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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 learned by practice.

EXAMPLE XIII.

x+y+x=26

Suppofing-y=4

X- 2=6

by addition 3x=36

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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 discovered by the application of the preceding Rules."

$85." If indeed there are more quantities required than equations given, then the question 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 question;" because some of thefe conditions may be inconfiftent with others.

CHAP. XII.

Containing fome General Theorems for the exterminating unknown Quantities in given Equations.

IN

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, b, 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 "oppofite" coefficients that are taken each from a different equation,

and

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, thofe equations are to be refolved by Direction 3.

EXAMPLE XII.

Suppofe

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y

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-321 3d

18 -32-32

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

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

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