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- 0 – ΣΗΜΕΙΩΣΕΙΣ ΑΛΓΕΒΡΑΣ Β ΛΥΚΕΙΟΥ ΕΠΙΜΕΛΕΙΑ :∆ΡΟΥΓΑΣ ΑΘΑΝΑΣΙΟΣ 0 ΣΥΣΤΗΜΑΤΑ ΑΛΓΕΒΡΑ B ΛΥΚΕΙΟΥ ΕΠΙΜΕΛΕΙΑ :∆ΡΟΥΓΑΣ ΑΘΑΝΑΣΙΟΣ
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### Transcript of 'Αλγεβρα Β λυκείου,συστήματα

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2. - 1 . x + y = . x = yy, xx xx (,0). y = x + yy. x + y = 0 0 : + = = = ( xx). 0 =0 x 0y x x+ = = = ( yy). =0 0 0 y y y 0 0 y x y ( )= = + ( y = x + , yy).: x + y = 0 0 . : ( ) + = x yx y + = : 1 3. - 2 0 0 ( ,y ) . . : + = + =x yx y 1) . (.. x). x y. y x.2) ( ). . , . . . . :i. .ii. () () , .. () () + () , () 0 () . () + () () (). , : 0 = = , = = +=+. : 2 4. - 3 1) .x + 2 y = 4 x = 4 2 y (2) 2 1 2(4 2 ) 1 8 4 1 3 1 8 3 9 3 9+ = + = + = = = = = =x y y y y y y y yy y y x = 4 2y : x = 4 2 3 x = 4 6 x = 2. (x,y)=(-2,3).+ = + = 3y = 9 3y = 9 y = 3. y : x + 2 3 = 4 x = 4 6 x = 2. (x,y)=(-2,3).3A(0,2)21(-2,3)(-1/2,0) + =2 4 (1)x yx y + =2 1 (2)+ = = 2 4 2 2 4 8(+ )1 2 -2 -1 01 2 3 4-1-2B(4,0)(0,-1)1. :( ) :93.32) .x y x yx y x y2 1 1 2 13) . .i. X+2y=4. x=0 0+2y=42y=4y=2 (0,2). y=0 x=4 (4,0). x+2y=4 .ii. 2x+y=-1. x=0 y=-1, (0,-1). y=0 1x= 2 (-1/2,0). 2x+y=-1 . : 3 5. - 4 . x=-2 y=3. (x,y)=(-2,3).2. ():+ = + = 3 3 (1)x yx y2 6 5 (2)1) .x + 3y = 3 x = 3 3y . (2) :2(3 3y) + 6y = 5 6 6y + 6y = 5 6 = 5 .2) .+ = = x 3y 3 2 2x 6y 62x 6y 5 1 2x 6y 5+ = + = (+) ( x)0x+0y=-110=-11 .( x, y ).3) . .i. X+3y=3. x=0 0 + 3y = 3 3y = 3 y = 1 (0,1). y=0 x = 3 (3,0). x+3y=3 .ii. 2x+6y=-5. x=0 5= = 6y 5 y6 (0,-5/6). y=0 5= = 2x 5 x2 (-5/2,0). 2x+6y=-5 . y 1y = x y = x +3 3 13 : 4 6. - 5 2 5 1 5y = x y = x y = x .6 5 26 6 3 6 -1/3. , .3. ():+ = + =2 2 (1)x yx y2 4 -4 (2)1) .x + 2y = 2 x = 2 2y . (2) 2(2 2 y) + 4 y = 4 4 4 y + 4 y = 4 4 = 4 . x, y . (..). x=, 2 + = = = .. 2y 2 2y 2 y2 2 = .(, y) (, )22) .+ = =2 2 2 2 4 4(+) ( x)x y x yx y x y+ = + = 2 4 4 1 2 4 40x+0y=00=0. x, y . 