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alter the declared relation between the known and unknown quantities, and if a possible solution exist in the case of but one of these modified forms of the question, that solution will always be furnished by the algebraic process, though for other forms of the question it may be clearly inapplicable or meaningless. This is a peculiar and a very important feature: it is equally impressed upon all algebraic researches, and enables the investigator not only to satisfy the demands of every question, within the province of the science he cultivates, but also informs him of the fact whenever those demands are incongruous or impracticable. He may implicitly trust himself to the guidance of his symbols with perfect confidence; so long as his deductions, step after step, are logically made, they can never lead him into error. If the question he is seeking to answer admits of an intelligible solution, to that solution he will be inevitably conducted: if it does not, his result will spontaneously declare to him the fact, and will at the same time suggest to him what modification of the question is necessary to convert it into a solvible form. It would almost seem that the symbols he employs, though but the creatures of his own contrivance, were endowed with intelligence, leading him to the truth, wherever it can be found, and saving him, by its own warning indications, from falling into error and absurdity. It is on this account that an algebraist takes no trouble, by a preliminary scrutiny of the question proposed to him, to discover whether the thing demanded of him be possible or impossible: he at once sets out on the search of it: if it exists, he finds it; if not, he arrives at unmistakable indications of its impossibility. A simple illustration will enable the learner more clearly to see the truth and bearing of these remarks.

Question. There are two numbers, such that one is double the other, but if 10 be added to each, then the one will be three times the other: what are the numbers?

Solution. Let x represent the smaller number, then 2x will represent the larger, by the first condition; and, by the second condition, we have the equation

2x-10=3(x+10); that is, 2x+10=3x+30

x=-20, and 2x=-40.

The sought numbers are, therefore, the two negative numbers -20 and -40; and the question is properly put, inasmuch as we have here obtained an answer to it which satisfies its conditions: for -40 is twice -20, and -40+10 is three times 20+10.

But suppose the question had been proposed in the following shape, namely: A father's age is double that of his son, but 10 years to come the father will be three times the age of the son: what are their ages?

The learner will at once perceive that the above is still the algebraical solution: the equation embodying the conditions of the question is the same in whichever of these two forms it be proposed. But, in the latter form of the question, the answer is absurd, as no one can be minus 20 years old. Now the absurdity of the answer is the indication of the absurdity of the question, and also the indication of the kind of absurdity in the language of it: the sign minus, in reference to the years, presenting itself where, if the question had been solvible, we should have had plus, suggests to us that time future has been absurdly put in the question for time past. Introducing the modification thus suggested, the question becomes this, namely: A father's age is double that of his son, but ten years ago the

father was three times the age of the son: what are their ages? As before, representing the ages of father and son by 2x and x, the equation is 2x-10=3(x-10), or 2x-10=3x-30

.. x=20, and 2x=40.

It is in this way that algebra discovers to us the practical impossibilities implied in questions like that above, and at the same time suggests the modification necessary to their intelligible solution: whenever the value of the unknown quantity presents itself with a minus sign, then this sign, being in direct opposition to plus, we infer that if the demand in the question be concerning time future, it can be satisfied only by correcting future, and writing past: if it concern a gain, the word must be replaced by loss; if height or altitude, the proper correction will be depth or depression, and so on; the minus sign in the result always implying that the thing is to be taken in the sense directly opposite to that expressed in the question, or assumed at the outset of the solution.

CHAPTER V.

SIMULTANEOUS SIMPLE EQUATIONS.

55. We have already seen that one simple equation is sufficient to enable us to determine the value of a single unknown quantity; for by the rules already taught, we may always isolate this unknown quantity on one side of the equation, and cause the remaining quantities to occupy the other side, so that these remaining quantities being all known, their equivalent, on the opposite side of the equation, becomes known. This would not be the case if the remaining quantities spoken of also contained an unknown quantity, since both sides of the equation would then be unknown. Hence, to determine two unknown quantities, x and y, a single equation is insufficient: two equations, therefore, must be necessary, and we shall shortly see that two are sufficient, provided these two are really independent, and not necessary consequences the one of the other; that is, provided they really express distinct conditions.

