HỆ PHƯƠNG TRÌNH HÀM SỐ - LIÊN HỢP - ĐẶT ẨN PHỤ_1

Header Page of 16 Tt c vỡ hc sinh thõn yờu Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page of 16 Header Page of 16 Tt c vỡ hc sinh thõn yờu GII HPT PHNG PHP HM S x10 x y x y Bi 1: Gii h phng trỡnh ( x, y ) x y Bi gii: iu kin: y y - - Xột x=0, t pt u suy y=0, thay x=y=0 vo pt th hai khụng tha (loi) ổ yử ổ yử - Xột x , chia v ca pt u cho x , ta c x x ỗ ữ ỗ ữ (1) ố xứ ố xứ ' Xột hm s f t t 2t , t Ta cú f t 5t 0, t 5 Vy hm s f t t 2t ng bin trờn Do ú (1) x ca h ta c: Xột hm s g ( y ) Ta cú g ' ( y ) y y x Thay vo pt th x y y (2) y y 1, y - 1 0, y - Vy g(y) ng bin trờn khong 2 y5 y ổ ỗ - ; ữ ố ứ M g(4)=6 nờn (2) y Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page of 16 Header Page of 16 Tt c vỡ hc sinh thõn yờu x x -2 Suy y x hoc y y xy ( x 1) x y x - y Bi 2: Gii h phng trỡnh y x y x x2 Bi gii: y x Bin i PT (1) x - y x - y y x x = y th vo PT (2) ta c: 3x x x x x x2 2x (-3x) (-3 x) f x f -3 x Xột f (t ) t t cú f '(t ) 0, t f l hm s ng bin nờn: x - x x 1 y5 y x th vo (2) 3( x 1) x x x x2 Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page of 16 Header Page of 16 Tt c vỡ hc sinh thõn yờu V trỏi luụn dng, PT vụ nghim ổ ố Vy h cú nghim nht: ỗ - ; - 1ử ữ 5ứ x3 y 3( x y ) y ( y - 2) 14 Bi 3: Gii h phng trỡnh sau 27 x 27 x 20 x y x - Bi gii: Phng trỡnh (1) x x - y y - 15 y 14 x x - y 32 - y Xột hm s: f (t ) t 3t liờn tc trờn R Ta cú f ' (t ) 3t vi t R hm s ng bin trờn R pt : f ( x ) f ( - y ) x - y y - x Th y = 2-x vo phng trỡnh (2) ta c 27 x x 20 x 43 x x 4(3 x 1) x 43 x Xột hm s: g (t ) t 4t liờn tc trờn R Ta cú g ' (t ) 3t hm s ng bin trờn R Suy ra: g (3 x 1) g (3 x 1) x x 27 x 27 x x x Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page of 16 Header Page of 16 Tt c vỡ hc sinh thõn yờu x y 27 x 27 x x 27 x 27 x 0(vn) Vy h phng trỡnh cú nghim (x;y)=(0;2) x ( x 1)( y - 2) x y y - Bi 4: Gii h phng trỡnh ( x - 8)( y 1) x, y ( y - 2) x - x - 4x Bi gii: iu kin: Xột phng trỡnh: t T phng trỡnh ta c phng trỡnh: ta cú thay vo phng trỡnh ta Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page of 16 Header Page of 16 Tt c vỡ hc sinh thõn yờu c Tip tc gii phng trỡnh Xột hm s Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page of 16 Header Page of 16 Tt c vỡ hc sinh thõn yờu Do ú hm s ng bin trờn T Gii phng trỡnh +) Vi +) Vi Vy h phng trỡnh ó cho cú nghim l: x x y y Bi : Gii h phng trỡnh 12 y - 10 y x3 ( x; y ) Bi gii: Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page of 16 Header Page of 16 Tt c vỡ hc sinh thõn yờu Ta cú: (1) x x (-2 y ) (-2 y ) (*) Xột hm s c trng f (t ) t t f '(t ) t t2 t t2 t2 t t t2 Suy f(t) l hm s ng bin trờn R T (*) suy ra: f ( x ) f ( -2 y ) x -2 y Thay vo phng trỡnh (2) ta c: 3x x x3 x x x x3 (**) Xột hm s g (t ) t 2t ta thy g(t) ng bin trờn R nờn t (**) suy x Vy h cú hai nghim l (-1; ); (0;0) x x3 x -1 y y y x Bi 6: Gii h phng trỡnh: x x2 - 2x 2 x - y ( x, y ) Bi gii: k: x - y Ta cú: x - y x - x - y2 y y2 y th vo PT (2) ta c Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page