Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/124

 L E B R A. G A 92 V. “ The fum of two quantities, and the whence iox=B>'. . . .v=2-. . and x—y—18. whence “ Examp. difference of their fquares, being given, to find the “ quantities.” Suppofe them to be x and y, their fum j—x8==^- and y=qo. . x—y—18=72. /, and the difference of their fquares d. a Then, To find the time, fay. If 8 miles give 1 hour, 72 miles —2fy+y* will give 9 hours. . thus, 8 : 1 :: 72 : 9. * d~s1—2sy lx'-y'=d 2iy~s%1—d Examp. III. “ Suppofe the dillance between London x—s—y1 “ and Edinburgh to be 360 miles; and that a courier fets y s —d “ out from Edinburgh, running at the rate of 10 miles l —2{H-/* x’X*^ xzd--y%, whence * and. *=2T j*4-d “ an hour; another fets out at the fame time from Lon—' “ don, and runs 8 miles an hour: It is required to know “ where they will meet ?” Direct. V. If there are thfee quantiSuppofe the courier that fets out from Edinburgh runs ties, there muft be three equations inunknown order to deterx miles, and the other y miles, before they meet, then ““ mine them, by comparing which, you may, in all cafes, (hall. “ find two equations involving only two unknown quan“ titles; and then, by Direct. 3d, from thefe two you “ may deduce an equation involving only one unknown “ quantity; which may be refolved by the rules of the “ laft chapter.” From three equations involving any three unknown 4 360—jr quantities, x, y, and z, to deduce two equations, involving only two unknown quantities, the following rule -—■=360—y will always ferve. Rule. “ Find three values of x from the three gif+j=36o 4 ven equations; then, by comparing the firft and fe0^=1440 cond value, you will find an equation involving only y and z ; again, by comparing the firft and third, you ,v=3 60—-y=2 00 will find another equation involving only y and z; laftly, thofe equations are to be refolved by Examp. IV. “ Two merchants were copartners; and, “ the fum of their flock was 300 1. One of their flocks Dir. 3. “ continued in company 11 months, but the other drew Examp. VL “ out his flock in 9 months; when they made up their Suppofe “ accounts they divided the gain equally, What x+y+z=i2- ri2— y—^ “ was each man’s flock?” Suppofe the flock of the firfl to be x, and the flock of the other to be ; then, x+2y+3Z=2o/then, > 20—2>—32/2d/ ^ by fuppof. £7^°° 12—y—2=20—2y—32 31 ^=77=300—JThefe two laft equations involve only y and 2, and h>’+9/=33oo are to be refolved by Direft. 3d, as follows, 20^=3300 C 2y—y+3Z—2=20—12=8 = 16 5... x=3 00—j=135. l y2z~8 Direct. IV. “ When in one of the given equa^ quantitydimenfion; is of oneyoudimenfion, 36—3y—62=24—2y—22 ““ intionsthe the otherunknown of a higher mu ft findanda I2=H-4Z “ value of the unknown quantity from that equation y whence y=z £$ 12—42 8-22 ... . 2d ift value value M“ where it is of one dimenfion, and then raife that value to the power of the unknown quantity in the other “ equation; and by comparing it, fo involved, with the 8—22=12—42 “ value you deduce from that other equation, you fhall 22=12—8=4 “ obtain an equation that will have only one unknown and 2=2 “ quantity, and its powers.” ^(=8—22)=4 That is, when you have two equations of different x(=i2—y—2)=6. dimenfions, if you cannot reduce the higher to the fame dimenfion with the lower, you muft raife the lower to This method is general, ,and will extend to all equathe fame dimenfion with the higher. tions that involve 3 unknown quantities: but there are often
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