RESOLUTION OF EQUATIONS. 



4. Given .r - y = «7 t0 find and 

 *' + / = * J . 



.v- — 2 .«.-_y +• /" = a', fquaring. 

 2 x~ + 2 y 7 = 2 b, doubling. 



x* + 2xy+y'=2b — a\ fubtr. 



* + y == %' ( 2 * - a! ) 



a + ,/(2* -a') 



a — ^/ (2 b — a 7 ) 



y = 



■* 2 



This method of rcfolution will apply in many problems, 

 and fometimes Caves conliderable labour. 



5- Given x> + xy = .T find „ and 



r + ■»■>' = b S . 



x*- 4- 2 jtj> + y~ = a + £, by adding. 

 x + y = J (a + b), by extract. 

 (x -t- J-) * =*/(« + *) = !" 



I* + j) j» = y V {" + b ) = h 



a h 



x = , and y = ; - . 



J(a + b) ' Vifl + b) 



6. Find three numbers in arithmetical progrefiion, whofe 

 fum is a, and fum of their fquares b. 



Let x — y, x and + j', be the numbers required. 



Then x — y + x+.v+y=$x = a 



(x-y)' + *' + (x + y) z = 3X ! + 2/=r b 



x* = — = b ~ 2y * 



9 3 



Whence a 1 = 3 b — 6y z 



Make y = a .v, then thefe become 

 a *' = .r' - z z x* ] 

 a * 2 — .v 1 4- a' a- 1 I 

 * =. I — x\ from the firft, 

 a 1 f a = 1, or a = - i + i v 5 ; 



whence a is a known quantity. Now, 



a = x + x 3 x, from the fecond, 



— I 4. X / r 



a 

 1 + a 



and 



= s/ 



lb — a 1 , a 



— , and x = — . 



6 3 



And a fimilar fubftitution, -viz. one which anfwers one of 

 the conditions of the queltion, may frequently be employed 

 to great advantage. 



7. Sometimes it will be convenient to fubftitute for the therefore .* = 

 fums and differences of numbers, as in the following ex 

 ample. 



Given * +, = «| t0 find* and «. 



.v* 4- / = 4J J 



1 - (i - h V 5) J 



3 T >; V 5 £ 



I ~ « + V5 ' 2 

 j> = a^ = — i-f-^^/j, as required. 



Sometimes it faves confiderable labour, to find the fum, 

 product, or difference, of the two quantities, inftead of tin- 

 quantities themfelves ; thus : 



9. Given x 1 + y x — x — y •= a 



*y + * + y = b\ 



by addition *' + xy 4- y- = a + b 



x 1 + 2 xy + y' = a -f b + xy 

 x + y = ^ (a + b + xy); 



but x -f y = b — .r _y ; 



therefore b % — 2bxy-* r x'y 7 = a + b+xy 



. 2}+ 1 //2 £ + i\" 

 whence xy = — + \/ I ) +a + b-b'. 



Now make xy = p a known quantity, and vre have, 

 from the fecond equation, 



p + x + y == by 



or x + y = b — p 

 xy =p 



whence «' — 2 xy + y- = (b — p)~ — 



x - y = V l(b-py - 4 pl 

 x' + y = b — p 



_ i-p+ ^ [(*-/)' -4/] 



2 



_b-p- x f\_{b-pY- 4/] 



Let x + y 

 and : 



lm jthenf v 



= ?u 4- « 



ji = 2n j (^ _y = wj — n 



(m + «) 4 + [m — «) 4 = b, 



or 2 m* 4- l2i» l n' 4- 2 «' = i ; 



but 2 m = a, or »i = - — ; therefore m is known, and the 

 2 



above becomes 



2 « 4 4- I2m"n' = J - 2 m' 

 n l 4- 6 m« «' = i{- w 4 



» a = - 3«* + V(i* + 8« 4 ) 

 and n = [3^ + .y (ii 4- 8m 4 )]*. 



Whence m and n being known, * and y are alfo known ; 

 for x = m 4- «, and _y = ;w — n. 



8. Sometimes it is advantageous to confider one of the 

 quantities as an unknown multiple of the other ; thus : 



Given xy — x 1 — y 1 = .r' ■+■ v 7 , to find x and y. 

 Vol.. XXX. 



Thefe, and a variety of other artifices peculiar to certain 

 equations, will occur to the practical analyfl ; of which, 

 numerous examples may be fcen in Bland's Algebraical 

 Problems, as alfo in Bonnycaftle's and Euler's Algebra. 



Resolution of a geometrical Problem algebraically. The 

 procefs in the former article is to be obferved throughout ; 

 but as it rarely happens we come at an equation in geo- 

 metrical problems by the fame means as in numerical ones, 

 there are fome farther things to be noted : ill, then, fup- 

 pofe the thing done, which was propofed to be done. 

 2. Examine the relations of all the lines in the diagram, 

 without any regard to known or unknown, in order to 6nd 

 which depends on which ; and from which being had, what 

 others are had, whether by fimilar triangles, or re&ang 

 Sec. 3. To obtain the fimilar triangles or reftang] , the 

 lines are to be frequently produced, till thej becon dither 

 directly or indirectly equal to given ones, orinte'fo-t others, 

 &c. Parallels and perpendiculars to be frequently drawn ; 

 points to be frequently connected ; and angles to be made 

 equal to others. Jt thus ymi do not arrive at a neat equa- 

 tion, examine the relation of the lines in another maimer. 



F Sometimes 



