We might also change the signs after reduction, and the equation

It follows from this, that we may transpose indifferently, to one member or to the other, all the terms involving the unknown quantity, observing merely to change the signs of the two members in the result, when the unknown quantity has the sign -.

58. Having undertaken, by means of letters, a general solution of the problem of art. 56, I will now examine a particular I suppose the first sum received by the labourer to be 46 francs, and the second 30, the other circumstances remaining as before; the equations of the question will then be

case.

multiplying this value by 5, in order to substitute it in the place of 5 y, in the second, we have

and the signs being changed agreeably to what has just been remarked,

If we substitute this value instead of x in the expression for y, it will become

Now how are we to interpret the sign, which affects the insulated quantity 14? We understand its import, when there are two quantities separated from each other by the sign and

when the quantity to be subtracted is less than that from which it is to be taken; but how can we subtract a quantity when it is not connected with another in the member where it is found? To clear up this difficulty, it is best to go back to the equations, which express the conditions of the question; for the nearer we approach to the enunciation, the closer shall we bring together the circumstances which have given rise to the present uncertainty.

I resume the equation

12x + 7y = 46;

I put in the place of x its value 5, and it becomes

Ꮖ

60+ 7y=46.

This equation, by mere inspection, presents an absurdity. It is impossible to make the number 46 by adding any thing to the number CO, which exceeds it already.

I take also the second equation,

8x+5y=30,

and putting 5 in the place of x, I find

40+5y=30;

the same absurdity as before, since the number 30 is to be formed by adding something to the number 40.

Now the quantities 12 x or 60 in the first equation, 8 x or 40 in the second, represent what the labourer earned by his own work; the quantities 7y and 5y stand for the earnings of his wife and son, while the numbers 46 and 30 express the sum given as the common wages of the three; we must see then at once in what consists the absurdity.

According to the question, the labourer earned more by himself, than he did by the assistance of his wife and son; it is impossible then to consider what is allowed to the woman and son, as augmenting the pay of the labourer.

But if, instead of counting the allowance made to the two la ter persons as positive, we regard it as a charge placed to the account of the labourer, then it would be necessary to deduct it from his wages; and the equations would no longer involve a contradiction, as they would become

we deduce from the one as well as from the other

and we conclude from it, that if the labourer earned 5 francs per day, his wife and son were the occasion of an expense of 2 francs, which may otherwise be proved thus.

For 12 days' labour he received

5 X 12 or 60 francs;

the expense of his wife and son for 7 days is

2 X 7 or 14 francs;

there remain then 46 francs.

For 8 days' labour he receives

5 X 8 or 40 francs;

the expense of his wife and son for 5 days is

there remain 30 francs.

2 X 5 or 10 francs,

It is very clear then, that in order to render the proposed problem with the first conditions possible, instead of the enunciation in article 56, we must substitute this;

A labourer worked for a person 12 days, having had with him the first 7 days, his wife and son at a certain expense, and he received 46 francs; he worked afterwards 8 days, during 5 of which he had with him his wife and son at an expense as before, and he received 30 francs. It is required to find how much he earned per day, and what was the sum charged him per day on account of his wife and

son.

Calling the daily wages of the labourer, and y the daily expense of wife and son, the equations of the problem will evidently be

and being resolved after the manner of those in art. 56, they will give

x = 5 francs, y = 2 francs.

59. In every case, where we find, for the value of the unknown quantity, a number affected with the sign, we can rectify the enunciation in a manner analogous to the preceding, by examining, with care, what that quantity is, among those, which are additive in the first equation, which ought to be subtractive in the second; but algebra supersedes the use of every inquiry of this kind, when we have learnt to make a proper use of expressions affected with the sign; for these expressions, being deduced from the equations of the problem, must satisfy those

equations; that is to say, by subjecting them to the operations indicated in the equation, we ought to find for the first member

a value equal to that of the second. Thus the expression

drawn from the equations

12x + 7y = 46,

8x+5y=30,

-14

7

must, consistently with the value of x = 5, as deduced from these same equations, verify them both.

The substitution of the value of x gives, in the first place,

60+ 7y= 46,

40+5y30.

It remains to make the substitution of 14 in the place of y;

7

and for this purpose we must multiply by 7 and by 5, having regard to the sign with which the numerator of the fraction

is affected.

If we apply the rule relative to the signs given in art. 42 for division, we have

besides, by the rule for the signs in multiplication, we find

and are verified, not by adding the two parts of the first member, but in reality by subtracting the second from the first, as was done above, after considering the proper import of the equations.

60. The problem in art. 58 does not admit of a solution in the sense in which it is first enunciated; that is to say, by addition, or regarding as an accession the sum considered with reference to the wife and son of the labourer; neither does the second enunciation consist with the data of the problem in art.

56.

If we were to consider in this case y, as expressing a deduction, the equations thus obtained

and the substitution of the value of x would immediately change

The absurdity of these results is precisely contrary to that of the results in art. 58, since it relates to remainders greater than the numbers CO and 40, from which the quantities 7y and 5 y are to be subtracted.

The sign minus, which belongs to the expression of y, implies an absurdity; but this is not all, it does it away also; for, according to the rule for the signs,

401050,

and are verified by addition; consequently, the quantities - 7 y and 5 y, transformed into + 14, +10, instead of expressing expenses incurred by the labourer, are regarded as a real gain. We are brought back then, in this case, also to the true enunciation of the question.

61. We perceive by the preceding examples, that there may be, in the enunciations of a problem of the first degree, certain contradictions, which algebra not only makes known, but points out also how they may be reconciled, by rendering subtractive certain quantities which had been regarded as additive, or additive certain quantities which had been regarded as subtractive, or by giving to the unknown quantities values affected with the sign

See then what is to be understood, when we speak of values affected by the sign, and of what are called negative solutions, resolving, in a sense opposite to the cnunciation, the question in which they occur.