XIII. DISCUSSION OF SIMPLE EQUATIONS.

203. A problem is said to be Indeterminate when its conditions are satisfied by an indefinitely great number of sets of values of the unknown quantities involved in it.

204. A problem is indeterminate when its conditions. furnish a smaller number of independent equations (Art. 193) than there are unknown quantities to be determined.

Suppose, for example, that the conditions of a problem involving three unknown quantities furnish but two independent equations; as

2x-3y+ z = 2,

(1)

By Art. 189, equation (4) is satisfied by an indefinitely great number of sets of values of x and y; and hence (1) and (2) are satisfied by an indefinitely great number of sets of values of x, y, and z.

And in general, if we have m independent equations involving m+n unknown quantities, we may eliminate m1 of the unknown quantities, and obtain a single equation involving the remaining n+1 unknown quantities; and the latter equation is satisfied by an indefinitely great number of sets of values of the unknown quantities involved in it.

205. A problem is said to be Impossible when its conditions are incompatible, and consequently cannot be satisfied.

206. A problem is impossible when its conditions furnish a greater number of independent equations than there are unknown quantities to be determined.

Suppose, for example, that the conditions of a problem involving two unknown quantities furnish three independent equations; as

2x=y+7,

3y=14-x,

x+y=16.

Solving the first two equations, we find x = 5 and y = 3. But the third equation requires that the sum of the values of x and y should be 16; and hence the problem is impossible.

If, however, the third equation had not been independent, but derived from the first and second, as

x+y=8,

the problem would have been possible; but the last equation, not being required for the solution, would have been redundant.

In general, if we have m+n independent equations involving m unknown quantities, we may find one or more sets of values of the unknown quantities which will satisfy m of the given equations; but since neither of these sets will satisfy the remaining n equations, the problem is impossible.

INTERPRETATION OF NEGATIVE RESULTS.

207. 1. The length of a field is 10 rods, and its breadth is 8 rods; how many rods must be added to the breadth so that the area may be 60 square rods?

Let x=

the number of rods to be added.

Then by the conditions, 10(8+ x)=60.

Therefore,

or,

80+10x=60,

x=-2.

By Art. 42, adding - 2 rods is the same thing as subtract

ing 2 rods.

Hence 2 rods must be subtracted from the breadth in order that the area may be 60 square rods.

In the arithmetical sense, the above problem is impossible; for the area of the field at present is 80 square rods, and it is impossible to make it 60 square rods by adding anything to the breadth.

If we should modify the problem so as to read:

"The length of a field is 10 rods, and its breadth is 8 rods; how many rods must be subtracted from the breadth so that the area may be 60 square rods?"

and let a denote the number of rods to be subtracted, we should find x = = 2.

2. A is 35 years of age, and B is 20; it is required to determine the epoch at which A's age is twice as great as B's.

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

But by Art. 28, -5 years after is the same thing as 5 years before the present date.

Hence the required epoch is 5 years before the present date.

If we had supposed the required epoch to be x years before the present date, we should have found = : 5.

From the discussion of the above problems we infer

1. That a negative result may be obtained in consequence of the fact that the problem is impossible in the arithmeti tal sense.

2. That a negative result may be obtained-in consequence of a wrong choice between two possible hypotheses as to the nature of the unknown quantity.

In the first case, it is usually possible to form a problem analogous to the given problem, whose conditions shall be satisfied by the absolute value of the negative result, provided we attribute to the unknown quantity a quality the opposite of that which had been attributed to it.

In the second case, a positive result may usually be obtained by attributing to the unknown quantity a quality the opposite of that which had been attributed to it.

In either case, the equations answering to the changed conditions may be derived from the old equations by simply changing the sign of the unknown quantity wherever it occurs.

Similar considerations hold in problems involving two or more unknown quantities.

VARIABLES AND LIMITS.

208. A variable quantity, or simply a variable, is a quantity which may assume, under the conditions imposed upon it, an indefinitely great number of different values.

A constant is a quantity which remains unchanged throughout the same discussion.

209. A limit of a variable is a constant quantity, the difference between which and the variable may be made less than any assigned quantity, however small, without ever becoming zero.

In other words, a limit of a variable is a fixed quantity to which the variable approaches indefinitely near, but never actually reaches.

Note It is evident that the difference between a variable and its limit is a variable which approaches the limit zero.

where a and b are any numbers (Art. 33, Note), and each denominator after the first is ten times the preceding denominator.

It is evident that, by sufficiently continuing the series, the absolute value of the denominator may be made greater than any assigned positive quantity, however great, and the absolute value of the fraction may be made less than any assigned positive quantity, however small.

That is, if the numerator of a fraction remains constant, while the denominator increases without limit in absolute value, the value of the fraction approaches the limit 0.

It is customary to express this principle as follows:

where a and b are any numbers, and each multiplier after the first is one-tenth of the preceding multiplier.

It follows from Art. 210 that, by sufficiently continuing the series, the absolute value of the multiplier may be made less than any assigned positive quantity, however small, and the absolute value of the product may be made less than any assigned positive quantity, however small.

That is, if the multiplicand remains constant, while the mul tiplier approaches the limit 0, the product approaches the limit 0.