of 16 Header Page of 16 Tt c vỡ hc sinh thõn yờu x -1 ổ x -1 ỗ ữ y y (*) (vỡ ố ứ y2 y y y ) Xột hm s f t t t trờn t f 't t2 t t 0, t , t t t t 0, t t ổ x -1 f t ng bin trờn , theo (*) ta cú f ỗ ữ f y ố ứ x y Vi x y thay vo (1) ta cú: y2 y y2 y y2 - y y x ổ5 3ử Vy h cú nghim x; y ỗ ; ữ ố2 4ứ x - y Bi 7: Gii h phng trỡnh y x y x xy x - 11 12 - x y - x Bi gii: iu kin x ,y0 Ta cú x - y 4( x - 2) y 4x - y Du = xy y=4x8 Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page of 16 Header Page 10 of 16 Tt c vỡ hc sinh thõn yờu y x y x 4x y Du = xy y=4x8 y x y x Du = xy y=4x8 Suy x - y Nh vy, pt(1) y = 4x Th vo pt(2) ta cú: x - x - 11 3x - 3x x - x - 3x - x - x - x - - 3x - x x - x - ổ 7ử ỗ x 2; ữ 3x x - 3x x - 3ứ ố 1 x - x - 4 3x x - 3x x - x - x - - x2 - x - () 1 (3) 3x x - 3x x - + pt () x - x - x i chiu iu kin ta cú x 13 - 13 x 2 13 ổ 13 ; 13 - ữ ố ứ H cú nghim ỗ + Xột pt(3) Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 10 of 16 10 Header Page 112 of 16 Tt c vỡ hc sinh thõn yờu a - b3 a a x y b x - y a b b Cõu 26 : Gii HPT : t ab x y a - b x y 72 xy 2 x - y 29 x - y H tr thnh : ab a -b 8a b ( a 2) - b - a 2 18( a - b ) 9b3 - 112b - 128 2 29 ab 18(a - b ) 29 ab b a b Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 112 of 16 112 Header Page 113 of 16 Tt c vỡ hc sinh thõn yờu b -4 b a x ; y -8 b Cõu 26 : Gii HPT : D thy x y y xy 2 x y y xy y khụng phi nghim ca h Khi ú h tng ng : ổ 7x 3x ỗ - x ữ - x - y y2 y ố y ứ 2 ổ2 ổ x ổ - x ổ x ỗy ữ ỗ y ữ 3ỗ y - x ữ ỗố y ữứ ố ứ ố ứ ố ứ t y - x a a - 3b (a b)(a - b - 3) b 3a x b y Vi a -1 b a b a 3a a -2 b Vi a - b a - 3a 11 (Loi) Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 113 of 16 113 Header Page 114 of 16 Tt c vỡ hc sinh thõn yờu Cõu 27 : Gii HPT : D thy x xy y y - 2 x x y x - x x khụng phi nghim ca h x.(1) - (2) x xy - 3xy Ly (3) (3) - (1) ta c : xy - y - xy y Xột thy y x khụng phi nghim ca h 4x - y x -1 Vi y0 Vi y thay vo khụng tha h 4x - x -1 thay vo (2) ta c : -1 5- y x 2 ( x x - 1) - -1 5 y x 2 Cõu 28 : Gii HPT : ( x y 3) x - y y 2 ( x - y )( x 4) y PT (1) ( x - y 1)( x y - x - y ) x - y x y Th x- y -x- y vo PT(2) ta c : Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 114 of 16 114 Header Page 115 of 16 Tt c vỡ hc sinh thõn yờu ( x - y )( x x - y - x - y ) y x ( x - y - 1) ( x - y )3 - ( x - y - 1)( x Ta c h mi : x - y x -1 x y x - y - y -2 Cõu 29 : Gii HPT : Th x - y x - y )0 x - y x xy y x xy y 1( x x y 2) x y -2 xy vo PT(2) ta c : -2 x y - x - y x y y x y - ( x y - 1) ( x y - y 1) x y x y y x ( y 1) x y y x2 y - x2 x y y x y xy -1 Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 115 of 16 115 Header Page 116 of 16 Tt c vỡ hc sinh thõn yờu D thy x khụng phi nghim ca h y -1 x2 - x x x -1 y -2 x y -1 y x 1- -2 ; ) i chiu iu kin ( x; y ) (-1;1);( Cõu 30 : Gii HPT : (2 x - 1) x y (6 - x - y ) - x 2 3x - xy - y -72 Xột thy x - x x y y -2 khụng tha PT(2) x y, - x PT(1) - ( x y ) - (8 - x) x y - 4x Xột hm s M f (t ) - t2 -6 f '(t ) - suy hm s nghch bin t t f ( x y ) f ( - x ) x y - 4x y - 5x Th vo PT(2) gii PT bc c bn Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 116 of 16 116 Header Page 117 of 16 Tt c vỡ hc sinh thõn yờu Cõu 31 : Gii HPT : Ta cú x2 y x 3 2 2( x y ) x 3( x y ) x.