If divided by the fame quantity, it may be truck out of them all." Thus, 3x + b = a + b .... 3x = a.... and x = If 3ax + 5ab8ac..3x+5b-8c.. and x= 3 8c-5b 3 2x + 8 = 16 .... and x=4. RULE VIII. $77. "Instead of any quantity in an equation you may fubftitute another equal to jt." y = 5x; then 15x+5* (= 20x) = 120, The further improvement of this Rule fhall be taught in the following chapter. CHA P. XI. OF THE SOLUTIONS OF QUESTIONS THAT PRODUCE SIMPLE EQUATIONS. SIM IMPLE equations are thofe "wherein the unknown quantity is only of one dimenfion:" In the folution of which we are to obferve the following directions. DIRECTION I. §78. "After forming a diftinct idea of the queftion propofed, the unknown quantities are to be expreffed by letters, and the particulars to be tranflated from the common language into the algebraic manner of expreffing them, that is, into fuch equations as fall exprefs the relations or properties that are given of fuch quantities." Thus, if the fum of two quantities must be 60, that condition is expreffed thus, x + y = 60. If their difference must be 24, that condition gives xy24. If their product must be 1640, then xy= 1640. If their proportion is as 3 to 2, then x:y :: 3:2, or 2x=3y; becaufe the product of the ex tremes tremes is equal to the product of the mean terms. $79. DIRECTION II. "After an equation is formed, if you have one unknown quantity only, then, by the Rules of the preceding Chapter, bring it to stand alone on one fide, fo as to have only known quantities on the other fide:" thus you shall difcover its value. EXAMPLE. A person being asked what was his age, anfwered that of his age multiplied by age gives a product equal to his age. was his age? of bis Qu. what $ 80. 48 3x2=48x, 3* = 48, "If there are two unknown quantities, then there must be two equations arifing from the conditions of the question: Suppose the quantities x and y; find find a value of x or y, from each of the equations, and then by putting these two values equal to each other, there will arife a new equation involving one unknown quantity; which must be reduced by the Rules of the former Chapter. EXAMPLE I. Let the fum of two quantities be s, and their difference d. Let s and d be given, and let it be required to find the quantities themselves. Suppose them to be x and y, then, by the suppofition, Let it be required to find two numbers whofe fum is s, and their proportion as a to b. Let the numbers be x and y, then shall A privateer running at the rate of 10 miles an bour, difcovers a fhip 18 miles off making way at the rate of 8 miles an hour: It is demanded how many miles the ship can run before she be overtaken? Let the number of miles the ship can run before the be overtaken be called x; and the number of miles the privateer must run before fhe come up with the fhip, be y; then fhall (by Supp.) .... y=x+18.... and xy:: 8:10, and xy-18. whence Icx=8y.... x= 4 y |