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$82.

DIRECTION V.

"If there are three unknown quantities, there must be three equations in order to determine them, by comparing which you may, in all cafes, find two equations involving only two unknown quantities; and then, by Direction 3, from these two you may deduce an equation involving only one unknown quantity which may be refolved by the Rules of the last Chapter.".

From three equations involving any three unknown quantities, x, y, and z, to deduce two equations

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equations involving only two unknown quantities, the following Rule will always ferve.

RULE.

"Find three values of x from the three given equations; then, by comparing the first and fecond value, you will find an equation involving only y and z; again, by comparing the first and third, you will find another equation involving only y and z;" and lastly, those equations are to be refolved by Direction 3.

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These two laft equations involve only y and z, and are to be refolved, by Direction 3, as follows.

2y—y + 32 — Z = 20 — 12 = 8

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83. This method is general, and will extend to all equations that involve three unknown quantities: but there are often easier and shorter methods to deduce an equation involving one unknown quantity only; which will be best Jearned by practice.

EXAMPLE XIII.

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$ 84. It is obvious from the 3d and 5th Directions, in what manner you are to work if there are four, or more, unknown quantities, and four, or more, equations given. By comparing the given equations, you may always at length discover an equation involving only one unknown quantity; which, if it is a fimple equation, may always be refolved by the Rules of the laft Chapter. We may conclude then, that "When there are as many fimple equations given as quantities required, thefe quantities may be difcovered by the application of the preceding Rules."

$ 85. "If indeed there are more quantities required than equations given, then the queftion is not limited to determinate quantities; but is capable of an infinite number of folutions." And, "If there are more equations given than there are quantities required, it may be impoffible to find the quantities that will anfwer the conditions of the queftion;" because fome of thefe conditions may be inconfiftent with others.

CHA P. XII.

CONTAINING SOME GENERAL THEOREMS FOR THE EXTERMINATING UNKNOWN QUANTITIES IN GIVEN EQUATIONS.

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N the following Theorems, we call thofe coefficients of the "fame order" that are prefixt to the fame unknown quantities in the different equations. Thus, in Theor. 2. a, d, g, are of the fame order, being the coefficients of x: alfo b, e, h, are of the fame order, being the coefficients of y: and thofe are of the fame order that affect no unknown quantity.

But those are called " oppofita" coefficients. that are taken each from a different equation,

and

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