2= , . ( , ) ( , )2y3) . .i. x+2y=-2. x=0 y=-1 (0,-1). y=0 x=-2 (-2,0). x+2y=-2 .ii.2x+4y=-4. x=0 y=-1 (0, -1). y=0 x=-2 (-2, 0). , . . , , 2( , )2 . : 5 7. - 6 4. (): x + y x y= 5 3 + = +1x y x y y2 3 2 () + + x y x y x y x y= = + = + = 15 15 3( ) 5( ) 3 3 5 5x y x y x y x y5 3 5 33 x 5 x + 3 y + 5 y = 0 2 x + 8 y = 0 x + 4 y = 0 x 4 y=0 () + + x y x y y x y x y y = + = + + = + 1 6 6 6 6 1 3( x y ) 2( x y ) 3 y62 3 2 2 3 23 x + 3 y 2 x + 2 y = 3 y + 6 3 x + 3 y 2 x + 2 y 3 y 6 = 0 x + 2 y=6 () =4 02 6x yx y + = .( ) = =4 0 1 4 0+x y x yx y x y+ = = 2 6 1 2 6( x)6 y = 6 y = 1 x 4y = 0 x 4 = 0 x = 4 () (x, y) = (4, 1).5. (3,2) (4,5). y=x+ (1). (1) , (1) 2 = 3 + 3 + = 2 5 = 4 + 4 + = 5 . + = + =3 24 5 + = + =3 2 1 3 24 5 1 4 5+ = = (+) ( )-=-3=3 3 + = 2 33 + = 2 = 2 9 = 7 . , (1) : y=3x-7. : 6 8. - 7 3) . + = + = x yx y . D = = . x D = = , D x . y D = = , D y .i. D 0 , x y D D= = .x yD Dii. D=0 0 0 x y D D , .iii. 0 x y D = D = D = , ====0 0 0 . 00 x + y= x + y= . (x, y)=(0, 0) .i. D 0 .ii. D=0 .1. (): = + =2 25 53 12yx y. = = + = + =2 x 25 5 y 2 x 5 y253 x y 12 3 x y12 D2 -5D = = 2 3( 5) = 2 + 15 = 17 0 .3 1 : 7 9. - 8 Dx Dy25 -512 1 x D = = = + =25 12( 5) 25 60 852 253 12 y D = = = 24 75 51 x y D D85 51= = = 5, = = = 3.x y17 17D D2. (): + = + = ( 1) x 2 y2( 1) x y3( 2). -1 2D = = ( 1) 2( 1) = ( 1)( 2). -1 -2 2D = = ( 2) 6( 2) = ( 2)( 6)x 3(-2) -1 -2D = = ( 1) 3( 2) ( 1)( 2) = ( 2)[3( 1) ( 1)] = 2( 2)( 1)y .-1 3(-2) 1. D 0 ( 1)( 2) 0 1 0 - 2 0 1 2 . (..) ( 2)( 6) 6( 1)( 2) 1x DD= = = , 2( 2)( 1)2= = = ( 1)( 2)y DyD.2. D=0( 1)( 2) = 0 1 = 0 2 = 0 =1 = 2 .i. =1 (1 2)(1 6) ( 1)( 5) 5 0 x D = = = . : 8 10. - 9 ii. =2 Dx = 0 Dy = 0 . (.. y ), . . =2 :(2 1) 2 0 2 0 + = + = + = + = + =2 0x y x y(2 1) 2 3(2 2) 2 0x yx y x y. x= = = = 2 22y x y y= .( , ) ( , )2x y3. ():2 2 = = 2x yx y. 2 - 2 3 2 2D = = = + = ( ) ( 1) - 2 2 -x D = = = + = 2 2 ( ) 2 2 2 ( 1)2 -2 2 3 2D = = 2 2 = 2 ( 1) = 2 ( 1)( + 1)y . 2 1. D 0 2 ( 1) 0 2 0 -1 0 0 1. .. 2 ( 1) 2x Dx= = = 2 ( 1)D + +2 ( 1)( 1) 2( 1)y Dy= = = , 2 ( 1)D.2. D = 0 2 ( 1) = 0 2 = 0 1 = 0 = 0 = 1.i. =0 2 0(0 1) 0, 2 0(0 1)(0 1) 0 x y D = = D = + = . : = =0 x 0 y20 x 0 y0 : 9 11. - 10 ( 0=2).ii. =1 D 2 1(1 1) 0 Dx = = Dy = 2 1(11)(1+1) = 0 . D=0,0 x y D = D = . =1 : = =x y 2x y 2 x, y . . =2 = =22x yx yx y. x= y = 2 x y = 2 y = 2 (x, y)= (, -2) .4. (): + + = + =( 1) 0x yx y2 8 0 (0, 0). . +1D = = 8 2 ( + 1) = 2 (4 ( + 1)) = 