cac phuong phap giai he phuong trinh
TRANSCRIPT
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Ch : H ph ng trnhVn 1: h ph ng trnh bc nht hai n
1) Gii v bin lun h sau: a) b)
+=+++
=+
1)1()1(
1)1()1(32
32
a ya xa
a ya xa
=++=++
b yba xba
a yba xba
)2()2(
)()(
2) Tm m h sau c nghim duy nht. Khi hy tm mt h thc lin h gia cc nghim khng phthuc vo tham s. Tm m nguyn nghim ca h l nghim nguyn.
a) b)
+=+=+
1
2
mmy x
m ymx
=++=+
423
12)1(
m ymx
m y xm
3) Cho h ph ng trnh:
+=+=+
ccay x
b yax2
a) Vi b=0 hy gii v bin lun h. b) Tm b vi a lun tm c c h c nghim.c) Tm b tn ti c sao cho h ph ng trnh c nghim vi mi a.
4) Cho h ph ng trnh c nghim. Chng minh rng a
=+ =+
=+
baycxacybx
cbyax3
+b3
+c3
=3abc. T chng t nu a, b, c
l di 3 cnh ca mt tam gic th: khi h trn c nghim th tam gic s u.5) Tm gi tr nh nht ca cc hm s sau y:
a) F=(x+2y-1)2+(2x+ay+5)2. b) G=(mx+y-1)2+(x+my-m+2)2.
6) Cho h:mx (2m 1)y 3m
(2m 1)x my 3m 2
+ + = + + = +
a. Tu thuc vo m xc nh nghim ca h.b. Tm m h c nghim duy nht tho mn:
a/ 2x-3y=1 b/ x+2y>5 c/ x>3; y
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c. Gi x, y l nghim duy nht ca h. Tm h thc lin h gia x, y khng ph thuc m
11) Cho h:2 1
3
+ = + + =
mx y m
x my m
a. Tu thuc vo m xc nh nghim ca h.b. Tm cc s nguyn m h c nghim duy nht x, y u l cc s nguyn.c. Gi x, y l nghim duy nht ca h. Tm h thc lin h gia x, y khng ph thuc m
12) Cho h:24 4
( 3) 2
+ = +
+ + = +
mx y m
x m y m 3
5
1
a.Tu thuc vo m xc nh nghim ca h.b. Tm cc s nguyn m h c nghim duy nht x, y u l cc s nguyn.c. Gi x, y l nghim duy nht ca h. Tm h thc lin h gia x, y khng ph thuc md. Tim m h c nghim duy nht tho mn x y
13) Cho h:2 1
3
+ = + + =
mx y m
x my m
a. Tu thuc vo m xc nh nghim ca h.b. Tm cc s nguyn m h c nghim duy nht x, y u l cc s nguyn.c. Gi x, y l nghim duy nht ca h. Tm h thc lin h gia x, y khng ph thuc m.
14) Cho h: 2 12 2
+ = + + = +mx y m x my m
a. Tu thuc vo m xc nh nghim ca h.b. Tm cc s nguyn m h c nghim duy nht x, y u l cc s nguyn.c. Gi x, y l nghim duy nht ca h. Tm h thc lin h gia x, y khng ph thuc m.
15) Cho h:( 1) 2 2
2 ( 1)
m x my
mx m y m
+ = + =
a. Tu thuc vo m xc nh nghim ca h.b. Tm cc s nguyn m h c nghim duy nht x, y u l cc s nguyn.c. Gi x, y l nghim duy nht ca h. Tm h thc lin h gia x, y khng ph thuc m
16) Cho h:(2 3) 3 2
5 (2 3) 5
m x my m
x m y
= + =
a. Tu thuc vo m xc nh nghim ca h.b. Tm cc s nguyn m h c nghim duy nht x, y u l cc s nguyn.c. Gi x, y l nghim duy nht ca h. Tm h thc lin h gia x, y khng ph thuc m
d. Tm iu kin h c nghim duy nht tho mn:1 2
3
x
y
<
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20) Cho h ph ng trnh:(m 2)x 2my m(2m 1)x y 2m 5
+ = = +
a. Gii h ph ng trnh vi m=3; m=2 5 ; m=37
b. Gii v bin lun nghim ca h ph ng trnh theo m.c. Tm m h c nghim duy nht tho mn h thc: 2x+3y=1.
