bai-giang-toan-hoc-298.pdf

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Bi ging s 2: GII H PHNG TRNH BC NHT BNG PHNG PHP TH

A. KIN THC TRNG TM

Phng php th: H phng trnh ax by ca x b y c

Ta c: ax by ca x b y c

a cy xb b

a x b y c

a cy xb b

a ca x b x cb b

T ta tm c nghim ca h phng trnh. Ch : Phng trnh 1ax b +) Nu 0a th phng trnh 1 c dng 0x b .

- Khi 0b th phng trnh 1 c dng 0 0x phng trnh c v s nghim. - Khi 0b phng trnh 1 v nghim.

+) Nu 0a th phng trnh 1 c nghim duy nht bxa

.

B. CC V D MU V d 1: Gii cc h phng trnh:

a) 2 74 3 4

x yx y

b) 3 3 8

42

a ba b

Gii

a) 2 74 3 4

x yx y

7 24 3 7 2 4y xx x

7 210 25y x

x

7 22,5

y xx

2,52

xy

Vy nghim ca h phng trnh l: 2,5;2 .

b) 3 3 8

42

a ba b

42

3 3 4 82

ab

aa

42

3 202

ab

a

42

403

ab

a

403323

a

b

Vy nghim ca h phng trnh l: 40 32;3 3

.

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V d 2: Cho h phng trnh ( 1) 4 2 (1)

( 1) 2 (2)m x my

x m y m

a) Gii h phng trnh vi 2m . b) Gii v bin lun h phng trnh. c) Tm m h c nghim duy nht tha mn 0, 0.x y d) Tm m h phng trnh c nghim , .x y

Gii:

a) Thay 2m vo h phng trnh ta c:

2 1 4 2 2

2 1 2 2

x y

x y

2 1 2 2 2 1 4 2 2

2 2 2 1

y y

x y

3 6 2 2 2

2 2 2 1

y

x y

2 23 6 2

2 2 2 1

y

x y

2 23 6 218 14 2

3 6 2

y

x

Vy nghim ca h phng trnh l: 18 14 2 2 2;3 6 2 3 6 2

.

b) 2 2 1x m m y . Th vo 1 ta c:

2 21 2 1 4 2 1 3m m m y my m y m m Vi 1m th 3 c dng 0. 0y ng y . H phng trnh c v s nghim tho mn 2 1.x y

Vi 1m , h c nghim duy nht

2

2

2 1 2 1 2 2 111

111

m m mx

mm

m m mymm

c)

12 2 10 11 1

1 20

10

01

mmx

m m mm

mmmymm

d) 1m , h phng trnh c nghim 2 1 2 1x y x y Vy h phng trnh c nghim nguyn 2 1,x y y

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4 2 241 1

1: 111

1 1

mxm m

m mmy

m m

(1),(2)

,x y nguyn 1 1 21 1 0

m mm m

Vy h phng trnh c nghim nguyn 2, 1,0m .

V d 3: Cho h phng trnh ( 1) 2 1 (1)3 1 (2)m x yx my

a) Chng t vi mi gi tr ca m h phng trnh lun c nghim. b) Tm m nghim ca h tho mn x y ln nht. c) Tm m h phng trnh c nghim 0, 0.x y

Gii:

a) 1 1

12

m xy

. Th vo 2 : 21 13 . 1 (3) 6 22

m xx m m m x m

V 2

2 1 236 02 4

m m m m

nn h phng trnh lun c nghim duy nht: 2

2

26

46

mxm m

mym m

.

b) 26 6 24

236 234

x ym m

nn x y nh nht 2423

khi 12

m

c) 2

2

2 06

2 44 0

6

mxm m

mmy

m m

(V 2 6 0m m m )

V d 4: Cho h phng trnh: 2

2 1mx y

x y

. Gii v bin lun h theo m .

Gii: 2

2 1mx y

x y

2 3 1

2 1 2

m x

x y

+) Xt phng trnh 1 : 2 3m x Nu 2 0 2m m th phng trnh 1 c dng 0. 3x (v nghim). Do h v nghim.

Nu 2 0 2m m th phng trnh 1 c nghim duy nht 32

xm

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+) Thay 32

xm

vo phng trnh 2 ta c: 2 1y x 6 12 m

42

mm

Vy vi 2m th h c nghim duy nht

3242

xmmym

.

