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Having the value of y, if we substitute it instead of y in the expression for x, this last will be known,

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To simplify this expression, we should, in the first place, perform the multiplication indicated upon the quantities

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and then reduce e to a fraction having the same denominator as the fraction which accompanies it, and perform the subtraction of this fraction (53); and it becomes

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* There might be some doubt as to the meaning of this expression; but it is obviated by attending to the bar denoting division, which is placed in the middle of the line. Thus, in the expression

x=

A

B'

A represents the dividend, whether integral or fractional, and

B the divisor, which may also be a whole number or a fraction. So also

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Suppressing the factor a, common to the numerator and denominator (38), we find

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are applied in the same manner as those, which we before found for literal equations, with only one unknown quantity; we substitute in the place of the letters, the particular numbers in the example selected.

We shall obtain the results in art. 56, by making

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77. The values of x and y are adapted not only to the proposed question; they extend also to all those, which lead to two equations of the first degree with two unknown quantities, since it is evident, that these equations are necessarily comprehended in the formulas,

the expression x =

fraction

C
B
divided by B, and the expression x =

signifies, that x is equal to the quotient of the

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It will be perceived by these remarks, that it is necessary to place the bars according to the result, which we propose to express.

ax + by = c,

dx + ey=f

provided the letters a, b, d, e, denote the whole of the given quantities, by which the unknown quantities x and y are respectively multiplied, and the letters c and f, the whole of the known terms, transposed to the second member.

Of the resolution of any given number of Equations of the First Degree, containing an equal number of unknown Quantities.

78. WHEN a question has as many distinct conditions, as it contains unknown quantities, each of these conditions furnishes an equation, in which it often happens, that the unknown quantities are involved with others, as we have seen already in the problems with two unknown quantities; but if these unknown quantities are only of the first degree, according to the method adopted in the preceding articles, we take in one of the equations the value of one of the unknown quantities, as if all the rest were known, and substitute this value in all the other equations, which will then contain only the other unknown quantities.

This operation, by which we exterminate one of the unknown quantities, is called elimination. In this way, if we have three equations with three unknown quantities, we deduce from them two equations with only two unknown quantities, which are to be treated as above; and having obtained the values of the two last unknown quantities, we substitute them in the expression for the value of the first unknown quantity.

If we have four equations with four unknown quantities, we deduce from them, in the first place, three equations with three unknown quantities, which are to be treated in the manner just described; having found the value of the three unknown quantitics, we substitute them in the expression for the value of the first, and so on.

See an example of a question, which contains three unknown quantities and three equations.

79. A person buys separately three loads of grain; the first, which contained 30 measures of rye, 20 of barley, and 10 of wheat, cost 230 francs;

The second, which contained 15 measures of rye, 6 of barley, and 12 of wheat, cost 138 francs;

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The third, which contained 10 measures of rye, 5 of barley, and 4 of wheat, cost 75 francs ;

x

It is asked, what the rye, barley and wheat cost, each per measure? Let a be the price of a measure of rye, that of a measure of barley,

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that of a measure of wheat.

To fulfil the first condition, we observe, that

30 measures of rye are worth 30x,

20 measures of barley are worth 20 y,

10 measures of wheat are worth 10 z;

and as the whole must make 230 francs, we have the equation 30 x + 20 y + 10 z = 230.

For the second condition, we have
15 measures of rye, worth 15 x,

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15x+6y+ 122 = 138.

For the third condition, we have

10 measures of rye worth 10 x,

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The proposed question then will be brought into three equations;

30 x + 20 y + 10 z = 230,
15x+6y+12 z = 138,

10x+5y + 4 z= 75.

Before proceeding to the resolution, I examine the equations, to see if it is not possible to simplify them by dividing the two members of some one of them by the same number (12), and I find that the two members of the first may be divided by 10, and those of the second by 3. Having performed these divisions, I have only to occupy myself with the equations

3x + 2y + z = 23,

5x + 2y + 4 z 46,

10x+5y + 4 z = 75.

As I can select any one of the unknown quantities in order to deduce its value, I take that of z in the first equation, because this unknown quantity having no coefficient, its value will be entire or without a divisor, as follows;

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This value being substituted for z in the second and third equa

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8 y = 75;

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10x+5y + 92 12 x

and reducing the first member of each, we find

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To proceed with these equations, which contain only two unknown quantities, I take in the first the value of the unknown and I obtain

quantity y,

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and by substituting this value in the second equation, it becomes

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The denominator, 6, may be made to disappear by the usual method, but observing that the denominator is divisible by 3, I can simplify the fraction by multiplying it by 3, agreeably to article 54 of Arithmetic. I have then

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The denominator 2 being made to disappear, it becomes

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and by substituting these values in the expression for that of z,

we obtain

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122 × 3 = 23

z = 23 3 X 4 6, or z = 5. It appears then, that the price of the rye per measure was 4 fr.,

that of the barley

that of the wheat

3,

5.

This example, while it illustrates the method given in the preceding article, ought to be attended to, on account of the abbreviations of calculation, which are performed in it.

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