2 (4 1) = 2 ( + 3) = 2 ( 3).2 80 +1D = = 8 0 0( + 1) = 0x ,0 8 0D = = 0 2 0 = 0y .2 0 1. D 0 2 ( 3) 02 0 3 0 0 3 , .. x y D D0 0= = = = = =0, y0 2 ( 3) 2 ( 3)D D.2. D = 0 2 ( 3) = 0 2 = 0 3 = 0 = 0 = 3 .i. =0 0 + y = 0 y=0 0 x + 8 y = 0 x (x, 0) x . : 10 12. - 11 ii. =3 3 + 4 = 0 3 + 4 = 0 3 + 4 =0 + = + = + = + =3 4 0x y x y x y6 8 0 2(3 4 ) 0 3 4 0x yx y x y x y. y= 44= = , 3x 4 3 .( ,)35. (): + = + = +( 1) 4y x( y ) x2 + = + = 4 ( 1) 4 y x yy x y () 2 2 + = + + =2 ( 1) 2. D.2 2D = = = = + .2-1 4( 1) 4( 1) ( 1)( 4) ( 1)( 2)( 2)-1 3 2x D = = = + +2 48 ( 2)( 2 4)2 -1 D = = 2( 1) ( 1) = ( 1)(2 ) = ( 1)( 2)y .-1 2 1. D 0 ( 1)( 2)( + 2) 0 1 0 2 0 +2 0 1 2 2 .. ( 2 )( 2 2 4) 2 2 4( 1 )( 2 )( 2 ) ( 1 )( 2)x DD + + + + = = = + +,( )( ) 1 2 1= = = ( )( )( ) + +1 2 2 2y DyD.2. D = 0 ( 1)( 2)( + 2) = 0 1 = 0 2 = 0 + 2 = 0 = 1 = 2 = 2 . : 11 13. - 12 i. =1 (1 2)(12 2 1 4) ( 1)(7) 7 0 x D = + + = = () .ii. =2 0, 0 x y D = D = y . =2 + 4 =2 + = + =4 2x y4 2x yx y. y= x = 2 4 y x = 2 4 (x, y)= (2- 4, ) .iii. =-2 ( )(( ) ( ) ) ( )( ) 2 2 2 2 2 2 4 4 4 4 4 16 0 x D = + + = + = .6. x, y + = =D D DD D D3x yx y , . D 0 :+ = =D D DD D D3 ( + )x yx y4DD = D D = D = D . 2 2 y y y D + D = D D = D D D = D. 2 4 22 x x x D 2DD D= x = = 2 y 1 xD Dy= = = . (x, y)= (2, -1).D D7. x, y 2 2 2 D 0 x y x y D + D = D D . x+y= 6 x, y. 2 2 2 x y x y D + D = D D (1) D 0 (1) 2 D :2 2 2 2 2 2 2 2 A D 0 + 2 2 2 x y x y x y x y x y x y D D D D D D D D D D D D= + = + = 2 2 2 2 D Dx y= y=D DD D D D D D D D D D( )x2 + y2 = 2xy x2 + y2 2xy = 0 x y 2 = 0 x y = 0 x = y . : 12 14. - 13 + = + = = =x y 6 x y 6x y x y 0 ( +)2 = 6 = 3 . 3 + y = 6 y = 3 (x, y)= (3, 3).8. x, y :2 2 2 4 2 5 x y x D + D + D = D + D . :i. ( ) ( ) 2 2 1 2 2 0 x y D + D + D =ii. x, y.i. ( ) ( ) 2 2 1 2 2 2 4 4 2 1 2 2 x y x x y D + D + D = D D + + D + D + D2 2 2 2 4 5 x y x = D + D + D D D + = 4 2 5 2 4 5 0 x D D D D + + = .ii. i. : ( ) ( ) 2 2 1 2 2 0 2 0 1 0 0 x y x y D + D + D = D = D = D = 2 1 0 x y D = D = D = . D = 2 0 x y D 1 D 0= = = = = .x y 0D 2 D 29. , 1 2 , .( ) 1 = 1 2:0y+ y= 2 + =3 1:2x yx y + = 1 :( ) -1 -D = = = +1 1 1( ) 2 -0 1 x D = = =2 0 2 : 13 15. - 14 -1 2Dy = = 0 2 = 2 1 D= 0 0 x y D D 0 -1+=0 x D = 2 0 -1+=0 (1). 