RESOLUTION OF EQUATIONS. 



We cannot, in a limited article like the prefent, give all 

 the methods that may be employed for exterminating the 

 unknown letters ; thefe being extremely various, and de- 

 pending much upon the practice and proficiency of the ana- 

 lyft himfelf, and the manner in which thofe quantities are 

 involved ; but the moft applicable and general methods are 

 the three following, viz. 



1. Find the value of one and the fame unknown quantity 

 in each equation, and put all thefe values equal to each 

 other, which will eliminate one of the quantities, and reduce 

 the number of equations to one lefs. Then do the fame in 

 thefe new equations ; and again in- the laft ; and fo on, till 

 there be but one equation and one unknown quantity, the 

 value of which mult be found by the proper rules, as above 

 referred to. 



2. Find the value of one of the unknown quantities in 

 one of the equations, in terms of the other quantities ; then 

 fubftitute this value for that quantity in all the other equa- 

 tions ; again, find the value of one of the remaining quan- 

 tities, and fubftitute its value as before ; and fo on, till there 

 remains but one equation and one unknown quantity, whofe 

 value is to be found as before. 



3. Multiply each of the equations by fuch numbers as 

 will render the co-efficients of one of the letters the fame in 

 all ; then, by adding or fubtra&ing thefe equations ac- 

 cording as the equal co-efficients have unlike or like figns, 

 the quantity whofe co-efficients were equal will difappear ; 

 which being repeated again upon the remaining quantities, 

 there will ultimately be found only one equation and one 

 unknown quantity. And it may be proper to obferve, that 

 in all thefe cafes, if any of the unknown quantities have frac- 

 tional co-efficients, the whole equation in which they are 

 found ffiould be multiplied by fuch a number as will convert 

 thefe fractions into integers. 



Thus, in the equations 



i* + tVj = 10 



-f* + 3y =95 

 multiply the firft by 1 J, and the latter by 2, gives 

 9 * -+- 2 y = 150 



x + 6y = 190 

 the folution of which, by each of the preceding rules, will 

 be as follows. 



Or, putting letters inftead of the above numerical co-effi- 

 cients, in order to render the folution more general, let there 

 be given 



ax + by = c\ toiiBixmi 

 dx + ey = /J J 



1ft method. B* = c - *T| by traction. 



by _ f - ey 



by divifion. 



a d 



it — db y = af — acy, by multiplication. 

 {ae — db) y = af — dc 

 af — dc 

 y ~ ae - db 

 And in the fame manner we find, 



ec - If 



db 



ad method, a x 

 dx 



+ b y - '] to find* and v. 

 + ey = fS 



e — by 



as above. 



dc 



db 



y 



+ ey = f, by fubftit. 



dc — dby + aey = af, by multiplication. 



{ae — db) y = af — de 



af — dc 

 y = - — — as before. 



ae 



ec 



db 

 If 



a e — db 



3 d method, ax + by = c\ tQ find . 



dx + c y = /J ■> 



dax + dby = dc, mult, by d. 



adx 4- acy = af, mult, by a. 



(db — ac) y = dc — af 



dc — af af — dc ■ 



- db 



' ~ db - ae ~ 

 _ h f- ec 



db — ae ae — db 



b J\ 



as before. 



This will, in fome meafure, illuftrate the preceding rules, 

 which we (hall not infill upon any further ; but, in the fol- 

 lowing examples, (hall avail ourfelves of any advantages that 

 the equations prefent, in order to arrive at the folution in 

 the eafielt manner poifible. 



Examples. 



1. Given x* 4- y' = a 1 . c , . 



T J , 5- to find * and y. 

 xy = b 3 



a* + y> — a 



2xy = 2b, doubling the 2d. 



x' + 2 * y + y' = a -t- 2 b, by addition. 

 „v' — 2 xy + y" = a — 2 b, by fubftit. 



«+*=«/ j fl + ^j}by«traftion. 

 x — y = J (a — 26) S ' 



J (a + 2 b) + ^ (a — 2 b) 

 x = : ■- — 



2 



y = 



/ (« + 2b) - (a -2b) 



2. Given x + y = al . c , , 



r J . \ to find x and y. 

 xy = b$ J 



x 1 •$• 2 x y -f y 1 = a 7 , by fquaring. 



4*j> = 4^> mult, by 4. 



x" — 2xy +/ = a 1 — 4 b, by fubftit. 

 .r — y = V {a r — 4*), by extract. 

 *■ + y = a, 1 ft equat. 



a + J [a-- - 4 3) 

 * = 



2 



a - J (a 1 - a b) 

 y = 



3. Given .r -y = a] tofindxZnd 



xy = bS J 



x* — 2 xy + y r =2: a', fquaring. 



4*j> = q.b, mult, by 4. 



*'■ + 2 xy -f- y 1 = a z -f 4 £, by adding. 

 x + y — V (a 1 + 4*), by extract. 

 x — y = a 



_ a+ ^(d 1 + 4^) 



2 

 _ a - ^/ (a' + 4 b) 

 y ~ 2 



4. Given 



