Now the parenthetical expressions in equation (9) are known quantities. Hence, to simplify the results,

DISCUSSION OF PROBLEMS INVOLVING SIMPLE EQUATIONS

180. The Discussion of a problem consists in attributing certain values and relations to the arbitrary quantities which enter the equation, and in interpreting the results.

181. When a problem has been solved in a general manner, we may proceed to make an unlimited number of suppositions upon the arbitrary quantities involved in the formulas, and thus obtain a variety of results. But our experience of algebraic equations would lead us to expect that the problem might not be rational, or possible, under every hypothesis. Now the principal object in the discussion of a problem is to examine the peculiar or anomalous forms which present themselves, and ascertain whether the problem is rational or absurd, or how it is to be understood, under the suppositions which lead to these peculiarities. We shall commence with the

INTERPRETATION OF NEGATIVE RESULTS.

1. What number must be added to a that the sum may be b? Let x represent the required number. Then by the conditions of the question,

This is a general solution, a and b being arbitrary quantities.

a result which satisfies the conditions; for, we perceive that 8 is the number which must be added to 20, or a, to make 28, or b. Second, suppose a = 20 and b = 12; then by the formula, x=12-20

a negative result.

-8,

In order to ascertain the meaning of the minus sign in this case, let us enunciate the question according to the supposition that gave this result; thus,

What number must be added to 20, that the sum may be 12?

Now as 20 is greater than 12, no number can be added to 20, arithmetically, to make 12. The problem is therefore impossible under the second hypothesis, if understood in an arithmetical sense.

We shall find, however, that if we change the words added to, and sum, to their opposites, the result will be a rational question, of which 8, the absolute value of x, is the answer. Thus,

What number must be subtracted from 20, that the difference may be 12? Ans. 8.

We observe, moreover, that the negative result, -8, will satisfy the equation of the problem, under the second hypothesis. Thus, 20+(-8)= 12; 20-8 = 12.

or,

That is, 12 is really the algebraic sum of 20 and -8.

2. A man dying left two sons, the elder of whom was a years of age, and the younger b years of age. In how many years after the death of the father was the elder son twice as old as the younger son?

Let x represent the number of years; then by the conditions,

Since a and b are arbitrary quantities, suppose a = 30 and b12. Then by the formula,

This result will satisfy the conditions arithmetically; for, if the elder son was 30, and the younger son 12 years old, at the death of the father, then in 6 years the age of the elder was 30+6 = 36 years, and the age of the younger was 12+6= 18 years.

Again, suppose a = 30 and b = 18. Then by the formula,

x = 30-36: = -6.

To interpret the negative result in this case, we observe that the problem under the second hypothesis is impossible, if understood in the exact sense of the enunciation. For, when the elder son was 30 and the younger son 18 years old, the younger son was already more than one half as old as the elder; and as their ages are equally increased by any lapse of time, it is evident that the elder

son could never become twice as old as the younger son, after the death of the father. Let us therefore modify the general problem as follows:

A man dying left two sons, the elder of which was a years of age, and the younger b years of age. How many years before the death of the father was the elder son twice as old as the younger?

If we let x represent the number of years, then the solution will be as follows:

α-x= 2(b-x);
x=2b-a.

Now suppose, as before, that a = 30 and b

new formula,

x 36-30 = 6,

(1)

(2)

18. Then by the

a result which will satisfy the modified conditions; for, six years before the death of the father, the age of the elder was 30—6— 24, and the age of the younger was 18–6:

12.

From the foregoing discussions we draw the following inferences: 1.—When the solution of a problem by a simple equation gives a negative result, the minus sign indicates that the problem is impossible, if understood in the exact sense of the enunciation.

2.-The impossibility thus indicated consists in adding a quantity when it should be subtracted; or in treating a quantity as reckoned or applied in a certain direction, when it should be reckoned or applied in an opposite direction.

3.—In all such cases, an analogous problem may be formed, involving no impossibility, by changing the terms of the absurd condition to their opposites; and the answer to the new question will be found by simply changing the sign of the negative result already obtained.

182. The foregoing discussions give a more extensive signification to the plus and minus signs, and lead to a more general view of positive and negative quantities, than was presented in a former section.

Let us recur to the problem of the two sons. In the solution of this problem, we employ the signs, + and —, in the statement, mere

ly to indicate addition and subtraction. But in the result, these signs have a very different use; they enable us to distinguish the circumstances or conditions of the quantities which they affect. Thus, under the first hypothesis, the period of time represented by x occurred after the death of the father, and in the result is found to be affected by the plus sign; but under the second hypothesis, the period represented by x occurred before the death of the father, and in the result is found to be affected by the minus sign.

Thus we perceive that plus and minus, in Algebra, are not symbols of operation merely, but also symbols of relation, serving to distinguish quantities in opposite conditions or circumstances.

It should be observed, however, that this enlarged use of the plus and minus signs is not entirely conventional or arbitrary, but is necessarily involved in the more extended signification given to the terms addition and subtraction, in Algebra. Indeed we shall never meet with a negative result in the solution of problems, so long as the language conforms, in the exact arithmetical sense, to the facts of the case.

EXAMPLES FOR PRACTICE.

1. What number is that whose fourth part exceeds its third part by 12? Ans. 144.

The question is impossible, if understood in an arithmetical sense. Let the pupil modify the enunciation, and solve the new problem.

2. A man when he was married was 30 years old, and his wife 15. How many years must elapse before his age will be three times the age of his wife? Ans. -7 years.

That is, their ages bore the specified relation 7 years before, not after, their marriage.

3. The sum of two numbers is s, and their difference d; what are the numbers? d

d

8

Ans. Greater,+; Less, 2 2

How shall the result be interpreted when s = 120 and d = 160? 4. Two men, A and B, commenced trade at the same time, A having 3 times as much money as B. When A had gained $400