2 :( ) 1 3D = = = +1 3 3-1 1 362 a x D = = ( ) 1 1D = = 2 1 1 = 2 + 1 = 3y D= 0 -1 2 x y D D 0 + 3= 0 y D = 3 0 +3= 0 (2). ( ) 4 + 1 = 0 3 + 3 1 = 0 3 4 = 0 3 = 4 = + = = = = = 33 0 3 3 3 310. ( ) 1 + = 1 1:+ = 1yx y( )( ) + + = 2+ 2 1:yy2 3 1= . . 1 :( ) +1 -D = = + = + + 1 1 11 1( )( ) 1 -D = = 1 1 = 1 x -1 1 : 14 16. - 15 ( ) +1 1Dy = = + 1 1 = 1 1 = 2 1 D= 01 -1 x y D = D = 0 + + 1=0, 1- = 0,- -2= 0.: + 1 = 0 = 1 = 1 2 = 0 = 2 = 2 2 + 1 + 1 = 0 . = 1, = -2 1 2 : + = + = + =3 y 5 x 3 y 5 1 x 3 y5( ) 2 + = + = = : +3 1 3 1 1 3 1x y x y x y0x + 0y = 40 = 4 . .1. ( ) 2 2=3 y2:8 2 3 217x y= . 2 = 2 = x , y : = = ( ) 3 2 8 24 8 16( +) = + = 8 3 17 3 24 9 51 = 35. 3 = 2 3 (35) = 2 3 + 35 = 2 3 = 33 = 11 . 2 = x 11 () 2 = y 35 (). () .2. ( )3 51 + = 2 +3 :y2 15 = +2 3x y : 15 17. - 16 . 1 1= = , +x 2 y 3. 2 y 3 . () :( ) 3 + 5 = 1 1 3 + 5 =1( +) = =2 5 5 10 5 2513 = 26 = 2 2 = 5 4 = 5 = 5 4 = 1 = 1 .1 1 1 = = = + =2 2 2 2, 52 2 2 = 2, 5 .1= + = = 1 3 1 43y yy+. () (x, y ) = (2,5 , -4 ) .3. ( ) =4 2 y11:6 5 15,5x y= . x = , y = : = = ( ) 4 2 11 3 12 6 33( +) = + = 6 5 15,5 2 12 10 311 = = . 4 224 2 = 11 :1 = = = = 3 = 3 = 34 2 11 4 12 321 1 1y = y = y = () 2 2 21 13, , 3,-2 2 ,1 13, , 3, .2 2 : 16 18. - 17 4. ( ) + =2 x 3 y5:3 x 5 y7 + = x = y = , y 0 : 2+3=5 3+5=7. + = = 2 3 5 3 6 9 15 ( ) ( +) + = + =3 5 7 2 6 10 14 = 1 =-1 : 2 + 3 = 5 2 + 3(1) = 5 2 3 = 5 2 = 8 = 4 . = 4 = 16 y = 1() . .+ = + = 2 x y2 2 y4 8 D, Dx, Dy :2 ( )( ) 2= = = + 4 2 22D( ) ( ) ( ) ( )( ) 2= = = = =4 8 2( )( ) 2( 2) 4 8 2 2 4 2 2 2 4D = 2 2 1 22 2( ) ( ) ( ) ( ) ( )( ) = = = = 2 4 8 2 2 4 2 2 2 8Dy 4 8 D = 0(2 )(2 + ) = 0 2 = 0 2 + = 0 = 2 = 2 = 2 = 2 : 17 19. - 18 1. 2 2 D 0 :2 2 1 2 2 2 1 2 2 (1 2 )( ) ( )( ) ( )( ) ( )( ) ( ) = = = = + + + 2 2 2 2 2x DxD( )( )( )( )( )( )( )( )D y 2 8 2 8 8 y = = = = + + +D 2 2 2 2 2 2. =2 2 + 2 =0 + = + =2 2 0y2 2 0x yx y . x= 2 + 2 y = 0 2 y = 2 y = (x, y) = (, -) . = + = 2 2 43. =-2 ( +) x yx y2 2 1 60x + 0 y = 20 0 = 20 . 