multiplying each numerator into all the denominators except its own-regarding each integral term as a numerator,—and omitting the given denominators Let the Equation be

3x

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4

28 OC
+
3 3

Multiplying the numerator of each fraction by the denominators of the other two, and the integral terms 7 and x by all the denominators, we obtain

27x-252-36x-336+12x.

The equality of the two members is not destroyed in thus clearing the Equation of fractions, because each of the terms connected by the signs and in the two members, is thus multiplied by all the denominators. (103) (113...4).

An Equation may also be cleared of fractions by multiplying its two members by the least common multiple of the denominators ;observing that a fractional term will be multiplied by multiplying its numerator into the quotient of said multiple the denominator.

In the given Equation the least common multiple of the denominators is 12. Multiplying by 12, we find

9x-84-12x-112+4x.

The advantage of this method is, that the new equation is found in its lowest terms:

Transposition and Addition of Terms.

(116.) Any term may be transposed from one side of an Equation to the other by changing its sign.-All the similar terms may thus be placed on the same side, and then added together.

In the last Equation 9x-84-12x-112+4x, by transposing -84 to the second member, and 12x and 4x to the first, we have

9x-12x-4x=-112+84;

And by adding together the similar terms,

-7x=-28.

The equality of the two members is not destroyed by transposing a term with its sign changed from one side to the other, because this is equivalent to adding the term with its sign changed to both sides. Thus by adding 84 to both members of the equation

9x-84-12x-112+4x,

the term -84 is canceled in the first member (28).

In like manner

by adding 12x to both members, 12x is canceled in the second member; so also with 4x. (113...2).

From the preceding principles it follows, that

Two equal terms with like signs on opposite sides of the sign be at once suppressed from the Equation.

Change of the Signs in an Equation.

=, may

(117.) All the signs in an Equation may be changed, to and to +, without affecting the equality of its two members.

This follows from the principle of Transposition, (116), since in transposing all the terms, the signs would all be changed, but the two members would still be equal.

In the Equation already found

-7x=-28,

we shall have, by Transposition,

28=7x, or 7x=28.

The only Transformation which remains towards finding the value of x in the equation at first assumed, is that of dividing by the coeffi cient of x, the unknown quantity. (113...5).

Dividing both members of the preceding Equation by the coefficient of x, we find

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We have thus found the value of x to be 4. This value may be verified by substituting it for x in the original equation.

We may now give

RULE XVII.

(118.) For the Solution of a Simple Equation containing but one unknown quantity.

1. Clear the Equation of fractions, if it contains any.

2. Transpose all the terms containing the unknown quantity to one side, and all the known terms to the other side, of the equation.

3. Add together all the similar terms in each member.

4. Divide both members by the coefficient of the unknown quantity;-observing that when the unknown quantity is found in two or more dissimilar terms, its coefficient will be the sum of its coefficients in those terms.

NOTE. When the sum of the terms containing the unknown quantity, after transposition, is negative, it will generally be expedient, though it is never necessary, to make it positive by changing all the signs in the equation.

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Clearing the equation of fractions, by multiplying it by the least common multiple of the denominator, which is 6, we have

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Remark. The student is apt to err in Clearing an Equation of its fractions, when, as in this Example, a fraction preceded by the negative sign has a polynomial numerator.

The sign

before the fraction in the second member above, denotes that the fraction is to be subtracted. When this fraction is multiplied by the 6, the product 3x+57 is subtracted by changing its signs. This gives the terms -3x-57 in the new equation.

EXERCISES.

Numerical Equations.

1. Given 4x-8-13-3x to find the value of x.

2. Given 7x+17=10x-19 to find the value of x.
3. Given 8x+6=36-7x to find the value of x.
4. Given 59-7x-4x+26 to find the value of x.
5. Given 20-4x-12-92-10x to find the value of x.

Ans. x=3.

Ans. x 12.

6. Given 8-3x+12=30-5x+4 to find the value of x.

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