Two such equations are said to be solved when values of x and y are found which, at the same time, satisfy both; and they are, therefore, called simultaneous equations. The determination of three unknown quantities requires, in like manner, three distinct equations, all of which the proper values of the three unknowns must equally satisfy: they are called simultaneous equations, in reference to the fact that they are simultaneously satisfied, by certain determinable values, of the

unknown quantities which they involve. And be the number of unknown quantities ever so numerous, if the conditions between them and the known quantities be such as to furnish an equal number of independent equations, a set of values of these unknowns always exists which will simultaneously satisfy all the equations; and, on this account, the equations are said to be simultaneous.

In the solution of simultaneous equations, the art is so to combine them with each other, or so to combine equations deduced from them, as to bring out a single equation involving only one of the unknown quantities: the other unknowns are then said to be eliminated; and the value of the one retained is then determined as in the preceding articles. The number of unknown quantities becomes thus diminished by one; and, consequently, the number of simultaneous equations becomes diminished by one: these form a new set of equations from which all the unknowns in them, but one, may in like manner be eliminated, and the remaining one determined as before: and so on, till the values of all the unknown quantities are found one after another. Or, without these repetitions of the same process, the determination of one unknown may in general be made the key for the discovery of the others by a few easy operations.

The way in which these operations are performed, as respects simultaneous simple equations, is now to be explained, or rather the ways: for there are three somewhat different modes of proceeding, either of which will accomplish the object sought.

56. To solve two simultaneous simple equations:

First Method. Find the value of one of the unknown quantities in terms of the other unknown and the known quantities, from the first equation: that is, proceed exactly as if there were but one unknown in the equation, and find the equal of it.

Do the same thing with the second equation: finding the value of the same unknown from it.

Put these two values, or expressions, for the same symbol, equal to each other: and we shall then have a simple equation containing only one unknown quantity; for the unknown at first dealt with will be eliminated.

The case of two unknown quantities becomes thus reduced to that of one, for the solution of which directions have already been given.

The truth of this rule, like that of all algebraical rules ex

pressed in words, will be seen by means of an example in which the known quantities are represented by general symbols, so as to render the reasoning applicable to every particular case. It. 'may be remarked that in such general investigations, algebraists contrive as much as possible to give a certain symmetry or semblance of appearance to those equations which are the same in all but the coefficients. They do so by representing the coefficients of the same unknown in the several equations by the same letter; and by way of distinction, they connect with these like letters different marks: thus, in treating two simple equations with two unknowns, if one were generally represented by ax+by=c, it would be unusual to employ totally unlike letters in the representation of the other, and to write it dx+ey=f, or mx+xy=p, &c.; the letters a, b, c, would be used for the known quantities in both equations, and for the purpose of showing that these equations are really distinct, a, b, c, in one of them would be marked thus, a', b', c', or a1, b1, c1, &c.; so that a pair of simple equations with two unknowns would be written in one or other of the following forms:

ax+by=c ax+by=c a1x + b1y = c1 &c.

a'x+b'y=c'

or,

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or,

ax+b2y=c2

the last is the form of notation we shall here employ.

If in each of these equations we transpose the term containing y, and then divide by the coefficient of x, we shall have, from the first,

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c2—b2y;

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so that equating these two expressions for the same thing, we have

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, or clearing fractions, ac-aby-ac2-aby; a2

an equation containing only the single unknown y, a being eliminated. By transposition and division, we get, for the solution of this equation, the value of y in terms of the coefficients; and y being thus known, if we substitute its value for that symbol, in either of the foregoing expressions for x, the value of the unknown, previously eliminated, will become also known, and we shall thus have

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which are the general formulæ of solution for every pair of equations of this kind, so that by substituting for the letters the corresponding numerical values of the coefficients, in any proposed case, the solution may be at once exhibited. The rule does not translate the formulæ, but it describes the process, which being simple in its character and easy of recollection, may be gone through in each individual case with but little trouble.

(1) 3x+2y=16, 4x-3y=-7.

From these, we have for x the expressions

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clearing fractions, 2x-80-y, 12x+12y+20x=30y—15x+2100.

In the first, the expression for one of the unknowns, y, is already exhibited: the second, after transposition, is

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