1. : + = =2 2 y2 63x y( )2 2 2 6 2 2 2 6 3 2 2 2 6 2 9 6 2 2 6 x + y = x + y = y + + y = y + + y + y= x y = 3 x = y + 3 x = y + 3 x = y+3( ) ( )3 2 6 3 0 3 2 2 1 0 2 2 1 0 1 2 0 y + y + = y + y + = y + y + = y+ = x = y + 3 x = y + 3 x = y + 3 x = y+3 : 18 20. - 19 + = = = = + = + =1 0 1 1y y yx y x x3 1 3 2 ( x, y ) = (2, 1) .2. : + = =2 2 5x yxy2( ) ( ) ( ) 2 2 5 2 2 5 2 4 5 2 9 9 x + y = x + y xy = x + y = x + y = x + y= xy = 2 xy = 2 xy = 2 xy = 2 xy=2 + = + = + = = = =9 3 3x y x y yxy xy y2 2 2 3x + y= xy=2 x, y 2 3 + 2 = 0 =1 = 2 . : ( x = 1 y = 2) ( = 2 y=1) . 3x + y= xy=2 x, y 2 (3) + 2 = 0 2 + 3 + 2 = 0 = 1 = 2 . ( = 1 y = 2) ( x = 2 y = 1) . + = =2 2 5x yxy2 (1, 2), (2, 1), (-1, -2), (-2, -1).3. :2 2 2 ( 2) + = + x y xyx y6+ = : 19 21. - 20 2 2 2 4 ( )2 2 2 4 x + y = xy + x + y xy = xy+ x + y = 6 x + y=6 36 2 xy = 2 xy + 4 4 xy = 4 36 4 xy = 32 4 xy = 32 xy=8 + = + = + = + = + =6 6 6 6 6x y x y x y x y x y x, y 2 6 + 8 = 0 = 4 = 2 . ( = 4 y = 2) ( = 2 y = 4) .4. 2 = 2 5 0xy y yy x 2x= + 4 3( )( ) 2 xy y 2 5 y = 0 y 2 x y 5 = 0 y = 0 2 x y 5 = 0 y=0 y = x 2 x + y = x 2 x + y = x 2 x + y = x 2 x+14 3 4 3 4 3 4 3 = = +2 5 0x yy x x( ) 2 24 3 ( ) ( ) = = = = = + + = = = =0 0 , 1, 0y y o y yy x x x x x x y2 4 3 2 4 3 0 1 3 ( , ) ( 3, 0) 2 x y 5 = 0 y = 2 x 5 y = 2 x 5 y = 2 x5 y = x 2 4 x + 3 2 x 5 = x 2 4 x + 3 x 2 4 x + 3 2 x + 5 = 0 x 2 6 x+ 8 =0 = = = = = = =2 5 1 32 4 2 4y x y yx x ( , y ) = (2, 1) ( x, y) = (4,3) . (1, 0), (3, 0), (2, -1) (4, -3). : 20 22. - 21 5. : + = + =2 2 297x yx y(1)( )2 2 29 2 2 29 49 2 29 2 20 2 20 x + y = x + y xy = xy = xy = xy= x + y = 7 x + y = 7 x + y = 7 x + y = 7 x + y=7= + =x y 1 0x y 7 x, y, 2 7 +10 = 0 = 2 = 5. (x=2 y=5) (x=5 y=2). (x, y)=(2, 5) (x,y)=(5,2). . x+ y+ z= . ( )+ + = yyy1 1 1 1 + + = 2 2 2 2 + + =3 3 3 3:. :+ + = yy1 1 1 1+ = = 2 2 2 3 3 ( ) 0 0 0 , y , . . : : 21 23. - 22 i. .ii. y.iii. , y x. . , . , . 1. ( )+ =53 + = : 2 103 223 + = 3 6 2 + = + = + = + =( )55 3 6 2 6 10 12 2 9 603 2 + = + = 4 3 18 26 6 33 6 2 , .