21) Cho h( 1)
( 1) 2
a x y a
x a y
+ = +
=
1
a. Tm iu kin ca a h c nghim duy nht. Tm nghim .b. Gi (x;y) l nghim duy nht ca h.
a/ lp biu thc gia x v y khng ph thuc vo tham s a. b/. Tm a 2x-3y=5
22)Cho h ph ng trnh:
+=+=+
1
1
mmy x
m ymx
a. Vi gi tr no ca m h c nghim duy nht tha mn iu kin 2+ x y b. Vi cc gi tr ca m tm c hy tm gi tr ln nht ca Z=x+y.
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Vn 2: h ph ng trnh i xng
I/ H ph ng trnh i xng li 1:A - H i xng loi I khng c tham s
1/ Gii cc h ph ng trnh sau:
1) . 2) 3) 4)
=++
=+
1
622
y xy x
x y y x
=+
=+
8
822
22
y x
x y y x
=++
=++
5
722
y xy x
xy y x
=++
=++
5
173333
y xy x
y x y x
5)
=+
=+
5
613
y x
x y
y x
6)
=++
=+
12
111
22 xy y x
x y 7) 8)
=+=+
13
52244
22
y x y x
y x
=++=++
21
72244
22
y x y x
xy y x
9) 10) 11) 12)
+=+=+
2233
1
y x y x
y x
=+=+
1
144
33
y x
y x
+=+=+
2255
33 1
y x y x
y x
=++=
15))((
3))((22
22
y x y x
y x y x
13) 14) 15)
=+++
=++
28)(3
1122
y x y x
xy y x
=+
=++
30
1122
xy y x
xy y x2 2
4 4
x xy y 3
x y 17
+ + =
+ =16)
2 2
3 3
x y xy 30
x y 35
+ =
+ =
17) 18)x y xy 5(x y)xy 2
+ + = + = 3 3
x y 2
x y 26
+ =
+ =19)
3 3x y 8x y xy 2
+ =
+ + =20)
+ =
+ = 2 2x y 1 2xy
x y 1
24) 2 2x xy y 1
x y xy 6
=
=25) 2 2 3 3
x y 4
(x y )(x y ) 280
+ =
+ + =26) 3 3
x y 1
x y 1
+ =
+ =
26) 27)2 2
2 2
x xy y 19(x y)
x xy y 7(x y)
+ + =
+ =
2
2
x(x 2)(2x y) 9
x 4x y 6
+ + =
+ + =28)
2 2x xy y 4
x xy y 2
+ + =
+ + =
29)2 2x xy y 3x xy y 1
+ + =
+ + =30)
2 2x y x y 2x xy y 1
+ + =
+ = 31)
+=+
=+
)(3)(2
6
3 23 2
33
xy y x y x
y x
2/ Gii cc h ph ng trnh sau: a)
=+
=+
35
30
y y x x
x y y xb)
=+
=+
7
5
xy y x
y xc)
=+
=++
4
28222
y x
xy y x
3/ Gii cc h ph ng trnh sau y:
1)
=+++
=+++
411
411
2222
y x y x
y x y x2)
x y 7 1y x xy
x xy y xy 78
+ = +
+ =
3)2x 2y 3y x
x y xy 3
+ =
+ =
4)
=++
=++
49)1
1)((
5)1
1)((
2222
y x y x
xy y x
5)2 2
2 2
1 1x y 5x y1 1x y 9x y
+ + + =
+ + + =
6)
1 1 7xyx y 2
3x y xy2
+ + =
+ =
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B - H i xng loi I c tham s
1/ Cho h ph ng trnh :
=++=+++
m y x xy
y x y x
)1)(1(
822
a/ Gii h ph ng trnh vi m=12. b/ Tm m h c nghim. 1616
33 m
2/ Tm tham s cc h sau c nghim: (c nghim duy nht b)
a) b)
=+=+
a y x
y x44
2
+=++=++
mm y x xy
m y xy x
)1()(
12
3/ Cho h ph ng trnh:
=++=+
4)(
)1(22
22
y x
a y x
a) Gii h vi a=1. b) Tm a h c 2 nghim.
4/ Cho h ph ng trnh:
=+
+=++
m y x xy
m xy y x22
1
a) Gii h vi m=2. b) Tm m h c nghim d ng.c) Tm m h c nghim duy nht.
5/ Bit rng x,y l nghim ca ph ng trnh sau: .
