Preparatory to the solution of problems, and to extended investigations of scientific truth, we commenced by explaining the reason and the manner of adding, subtracting, multiplying, and dividing algebraic quantities, both whole and fractional, that the mind of the pupil need not be called away to the art of performing these operations, when all his attention may be required on the nature and philosophy of the problem itself.

For this reason we did not commence with problems, as Colburn and some others have done.

Analytical investigations are mostly carried on by means

OF EQUATIONS.

(Art. 35.) An equation is an algebraical expression, meaning that certain quantities are equal to certain other quantities. Thus, 3+4=7; a+b=c; x+4=10, are equations, and express that 3 added to 4 is equal to 7, and in the second equation that a added to b is equal to c, &c. The signs are only abbreviations for words.

The quantities on each side of the sign of equality are called members. Those on the left of the sign form the first member, those on the right the second

In the solution of problems every equation is supposed to contain at least one unknown quantity, and the solution of an equation is the art of changing and operating on the terms by means of addition, subtraction, multiplication, or division, or by all these combined, so that the unknown term may stand alone as one member of the equation, equal to known terms in the other member, by which it then becomes known.

Equations are of the first, second, third, or fourth degree, according as the unknown quantity which they contain is of the first, second, third, or fourth power.

ax+b=3ax is an equation of the first degree or simple equation.

ax2+bx=3ab is an equation of the second degree or quadratic equation.

ax2+bx+cx=2a4b is an equation of the third degree. ax2+bx+cx2+dx=2ab5 is an equation of the fourth

degree.

We shall at present confine ourselves to simple equations. (Art. 36.) The unknown quantity of an equation may be united to known quantities, in four different ways: by addition, by subtraction, by multiplication, and by division, and further by various combinations of these four ways as shown by the following equations, both numeral and literal:

5th. x+6-8+4=10+2-3 x+a-b+c=d+c, &c. are equations in which the unknown is connected with known quantities by both addition and subtraction.

Equations often occur in solving problems in which all of these operations are combined.

(Art. 37.) Let us now examine how the unknown term can be separated from other terms, and be made to stand by itself. Take the Ist equation, or other similar ones

remainders must be equal. (Ax. 2.) Now we find the term added to x, whatever it may be, appears on the other side with a contrary sign, and the unknown term a being equal

to known terms is now known.

Take the equations

Add equals to both memb.
Sums are equal

x-8-12

8 8

x-c-d

c=c

x=12+8 x=d+c (Ax. 1.)

Here again the quantity united to x appears on the opposite side with a contrary sign.

From this we may draw the following principle or rule of operation:

Any term may be transposed from one member of an equation to the other, by changing its sign.

Now 20x=80. axe. If we divide both members by the coefficient of the unknown term, the quotients will be

equal. (Ax. 4.) Hence x==4.

20

e

x=.

a

That is, the unknown term is disengaged from known

terms in this case by division.

Multiply both members by the divisor of the unknown term, and we have x=16×4. x=gd+ad. Equations which must be true by (Ax. 3.) and here it will be observed that æ is liberated by multiplication.

From these observations we deduce this general principle: That to separate the unknown quantity from additional terms we must use subtraction; from subtracted terms we must use addition; from multiplied terms we must use division; from division we must use multiplication.

In all cases take the opposite operation.

(Art. 38.) In many practical problems, the unknown term is often combined with known terms, not merely in a simple manner, but under various fractional and compound forms. Hence, rules can only embody general principles, and skill and tact must be acquired by close attention and practical application; but from the foregoing principles we draw the following

GENERAL RULE. Connect and unite as much as possible all the terms of a similar kind on both sides of the equation. Then, to clear of fractions, multiply both sides by the denominators, one after another, in succession. Or, multiply by their continued product, or by their least common multiple, (when such a number is obvious,) and the equation will be free of fractions.

Then, transpose the unknown terms to the first member of the equation, and the known terms to the other. Then unite the similar terms, and divide by the coefficient of the unknown term, and the equation is solved.

EXAMPLES.

1. Given x+x+3-7=6-1, to find the value of x. Uniting the known terms after transposition agreeably to the rule of addition, we find x+x=9. Multiply every term by 2, and we have 2x+x=18. Therefore x=6.

2. Given 2x+x+x-3=4b+3a, to find x. N. B. We may clear of fractions in the first place before we condense and unite terms, if more convenient, and among literal quantities this is generally preferable.

In the present case let us multiply every term of the equation by 12, the product of 3×4, and we shall have 24x+9x+4x-36a 48b+36a. 37x=48b+72a.

Transpose and unite, and
Divide by 37, and x= 37

48b+36a

3. Given x+x+x=39, to find the value of x.

Here are no scattering terms to collect, and clearing of fractions is the first operation.

By an examination of the denominators, 12 is obviously their least common multiple, therefore multiply by 12. Say 12 halves is 6 whole ones, 12 thirds is 4, 12 fourths is 3, &c.

Hence,

Collect the terms,

Divide by 13, and

6x+4x+3x=39×12

13x=39×12

x=3×12=36, Ans.

N. B. In other books we find the numerals actually multiplied by 12. Here it is only indicated, which is all that is necessary. For when we come to divide by the coefficient of x, we shall find factors that will cancel, unless that coefficient is prime to all the other numbers used, which in, practice is very rarely the case.

4. Given x+x+x=a, to find x.

This example is essentially the same as the last. It is

identical if we suppose a=39.

Now if a be any multiple of 13, the problem is easy and