( ) + = + = + = + =5 12 12 12 12 60+12 2 9 60 1 12 2 9 6014 + 21 = 0 : 22 24. - 23 ( ) + = + = + = + =5 4 4 4 4 20+4 3 18 1 4 3 185 + 7 = 2 ( )5 + = + = 14 21 05 7 2 + = + = = + = + = 14 21 0 5 70 105 0 : ( +)5 7 2 14 70 98 287 = 28 = 4 14 + 21 = 0 14 + 21 4 = 0 14 = 84 = 6 . : + 6 4 = 5 = 3 . () : (3, 6,4).2. ( ) = 2 3x y zx y zx y z + = : 3 8 14 92 3 5 7 + + =( ) + = = = 2 3 0 2 3 2 3x y z x y z x y zx y z y z y z y z y zx y z y z y z y z y z + + = + + = + + = : 3 8 9 14 3(2 3 ) 8 9 14 6 9 8 9 142 3 5 7 2 2 3 3 5 7 4 6 3 5 7 + + = ( )+ + = + + = = = = = = = = = = = = = 2 3 2 3 2 1 3 2x y z x y z x z xy y y yy z z z z14 14 1 1 17 7 7 1 7 0 03. ( ) 5 + = yyy = : 2 11 + =3 7 3 : 23 25. - 24 x .( ) + = + = = =5 2 2 2 2 10+y yy y2 11 1 2 11y 3 = 1( ) + = + = + = + =5 1 5+y yy y3 7 3 1 3 7 32y + 6 = 2 :( )5 + = yyy = : 3 1 + = 2 6 2 = + = 3 12 6 2yy :( ) 3 = 1 2 2 6 =2 + = + = +y y 2 y 6 2 1 2 y6 20 y + 0 = 0 0=0 y, . (y, )= (3+1, ) . y : y + = 5 = 5 + y = 5 + 3 +1 = 6 + 2 . () (x, y, )= (2+6, 3+1, ) .4. ( ) + = 5 y3 4: 3 y23 2 y2 2 + = + = x , . : 24 26. - 25 5 y + 3 = 4 1 5 y+ 3 =4 + y + = + y = ( ) ( )3 2 5 5 15 5 1014 y 2 = 6 7 y = 3 5 y + 3 = 4 3 15 3 y+ 9 =12 + = + = ( ) ( +) 3 x 2 y 2 2 5 15 10 y10 107 y = 2 :( ) + = 5 x y3 4: 7 3 = yy =7 2 = =7 y37 y2( ) = = = = 7 3 1 7 3+y yy y7 2 1 7 20 y 0 = 5 0 = 5 , () .5. ( ) + + = 4 3 y17 0: 5 4 y22 04 2 y19 0 + + = + + = . . x :( )( ) + + = = + + = + + =4 3 17 0 5 20 15 85 0+x y y 5 4 y 22 0 4 20 x 16 y88 0y + 3 = 0( ) + + = = + + = + + =4 3 17 0 1 4 3 17 0+y yy x y4 2 19 0 1 4 2 19 0y + 2 = 0 : 25 27. - 26 () :( ) + + = 4 3 17 0 + = 3 02 0yyy + =+ = + =3 0yy ( +)2 05 = 0 = 0 () 4 + 3 y + 17 = 0 4 + 3 y+ 17 = 0 4 + 3 0 + 17 0 =0 y + 3 = 0 y = 0 y= 0 = 0 = 0 = 0 = = = = = = 4 0 00 00 0y y () (x, y, )= (0, 0, 0).6. ( ) + = 5 5 0: 10 5 2 0yyy + + = + =5 15 9 0 . . x , :( ) + = + = + + = + + =5 5 0 2 10x 10y 2 0+x yx y10 5 2 0 1 10x 5y 2 05y + 4 = 0( ) + = + = + = + =5 5 0 1 5 5 0+x y y 5 15 y 9 0 1 5 15 y9 010y 8 = 05y 4 = 0 . () : : 26 28. - 27 ( )+ = 5 x 5 y0 = 5 y4 05 y4 0 = 5 4 =0 = =5 4 0y5 4 0yy ( ) 4y , = ,5 . y, () :4 3 + = + = = = () 5 5 0 5 3 0 5 3 .5 53 4 , , 5 5 .x y = = 7. ( ) 2 3 4 + =2 x y2 18 =x y= = = = = = x, y, 2 3 4 .2 3 4y :( ) ( ) 18 + = + = = = =2 2 3 2 4 18 4 3 8 18 9 18 29 x = 2 2 = 4, y = 3 2 = 6, =4 2=8 () (x, y, )= (4, 6, 8).8. i) 1 14 + = x y1 19+ = y1 13+ = xii) = 2 x 3 y196 x 2 y6 = i) . : : 27 29. - 28 1 1 1= = = , , , 0, y 