+=+=+
6222 m y x
m y x
a) Gii h ph ng trnh vi m=1. b) Tm cc gi tr ca m h ph ng trnh c nghim. s: 2 m 2 c) Hy tm gi tr ln nht v nh nht ca F=xy+2(x+y)
6/ Tm m h sau c nghim:
a)
+=+
=++
64
112 mm y x
m y xb)
=+
=+
m xy y x
m y x
7/ Cho cc s a, b, c tho mn: Chng minh rng:
=++=++
1
2222
cabcab
cba
34
,,34 cba
8/ 2 2x y 1x y k
+ =
=Tm k h c nghim duy nht
9/ Cho hx y a
x y xy a
+ =
+ =
a) Gii h ph ng trnh vi a=4.b) Tm cc gi tr ca a h ph ng trnh c nghim. s: 041 = aa
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10/ Cho h5(x y) 4xy 4
x y xy 1 m+ =
+ = Tm cc gi tr ca m h ph ng trnh c nghim. s: 1;
41 mm
11/ Cho h: 2 2 2x y m 1
x y xy 2m m 3
+ = +
+ =
a) Gii h ph ng trnh vi m=3.b) Chng minh rng h c nghim vi mi m.
12/ Cho h:2 2x 2x y 2y 2m
x(x 2).y(y 2) 5m 6
+ + + =
+ + =
Tm cc gi tr ca m h ph ng trnh c nghim. s:5 m 2;m 37
13/ Cho h:
1sin x sin y
2cos2x cos2y m
+ =
+ =Tm m h c nghim.
14/ Cho h 2 2x xy y m 1
x y xy m
+ + = +
+ = 10 m ;m 24
<
a) Gii h ph ng trnh vi m=2.b) Tm cc gi tr ca m h ph ng trnh c t nht 1 nghim tha mn:x 0;y 0> >
15/ Cho h: Tm k h c nghim s:cosx cosy 1
cos2x cos2y 2k 2+ =
+ = 1 k 12
16/ Cho h:x y xy m 22 2x y xy m 1
+ + = +
+ = +Tm m h c nghim s:
3m ; m4
1
17/ Cho h:2 2x y 1 k( x y 1) 1
x y xy 1
+ + =
+ = +
a) Gii h ph ng trnh vi k=0.b) Tm cc gi tr ca k h ph ng trnh c nghim duy nht.
18/ Cho h:3 3x y m(x y)x y 1
= + =
. Tm m h c 3 cp nghim phn bit v x1,x2,x3 lp thnh cp
s cng trong 2 gi tr c tr tuyt i ln hn 1.
19/ Cho h:2 2
4 4
x y m
x y 3m 2
+ =
+ = .Tm m h c nghim.
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II/ H ph ng trnh i xng loi 2:A - H i xng loi II khng c tham s.
1/ Gii h ph ng trnh sau:
1) . 2)
==
x y y
y x x
3
32
22
2
x 1 y
y 1 x
=
= . 3) 4)
==
23
233
3
x y
y x2
2
2x y 4y 5
2y x 4x 5
= +
= +
5)
2
2
12x yy12y xx
= +
= +
6)
2
2
1x 1 y41y 1 x4
=
=
7)
+=
+=
x x y
y y x
123
123
2
2
8)2
2
2yx1 y
2xy
1 x
=
=
9) 10) 11) =
=
2 2
2 2
x y 4 y
y x 4 x
3
3
x 3x 8
y 3y 8
= +
= +
y
x
2 2
2 2
2x y 3x 2
2y x 3y 2
=
= 12)
3
3
y 1 2
x 1 2
y
x
+ =
+ =
13)2
2
32x y x
32y xy
+ =
+ =
14)yx 3y 4 xxy 3x 4y
=
=
15)
1 32x y x
1 32yx y
+ =
+ =
16) + =
+ =
3 2
3 2
y x y 40
x xy 40
x
y
2/ Gii cc ph ng trnh sau:a) x3-3. 2233 =+ x . b) x3-2. 1123 =+ x . c) (x2+3x+4)2+3(x2+3x+4)=x-4.
Ch 1: Mt s h i xng loi 2 nh ng khng th gii theo ph ng php trn khi c dng cn, lc tabnh ph ng 2 v a v h i xng loi 2 khc t ng ng.