V d 5: Cho h phng trnh ( 1) 2 2 (1)2 ( 1) 1 (2)m x mymx m y m

a) Gii v bin lun h phng trnh. b) Trong trng hp h c nghim duy nht, tm h thc gia ,x y khng ph thuc .m c) Tm cc gi tr m sao cho , .x y

Gii:

a) Nu 0m h phng trnh: 2 21 1

x xy y

Nu 2 1

0 : (1)2m x

m ym

. Th vo 2 23 1 1 2 1 3m m x m

Vi 13

m th 3 tr thnh: 160.9

x (V nghim) H v nghim.

Vi 1m th h c nghim 1x y

Vi 131

m

m

th h c nghim duy nht:

2 13 1

13 1

mx

mm

ym

b) Vi 131

m

m

th

2 43 3 3 1 1

1 4 43 3 3 1

xm

x yy

m

khng ph thuc vo .m

c)

43 23 1

43 13 1

xm

ym

Nu , 3 , 3 3 1x y x y m (4). C 3 gi tr m tho mn:

1 1, 0

0 2, 11 0, 1

m x ym x ym x y

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V d 6: Cho h phng trnh:

2 2 5 1

1 2

m x y

mx y

. Tm m h c nghim nguyn.

Gii: T 2 ta c: 1y mx . Thay vo 1 ta c: 2 2 1 5m x mx

3 2 7mx x

23 2 73

m x m

73 2

xm

Thay vo 1y mx 7 . 13 2

y mm

4 23 2

mym

.

x 73 2m

3 2m 7 7; 7;1; 1

+) 3 2 7 3m m

+) 53 2 73

m m (loi)

+) 13 2 13

m m (loi)

+) 3 2 1 1m m

Thay 3m vo 4 23 2

mym

2y (tha mn).

Thay 1m vo 4 23 2

mym

6y (tha mn).

Vy vi 3m hoc 1m th h c nghim nguyn.

V d 7: Cho h phng trnh:

2

2

1

2 2 2 2

mx y m

x my m m

a) Chng minh rng h phng trnh lun c nghim duy nht vi mi m .

b) Tm m biu thc 2 3 4x y nhn gi tr nh nht. Tm gi tr .

Gii:

a) Do 2 0m m 2 2 0m m hay 2 2 0m m .

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Vy h phng trnh lun c nghim duy nht vi mi m .

b) Rt y t 1 ta c: 2 3y mx m . Th vo 2 ta c: 2 22 2 2x m mx m m m

2 3 22 2 2x m x m m m 2 3 22 2 2x m x m m m 2 3 22 2 2x m m m m

2 2 22 1 2 2 0x m m m do m 1x m

Thay vo 23 1y m m m m . Thay 1x m v y m vo

2 5 5min 3 4 4 2x y khi m ta c:

2 21 3 4 5 5m m m m 2 25 5 5 5 0

2 4 4 2m do m

Vy 2 5 5min 3 4 4 2x y khi m

V d 8: Cho h phng trnh: 2 3 5

3 4mx yx my

a) Chng minh rng h lun c nghim duy nht. b) Tm h thc lin h gia ,x y khng ph thuc m .

Gii: a) h c nghim duy nht ta xt hiu: 22 .3 3. 1 6 3 0m m m m Vy 26 3 0m m . Hay h lun c nghim duy nht.

b) Rt m t 1 ta c: 5 32

ymx

thay vo 2 ta c: 5 33. 4

2yx

x

2 22 8 15 9 0x x y y

y chnh l h thc lin h gia ,x y khng ph thuc m . C. BI TP T LUYN Bi 1: Gii cc h phng trnh bng phng php th:

1. 3 2 42 5

x yx y

S: 2;1

2. 4 2 36 3 5

x yx y

S: V nghim

3. 2 3 54 6 10

x yx y

S: V s nghim

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4. 3 4 2 05 2 14

x yx y

S: 2;2

5. 2 5 33 2 14

x yx y

S: 4; 1

6. 4 6 910 15 18

x yx y

S: V nghim

Bi 2: Cho h phng trnh: 2

2

13 4

x ay a aax y a a

. Tm a h phng trnh c nghim 0, 0x y .

S: 1 0a

Bi 3: Cho h phng trnh: 2 5

3x ymx y

. Tm m h c nghim 0, 0x y .

