175. A DETERMINATE PROBLEM is one in which the given conditions furnish the means of finding the required quantities.

A determinate problem leads to as many independent equations as there are required quantities (Art. 162).

176. An INDETERMINATE PROBLEM is one in which there are fewer imposed conditions than there are required quantities, and, consequently, an insufficient number of independent equations to determine definitely the values of the required quantities.

177. An IMPOSSIBLE PROBLEM is one in which the conditions are incompatible or contradictory, and consequently cannot be fulfilled.

178. A determinate problem, leading to a simple equation involving only one unknown quantity, can be satisfied by but one value of that unknown quantity.

For, after transposing and uniting, if the coefficient of x is represented by a, and the second member by b, the equation will take the general form,

Suppose now, if possible, this equation has two roots, r and r', then, by substituting successively these values

in (1),

Or,

Therefore, by subtracting (3) from (2), we have

But the last equation is impossible, since by supposition r is not zero, and a is not zero. That is,

r

A determinate simple equation can have but one root.

179. An indeterminate problem, or one leading to a less number of independent equations than it has unknown quantities, may be satisfied by any number of values.

For example, suppose a problem involving three unknown quantities leads only to two equations, which, on combining, give

Thus, we may find sets of values without limit that will satisfy the equation. Hence,

An indeterminate equation may have an infinite number of solutions.

180. When a problem leads to more independent equations than it has unknown quantities to be determined, it is impossible.

For, suppose we have a problem furnishing three independent equations, as,

But (3) requires their product to be 16, which can not be satisfied; hence the problem is impossible.

If, however, the third equation had not been independent, but derived from the other two; as,

xy = 12;

then the problem would have been possible, but the last equation, not being required for the solution (Art. 162), would have been redundant.

INTERPRETATION OF NEGATIVE RESULTS.

181. In a NEGATIVE RESULT, or a result preceded by the sign, the negative sign is regarded as a symbol of interpretation.

Its significance when thus used it is now proposed to investigate.

1. Let it be required to find what number must be added to the number a, that the sum may be b.

Let
Then,

Whence,

x = the required number. a + x = b.

Here, the value of x corresponds to any assigned values of a and b. Thus, for example,

which satisfies the conditions of the problem; for if 13 be added to 12, or a, the sum will be 25, or b.

But suppose
Then

a 30 and b 24.
x=24 30 - 6,

which indicates that, under the latter hypothesis, the problem is impossible in an arithmetical sense, though it is possible in the algebraic sense of the words "number," "added," and "sum."

The negative result, -6, points out, therefore, either an error or an impossibility.

But, taking the value of x with a contrary sign, we see that it will satisfy the enunciation of the problem, in an arithmetical sense, when modified so as to read:

What number must be taken from 30, that the difference may be 24?

2. Let it be required to find the epoch at which A's age is twice as great as B's, A's age at present being 35 years, and B's 20 years.

Let us suppose the required epoch to be after the present date.

Then

x= the number of years after the present date, and 35+ x = 2 (20 + x) ;

whence, x = - 5,

a negative result.

On recurring to the problem, we find it is so worded as to admit also of the supposition that the epoch is before the present date, and taking the value of x obtained, with the contrary sign, we find it will satisfy that enunciation.

Hence, a negative result here indicates that a wrong choice was made of two possible suppositions which the problem allowed.

From the discussion of these problems we may infer:

1. That negative results indicate either an erroneous enunciation of a problem, or a wrong supposition respecting the QUALITY of some quantity belonging to it.

2. That we may form, when attainable, a possible problem analogous to that which involved the impossibility, or correct the wrong supposition, by attributing to the unknown quantity in the equation a QUALITY DIRECTLY OPPOSITE to that which had been attributed to it.

In general, it is not necessary to form a new equation, but simply to change in the old one the sign of each quantity which is to have its quality changed.

Interpret the negative results obtained, and modify the enunciation accordingly, for the following

PROBLEMS.

3. If the length of a field be 10 rods, and the breadth 8 rods, what quantity must be added to its breadth that the contents be 60 square rods?

Ans.

2 rods.

4. If 1 be added to the numerator of a certain fraction, its value becomes ; but if 1 be added to the denominator, it becomes . What is the fraction?

Ans.

- 5 9°

5. The sum of two numbers is 90, and their difference is 120; what are the numbers?

Ans. 105 and

- 15.

6. A is 50 years old, and B 40; required the time when A will be twice as old as B.

7. A and B were in partnership, and A had 3 times as much capital in the firm as B. When A had gained $2000, and B $750, A had twice as much capital as B. What was the capital of each at first?

Ans. A was in debt $1500, and B $500.

8. A man worked 14 days, his son being with him 6 days, and received $39, besides the subsistence of himself and son while at work. At another time he worked 10 days, and had his son with him 4 days, and received $28. What were the daily wages of each?

Ans. The father's wages, $3; the son's, 50 cts. That is, the father earned $3 a day, and was at the expense of 50 cents a day for his son's subsistence.

9. A worked 10 days, B 4 days, and C 3 days, and their wages amounted to $23; at another time, A worked 9 days, B 8 days, and C 6 days, and their wages amounted to $24; a third time, A worked 7 days, B 6 days, and C 4 days, and their wages amounted to $18. What were the daily wages of each?

Ans. A's, $2.00; B's, 0; and C's, $ 1.00.