0, 0.y 49 +3 + = ( ) ( ) + = + = :2 + 2 + 2 =16 + + = 8 ( + + ) ( + ) = 8 4 + + = 4 = 4 1 1= =441( + + ) ( + ) = 8 9 + + = 1 = 1 = = 1 1( + + ) ( + ) = 8 3 + + = 5 = 5 1 1= =55yy () ( ) 1 1x , y , = 1, ,5 4 .ii) x = y = , y 0 : 2+3=5 3+5=7. = + = 2 3 19 3 6 9 57 ( ) ( +) = =6 2 6 1 6 2 6 = 9 =9 : 2 3 = 19 2 3(9) = 19 2 27 = 19 2 = 8 = 4 . x = 4 x = 16 y = 9 x = 81 : 28 30. - 29 - .1) x3 y = 4 . 2) ( 2 9)x + ( 3) y = 8 . x3) ( x, ), 0 x + y = . 4) 0x y = 0 (,0), . 5) (x 2)2 x2 + 2y = 0 . 6) x = 5 (5, ), 7 57) ( , ),3 7 3x = 5y . 8) y = 2 . 9) y + x = (0, ) 10) + = + =2 x y12 x y2 . 11) + = + =ax yx y z 1 2 , 1 2 : ax + y = , : x + y = z . 12) 1 2 : y x = 2 , : y + 4x = 0 . 1. (1, 1) , (4 ,-10) , (3,7),2+2 2 3( , 2 1)3aa+ 3x y = 22. :i) = + =2 03x yx yii) = =2 x y12 x y33. :i)+ = + = 3 x y102 x 3 y25ii)3 + = = 2 1x y2y x6 2 8 : 29 31. - 30 4. :i) = + =4 x 3 y47 x 3 y26ii) 3 x + 1 4 y 10 = 18 2 7 2 + 6 =0x y x2 5iii)+ + = + + = (6 x 1)( y 2) (2 x 1)(3 y1) 33 x (4 y 1) (2 x 4)(6 y1) 28iv)3 24 = x y4 37 = x yv) = 2 x 3 y196 x 2 y6 = 5. , , ( 2 1) x + (2 5) y + 1 = 0 .6. , (5 3)x + 2 4 + 6 = 0 1974 . + + + = ( 1) x ( 2 ) y52 4 1 47. ( )x + y= + .8. + = + =2 21x yx y, , x y= x y=4 3, .9. , , :( 3) ( 1) 1 x + y= x + (2 + 1) y=7 (x,y)=(2,-3).10. :2 + = 5 + = ( 1)x y( 1) 7Sx a y( 1) 2= =( 2) a x y 2( 2) 2Sx y : 30 32. - 31 S1 .i) S2 .ii) S2.11. 2 18 3 2 6 90 x y y x D + D + D = D + D .12.(1): 5x-y=17(2): y-2x=1(2): 3x+2y=5 i) = =5 17x yy x2 1ii) = + =5 x y173 x 2 y5iii) = + =2 1y xx y3 2 5 : 31 33. - 32 13. : 1 : x y = 1 2 : x y = 1 1 2 , .14. x, y ( ) ( ) ( )2 2 2 4 2 4 0 x y D + D + D + = . .15. :i)3 21 0ii)4 24 3iii)1 52 62-4316. :i) ii) iii) + +17. 2 2 2 22 4x y y y+y 2 x 2 x 2 y22 222021yx= ,y.18. :(5x 6) + 2 (x 1) = 3x 4 , , ) =) , . : 32 34. - 33 19. :i)+ + = + = =2 113 2 21x y zy zzii)+ = 2 x y 3 z03 x 2 y 4 z13 x y z5+ + = = iii)345+ = x yx+ = + y=( : )20. , , x y D D D x y :228 + =D DxD DD D+ = x y + = y ( x,y) .21. :2 x + y= 5 x xy y= i) 2 22 1ii)+ + = = 3 1 02x yxyiii) + = =2 2 10x yxy3iv) = 2 3 1y xy x+ = 2 3-22. 3 1050 1 4 . 1 4 ; : 33 35. - 34 23. + 6 2, , 7 3+ , 2 2 : + 2 3 5 1, 2 ,+ , 3 .24. 12. , 18. . : 34