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But when the two fractions have not the same denominator, we must reduce them to this form by Problem IV.

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It is now plain that the quotient must be represented by the division of ad by bc, which gives

ad
bc'

the same result as obtained by the above rule.

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SECTION VII.

EQUATIONS OF THE FIRST DEGREE.

(87.) An equation is a proposition which declares the equality of two quantities expressed algebraically. Thus, x-4-a-x is a proposition expressing the equality of the quantities x-4 and a-x.

(88.) Equations are usually composed of certain quantities which are known, and others which are unknown. The known quantities are represented either by numbers or by the first letters of the alphabet, a, b, c, etc.; the unknown quantities by the last letters, x, y, z, etc.

(89.) A root of an equation is the value of the unknown quantity in the equation.

(90.) Equations are divided into degrees, according to the highest power of the unknown quantity which they contain.

Those which contain only the first power of the unknown quantity are called equations of the first degree.

As

ax+b=cx+d.

Those in which the highest power of the unknown quantity is a square, are called equations of the second degree, etc.

QUEST.-What is an equation? How are known and unknown quan. tities represented? What is a root of an equation? What is an equa tion of the first degree? What is an equation of the second degree?

Thus 3x-2x=40 is an equation of the second degree.

(91.) To solve an equation is to find the value of the unknown quantity; or to find a number which, substituted for the unknown quantity in the equation, proves the two members of the equation to be equal to each other.

(92.) The following principles are regarded as selfevident, and are called axioms :

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.

(93.) The unknown quantity may be combined with the known quantities in the given equation by the operations of addition, subtraction, multiplication, or division.

We shall consider these different cases in succession. I. The unknown quantity may be combined with known quantities by addition.

Let it be required to solve the equation

x+5=25.

If from the two equal quantities x+5 and 25 we subtract the same quantity 5, the remainders will be

Name the axioms

QUEST.-What is meant by solving an equation? employed in algebra. When known quantities are added to the un known quantity, how do we solve the equation?

equal, according to the last Article, and we shall have x+5-5-25-5,

or

x=20, the value required.

So, also, in the equation

x+a=b,

subtracting a from each of the equal quantities x+a and b, the result is

x=b-a, the value required.

(94.) II. The unknown quantity may be combined with known quantities by subtraction.

Let the equation be

x-5=15.

If to the two equal quantities x-5 and 15, the same quantity 5 be added, the sums will be equal according to Art. 92, and we have

or

x-5+5=15+5,

x=20, the value required.

x-a=b,

So, also, in the equation

adding a to each of these equal quantities, the result x=b+a, the value required.

is

From the preceding examples we conclude that We may transpose any term of an equation from one member to the other by changing its sign.

We may change the sign of every term of an equation without destroying the equality.

This is, in fact, the same thing as transposing every term in each member of the equation.

QUEST.-When known quantities are subtracted from the unknown quantity, how do we solve the equation? How may the terms of an equation be transposed? What change may be made in the signs of the terms?

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