3/ Gii cc h ph ng trnh sau :
a)
=++=++
725
725
x y
y x. b)
=+=+
22
22
x y
y x
c)x 9 y 7 4
y 9 x 7
+ + =
+ + = 4d)
x 1 7 y 4
y 1 7 x
4
+ + =
+ + =
Ch 2: Mt s h i xng loi 2, s dng php t n ph s c 1 h i xng loi 2 n gin hn.4/ Gii cc h ph ng trnh sau:
a)
=+
=+
11
114
4
x y
y x. b)
2 2
2 2
x x y 1 x y x y 1 y 2
x x y 1 x y x y 1 y 18
+ + + + + + + = + + + + + + + + + =
5/ Gii cc h ph ng trnh:
1)5 5
9 9 4
x y 1
x 4y x y
+ =
+ = +2)
2 2
3 3
x y 1
x y 1
+ =
+ =3)
3 3
6 6
x 3x y 3y
x y 1
=
+ =4)
4 4
6 6
x y 1
x y 1
+ =
+ =
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B - H i xng loi II c tham s
1 / Gii bin lun h pt:2
2
x y m
y x
+ + =
+ + = m.
2/ Cho h ph ng trnh: a/ Gii h ph ng trnh vi m=-1.
=+=+
)1()1(
2
2
xm y xy ym x xy
b/ Tm m h c nghim duy nht.
3/ Cho h ph ng trnh: . a/ Gii h pt khi m=0.
+=+=
mx x x y
my y y x232
232
4
4
b/ Tm m h pt c nghim duy nht.
4/ Tm a0.
b/. iu c ng khng khi a
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13) Cho h ph ng trnh :2
2
y (x y) 2m
x (x y) 2m
+ =
+ =
a. Gii h vi m=0 b. Tm m h c nghim.
14) Tm a h sau y c nghim duy nht:2
2
(x 1) y a
(y 1) x a
+ = +
+ = +
15) Cho h ph ng trnh: x 1 y 2 my 1 x 2
+ + =
+ + = m
a/. Tm m h ph ng trnh c nghim. b/. Gii h ph ng trnh vi m=9
Vn 3: h ng cp (thun nht)
A - H ng cp khng c tham s1 Gii h:
1) 2) 3)
=++=++
1732
112322
22
y xy x
y xy x
==+
15395
3845322
22
y xy x
y xy x
=++=+
7223
14222
22
y xy x
y xy x
4)
=+=+
2
032 22
y y x x
y xy x5)
2 2
2 2
x 4xy 2y
2x xy 3y 4
3+ =
+ =6)
2 2
2
x 4xy y
x 3xy 4
+ =
=
1
92
7) 8)23 3 8
=
+ = x y x y
2 22 3
2 22 2
+ + =+ + =
x xy y x xy y
9)2 2
2 22 3 13 11
+ + = + =
x xy y x xy y
2
2) Gii ph ng trnh: 3 23 23 2 42003)2(2002)2( =++ x x x
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3) Gii h ph ng trnh: a) b) c)
=
=
2)(
192
33
y y x
y x
=+
=
y y x x
x y x y
10)(
3)(222
22
=+
=+222
22
51
6
x y x
x xy y
4) Gii hpt:
=
++
=++
32
12
0)2(6)4(5)2( 2222
y x y x
y x y x y x
B - H ng cp c tham s
1) Tm a h sau c ngim
=++
=+
+=
1
02
)1(
223
233
xy yax x
y x
aay x
2) Cho h ph ng trnh :2 2
2
x 4xy y k
x 3xy 4
+ =
=. CMR h c nghim vi mi k.
3) Gii v bin lun nghim h ph ng trnh2
2 2x xy 22x 4xy 2y m =
+ =
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4) Gii v bin lun nghim h ph ng trnh2
2 2
y 2xy 3
x xy y m
=
+ + =
5) Tm iu kin h ph ng trnh sau c nghim:2 2 2 2
+ = + = x y m
x y x
6) Cho h ph ng trnh:2 2
2 2
3 2 11
2 3 17
+ + =
+ + = +
x xy y
x xy y ma. Gii h vi m=0
b. Tm m h c nghim.
7) Tm iu kin ca a h sau y c nghim:2 2
2 2 4 3 2
2 3 8
2 3 4 4 12 105
x xy y
x xy y a a a
=
+ = + +
8) Cho h ph ng trnh: a. Gii h vi k=6.
=
=+
43
42
22
xy y
k y xy x
b. Chng minh rng vi mi k h c ngim.