S: m

Bi 4: Cho h phng trnh:

2

2

3 3 6

1 2 13

mx n y

m x ny

a) Gii h phng trnh vi 2; 1m n . S: 5; 6

b) Gii h phng trnh vi 1; 3m n . S: 28 3;5 10

Bi 5: Cho h phng trnh: 2nx y n

nx ny n

. Gii v bin lun h theo n .

S: +) 1n : V nghim

+) 1n : Nghim l 1 2 ;1 1

n nn n

Bi 6: Gii cc h phng trnh sau

a) 12 10

14 43

x y

x y

S: V nghim

b) 2 6

1 2 3 22

x y

x y

S: V s nghim

Bi 7: Gii cc h phng trnh sau:

a) 3 ( 3 1) 5

( 3 1) 2

x y

x y

S: 8 3 3;5 3 3

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b) 2

2

( 1)( 1) 1

a x a y aax a y

( a l tham s) S: ;1a

Bi 8: Xc nh m v n bn ng thng sau y ng qui: 1 : 2 6d x y , 2 : 2 3d x y ,

3 : 1d mx ny , 4 : 13d nx my . S: 61 47;9 9

m n

Bi 9: Cho h phng trnh: 1 3 12 5m x my mx y m

a) Gii v bin lun h theo m . S: +) 1m : V s nghim: 2 4y x +) 1m : Nghim l 1; 3m m b) Vi cc gi tr nguyn no ca m th h c nghim duy nht ;x y sao cho 0, 0x y . S: 0;1;2m c) nh m h c nghim duy nht ;x y m 2 2P x y t gi tr nh nht. S: min 8 1P m d) Xc nh m h c nghim duy nht ;x y tho mn 2 2 0x y . (Hoc: sao cho ;M x y nm trn parabol 20,5y x ). S: 1; 5m m e) Chng minh rng khi h c nghim duy nht ;x y th im ;D x y lun lun nm trn mt ng thng c nh khi m nhn cc gi tr khc nhau. S: 4y x


2 1x mymx y

a) Gii h phng trnh trn khi 2m . S: 11;2

b) Tm cc s nguyn m h c nghim duy nht ;x y m 0x v 0y . S: 3; 2; 1;0m c) Tm cc s nguyn m h c nghim duy nht ;x y m ,x y l cc s nguyn. S: m d) Tm m h c nghim duy nht ;x y m S x y t gi tr ln nht.

S: max5 02

S m


2 8m x y

mx y

. Tm m h phng trnh c nghim nguyn.

S: 5;7;4;8;2;10m

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2

3 3 2 12

mx y m mx my m

. Tm h thc lin h gia ,x y khng ph thuc

vo m . S: 2

3 2 2 3 2 226 3 4 6 3 4x y x yx yx y x y

Bi 13: Cho h phng trnh: 1 31 12

m x y

x y

a) Gii h phng trnh khi 32

m . S: 11;2

b) Tm m h phng trnh c nghim 2; 2 . S: 32

m


1

1 2

m x y m

x m y

. Tm h thc lin h gia ,x y khng ph thuc vo m .

S: 2 23 2 0x x y y


2

5 123 6 2

x ay a aax y a a

. Tm h thc lin h gia ,x y khng ph thuc vo

a . S: 2

30 2 30 25 123 6 73 3 6 73

x y x yx yx y x y


2 11 2

mx my mx m y

a) Chng minh nu h c nghim ;x y th im ;M x y lun thuc mt ng thng c nh khi m thay i. S: 1y x b) Tm m im M thuc gc phn t th nht. S: 1m c) Xc nh m im M thuc ng trn c tm l gc ta v bn knh bng 5 .

S: 11;2

m m


3 2 83 1 1

x ymx x y m

. Tm m h c nghim ;x y tha mn:

4 2 6x y . S: 1 7m


2 3 5x my

x my

. Tm m h c nghim duy nht tha mn

2 1 10 4 5m x my m . S: 12m

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3 7m x y

mx y

a) Gii h phng trnh vi 1m . S: 1 5;2 2

b) Tm m 0, 0x y . S: 3m


mx my mmx y m

. Tm m nghim ca h tha mn iu kin 0, 0x y .

S: 102

m

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bai-giang-toan-hoc-298.pdf