(1) - (2) y - xy x 3( x y ) - x y (2 y - x - 3) ( x - 1)( x - 5) M y - x - x (1 - x)(3 x) ( x - 1)( x 3)(2 y - x - 3) ( x - 1)(5 - x) Vi x y Vi ( x 3)(2 y - x - 3) (5 - x) y 5- x x2 x x 2( x 3) x3 Th vo PT(1) ta c : ổ x2 x x 2x ỗ ữ x ố ứ 2 ( x x)( x 3) ( x x 7) 3( x 3) 2 ( x 1) ( x x 11) x -1 y Cõu 32 : Gii HPT : Ly x xy x y ( x 3) x x y y (1) - (2) x - y x y y Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 117 of 16 117 Header Page 118 of 16 Tt c vỡ hc sinh thõn yờu ( x - y )( x y ) y x2 Th vo PT(1) ta c : x3 x 3x x -1 y 2 ( x y )(1 2 ) x y Cõu 33 : Gii HPT : ( xy - 1) x - y 2 ( x ) ( y - ) x y ( x 1)( y - 1) xy 2 ( x ) ( y ) x y ( x )( y - ) x y Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 118 of 16 118 Header Page 119 of 16 Tt c vỡ hc sinh thõn yờu t b a b -2 x x a a b a -1 y - b ab b y a b -1 a -2 Xột tng TH gii x, y n gin ( x 1) ( y 1) xy Cõu 34: Gii HPT : 2 ( x 1)( y 1) 10 xy 1 ( x x )( y y ) -9 ( x )( y ) -10 x y t b x a (a 2)(b 2) -9 a b -3 a -5 x b -5 ab 10 ab 10 y b y a Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 119 of 16 119 Header Page 120 of 16 Tt c vỡ hc sinh thõn yờu Do vai trũ ca x, y Cõu 35 : Gii HPT : Ta thy Xột l nh nờn ta xột TH ca cp a, b ri hoỏn i li ( x y ) x y y xy y 2 2 x x ( x y ) y xy - x x tha l mt cp nghim ca h y x2 y2 Dựng nh thc ta cú : D x2 y2 2 Dx y ( x y ) D - x( x y ) y Dx x 2y D x -2 x x y (Loi) y Dy - x D Kt hp ( x; y ) (0;0) Cõu 36 : Gii HPT : Xột h mi : l nghim nht ( x y ) xy xy - y ( x y ) xy xy x - y - 2.(1) - (2) 3.(1) - 2.(2) Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 120 of 16 120 Header Page 121 of 16 Tt c vỡ hc sinh thõn yờu x - y - (2 xy - 5) - x xy ( x - y ) xy y - x 15 x - y - (2 xy - 5) - x xy ( x - y - 3)( xy 5) 3( x - xy 5) (2 xy - x xy 5)( xy 5) 3( x - xy 5) ( x - xy 5)(2 xy xy 3) x xy xy xy Cú xy xy x2 - x xy xy x2 Th vo PT(1) ta c x2 - 3( x - 5) 2( x - 5) 2x x - 3x - 13 x 15 x -3( L) x y x y -2 Cõu 37 : Gii HPT : y x xy - y 2 xy x y x - y Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 121 of 16 121 Header Page 122 of 16 Tt c vỡ hc sinh thõn yờu Ly (1) - (2) xy ( y x - 1) (3 y - 1)2 Li cú xy ( x y - 1) 2(3 y - 1) xy,( y x - 1) l nghim ca PT X - 2(3 y - 1) X (3 y - 1)2 X y - xy y - x y -1 3y -1 Thy Th y0 x khụng phi nghim ca h 3y -1 vo PT(2) ta c : y 3y -1 y y y - y y - y -1 x y Cõu 38 : Gii HPT : 3x - y 3x y 3x y x2 - y 3x - y x2 - y t 2 2a(b 1) b(a 1) 8ab x y a 2b a a b 2a b 8ab 2 x - y b 2a b ab 2b a ab 2b(a 1) a(b 1) 7ab ( ab ) 2a(b 1) b( a 1) 2a(b 1) 3b(a 1) 2 2b( a 1) a(b 1) Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 122 of 16 122 Header Page 123 of 16 Tt c vỡ hc sinh thõn yờu M cú 3a(b2 1) 3b(a 1) 15ab b 5a (b 1) 15ab b - 3b 3- b b2 Vi b a 2a 2a a 3b x y x - y Vi b 3- b2 a 2a 2a a 3b x y 3- x - y Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 123 of 16 123 Header Page 