3. Given 3ax-4ab=2ab-6ac, to find the value of x in terms

This result may be verified in the same manner as the preceding.

4. Given 3x2-10x=8x+x2, to find the value of x.

9. Given 5ax-2b+4bx=2x+5c, to find x.

(105.) An equation may always be cleared of fractions by multiplying each member into all the denominators according

to Art. 103. But sometimes the same object may be attained by a less amount of multiplication.

Thus, in the preceding example, the equation may be cleared of fractions by multiplying each term by 16, instead of 16x8 ×2, and it is important to avoid all useless multiplication. In general, it is sufficient to multiply by the least common multiple of all the denominators. See Art. 86.

5 4 2 3

20. Given ab -ас -cx=ac+2ab-6cx, to find the value

4

of x.

70ab-3ac

Ans. x=

320c

SOLUTION OF PROBLEMS.

(106.) The solution of a Problem by Algebra consists of two distinct parts:

E

1. To express the conditions of the problem algebraically ; that is, to form the equation.

2. To solve the equation.

The second operation has already been explained, but the first is often more embarrassing to beginners than the second Sometimes the statement of a problem furnishes the equation directly; and sometimes it is necessary to deduce from the statement new conditions, which are to be expressed algebraically. The former are called explicit conditions; and those which are deduced from them, implicit conditions.

It is impossible to give a general rule which will enable us to translate every problem into algebraic language. The power of doing this with facility can only be acquired by reflection and practice.

The following directions may be found of some service.

Denote one of the required quantities by x; then, by means of this letter, with the algebraic signs, perform the same operations which would be necessary to verify its value if it was already known.

Problem 1. What number is that, to the double of which if 16 be added, the sum is equal to four times the required number?

Let x represent the number required.
The double of this will be 2x.

This increased by 16 should equal 4x.

Hence, by the conditions, 2x+16=4x.

The problem is now translated into algebraic language, and it only remains to solve the equation in the usual way. Transposing, we obtain

To verify this number, we have but to double 8, and add 16 to the result; the sum is 32, which is equal to four times 8, according to the conditions of the problem.

Prob. 2. What number is that, the double of which exceeds its half by 6?

To verify this result, double 4, which makes 8, and diminish it by the half of 4, or 2; the result is 6, according to the conditions of the problem.

Prob. 3. The sum of two numbers is 8, and their difference 2. What are those numbers ?

Then x+2 will be the greater number.

The sum of these is 2x+2, which is required to equal 8. Hence we have

By transposition

and

Also,

b.

Verification. 5+3=8)

2x+2=8.

2x=8-2-6,

x=3, the least number.

x+2=5, the greater number.

6-3=2} according to the conditions.

The following is a generalization of the preceding Problem. Prob. 4. The sum of two numbers is a, and their difference What are those numbers?

Let x represent the least number.

Then x+b will represent the greater number.

The sum of these is 2x+b, which is required to equal a.

Hence we have

As these results are independent of any particular value attributed to the letters a and b, it follows that

Half the difference of two quantities, added to half their sum, is equal to the greater; and

Half the difference subtracted from half the sum is equal to the less.

they may be regarded as comprehending the solution of all questions of the same kind; that is, of all problems in which we have given the sum and difference of two quantities.

= 10; their difference = 6; required the numbers.

Given the sum of two numbers,

Prob. 5. From two towns which are 54 miles distant, two travelers set out at the same time with an intention of meeting. One of them goes 4 miles and the other 5 miles per hour. In how many hours will they meet?

Let x represent the required number of hours.

Then 4x will represent the number of miles one traveled, and 5x the number the other traveled;

and since they meet, they must together have traveled the whole distance.

Proof. In 6 hours, at 4 miles an hour, one would travel 24 miles; the other, at 5 miles an hour, would travel 30 miles. The sum of 24 and 30 is 54 miles, which is the whole distance. This Problem may be generalized as follows:

Prob. 6. From two points which are a miles apart, two bodies move toward each other, the one at the rate of m miles per hour,