124 of 16 Tt c vỡ hc sinh thõn yờu GII BT PHNG TRèNH , PHNG TRèNH NGHIM KẫP NH GI PHN SAU BNG CASIO THY QUANG BABY Bi : x - x - x x - 2( x - 2) x - Bi : x x - 2( x - 2) x - x - x 24 x - 29 Bi : x x - ( x - 1) x - 32 x 114 x 99(1) BI : x x x 11 x , iu kin : x - Bc , dũ nghim ta c nghim : x = , x = -1 x x x 11 x 2( x - x - 2) - x - x - x - x 11 1 ( x - x - 2) x x x 11x Bc , ta chng minh biu thc ngoc luụn dng x 5x Ta t : f ( x) - - x 11x Chng minh f ( x ) , bng vic s dng casio , chc nng Table nh sau : Nhp mode , , g ( x) biu thc x 5x , start -6/5 , end , step 0,2 thy g(x) ln hn 1,25 , vy ta tỏch , cũn li ta cú x 5x x 11x Vic cũn li cỏc em bin i tng ng thụi Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 124 of 16 124 Header Page 125 of 16 Tt c vỡ hc sinh thõn yờu x 5x Bc : Kt lun : ( x - x - 2) - - x -x-20 x 11x Nghim : -1.25 x -1 hoc x Video hng dn : https://www.youtube.com/watch?v=fGcVf77I-9g x -2.5 x -1 Gii bi : iu kin : CHNG TA GII MT BPT CNG GING NH GII MT PHNG TRèNH , GM CC BC SAU : Bc : Casio ta tỡm c nghim kộp , cỏc em bm Shift + Cal thỡ s thy Eror , nhng thc t khụng phi l vụ nghim , cỏc e th nht : Mode , , f(x) = VT VP , Start -2,5 , end -1 , step 0,2 Cỏc em s thy x = -2 thỡ f(x) = v khụng i du , vy ta s cú f(x) = cú nghim kộp BC 1: Dề NGHIM : Dựng casio phỏt hin nghim kộp (qua chc nng Table ) : x = -2 Tip n chỳng ta to lien hp cho cỏc cn : x ax b : s dng iu kin ng tip xỳc : f ( x) g ( x) -2a b xột ti im cú x = - ta cú h : f '( x) g '( x) a Vy ta cú c liờn hp : x - ( x 3) , lm hon ton tng t ta s cú : x - (2 x 1) (1) x( x - x - 3) - ( x - 1)( 3x - x 1) 25( x x 4) 5x x -1 ( x x 4) 25 ( x 2)2 f ( x) 3x - - x - 2x x 5x 5x f ( x) 25 2x x 3x - - x - BC : X Lí BIU THC TRONG NGOC : Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 125 of 16 125 Header Page 126 of 16 Tt c vỡ hc sinh thõn yờu Xột : f ( x) 5x x -1 25 , ta chng minh nú luụn dng vi mi x thuc 2x x 3x - - x - xỏc nh +)u tiờn ta khng nh rng : x thỡ f ( x ) , ch ny em c dung table , mode , star , end 100 , step 10 xem , s thy -2.5 x -1 ,dung +)Xột f ( x) chc nng table ta thy 5x x -1 25 (bm mode , nhp f(x) , start -2,5 , end -2 , step 2x x 3x - - x - 0,2) Chn riờng hm g ( x ) - x -1 (bm mode , nhp f(x) , start -2,5 , end -2 , step 0,2) thỡ ta 3x - - x - thy g ( x ) -25 , vy ta s cú 5x 25 , cỏi ny cỏc em d chng minh bng bin i 2x x tng ng Vy trờn xỏc nh thỡ f ( x ) , vy (1) ( x 2) x -2 ỏp sụ : x = -2 Facebook cỏ nhõn : https://www.facebook.com/quang.manngoc CC EM HC TON KHễNG THY TIN B , THY QUANG S GIP CC EM THAY I Footer Page 126 of 16 126 ... x - x x3 x - x - ( x 2)(3 - x) - x 3- x ( x 2)(3 - x) 2 (- x x 2) x - x - x3 x - x - x 3- x ( x - x - 2) x ( x 2)(3 - x) x 3- x ( x 2)(3 - x) - x 3- x ... - y Nh vy, pt(1) y = 4x Th vo pt(2) ta cú: x - x - 11 3x - 3x x - x - 3x - x - x - x - - 3x - x x - x - ổ 7ử ỗ x 2; ữ 3x x - 3x x - 3ứ ố 1 x - x - 4 3x x -. .. x - - 2 x - x - 11x 11 x - 11x 11 ** 2 4 x - x - 11x 11 8x - x4 121x2 121 - 44 x3 44 x - 242 x x - 44 x3 165 x - 250 x 125 x - x - 40 x 125 x - 125 x - x - x -