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Vn 4: H ph ng trnh khc
A - H ph ng trnh khc khng cha tham sGii h
1) 2)2 2
2 2
3 4 1
3 2 9 8
x y x y
x y x y
+ + =
= 302 22 3
| | | | 2
+ =
+ = x xy y
x x y y3)
2
4
0
x x y
y
x xy y
+ + =
+ =
4)2 2 2
2 2 2 2
3 2 x y xy x y
x y xy x y
+ + =
+ =
5) 6)1
4
9
x xy y
y yz z
z zx x
+ + = + + = + + =
163
92
x xy
y
y xy
x
=
=
7)9
27
1 1 11
x y z
xy yz zx
x y z
+ + = + + =
+ + =
8)
3 3 3
2
2
1 2
2
x y y
x x y y
+ =
+ =
9) 10) 11) 12)2
2
4 2
2 2
x xy
y xy
+ =
+ =
3
3
6
3 3
5 5
2 ( ) 6
30 32
x y xy x y
x y xy
+ + + =
+ + =
( ) 30
( ) 120
x y xy
x y xy
= + =
2
2 2
3 4 2 6
4 4 12
x y xy
x y x y
+ + =
+ + + =
13) 14)3 3
9( ) x y x y xy
+ = + =
( 2 ) 2(2 ) 2 x y xy y x
xy xy y x
+ = + + = + 15)
2
2 33 5
x y x y x y xy
6+ + =
+ + =16)
3 3
3 32 22 2 x y y x x y y
+ = + + = + x
17)2 2
2
4
2 9 7 5
x y x y
x xy x y
+ + + =
+ + = +
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15) Tm k h c ng 4 nghim:2 2 2(2 ) 4
( 4)( 2)
x y x y
xy x y k
+ + =
=k
16) Tm m h c ng 2 nghim:1
x y m
x y m
+ =
= +
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Vn 5: H ph ng trnh v t A - h ph ng trnh v t khng c tham s
Gii h:
1)
++=+
=
2
3
y x y x
y x y x2)
=++
=+++
65)(
185)(222
2222
y x y xy x
y x y xy x3)
=+
=+
xy x y y x
xy x y y x
11
2)1()1(
4)
=++
=+++
12
527
y x y x
y x y x5)
1 1
2 2 2
x y
x y y
+ =
+ = 6)
5 5
4
x y
x y
+ + + =
+ =
6
7)2 2
4
128
x y x y
x y
+ + =
+ =8)
1
1
x y
x y
+ =
+ =9)
2 2 2 2
2
4
x y x y
x y x y
+ =
+ + =
10)2
1
x y x y
x y y x
+ + =
+ =11)
3
3
31 2
31 2
x y x
y x
y
+ = +
+ = + 12)
2 2
1 1
2 2
x y x y
1 x y x y x
+ + =
+ =
13)2 2
2 2
2
2
x y x y
x y y x + =
+ =14)
24
2 4 2
1 8 4
2 2 2 2
x x y
x x y y
+ + + =
2 + + + = 15)
4
24
32 24 6
32 3
x x y
x y y + =
+ =
16)( 3
1
y x x
x y x
+ + =
+ = +
) 317)
33
3
y x y x
x
x y x x
+ + + =
+ + = +18)
2 2
2
1
x y x y
x y
+ + =
+ =
19)2 21
1
x y x y x y
x y
+ + = +
+ =20)
1 1
1 1
x x y
y y x
+ + + =
+ + + =21)
2
2
2 2
2 2
x y
y x
+ =
+ =
22)2 2
1
x x x y
x y x y
+ + + =
+ = +
y23)
2
11
1 1
x x y y
xy y x
+ + =
1+ + + =
24)1 3 3
12 8
x x y y
x y y
+ + + =
+ + =
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b - h ph ng trnh v t c cha tham s
1) Gii bin lun cc h: 1)
==++
a y x
a xy y x2)
=+
=+
m x y
m y x
12
12
2) Tm tham bin cc h sau c nghim: 1)
=++
=++
11
1
x y
m y x. 2)
=+=+++
a y x
a y x
3
21
3) Cho ph ng trnh: ( 021
21
=++
=++m
m x y
m y x) a, Gii ph ng trnh vi m = 9
b, Xc nh m h c nghim.
4) Tm a h sau c ng 4 nghim:2 2
1 1 7
49 4 2 1
x y
y x a x
=
+ + =
( Ph ng php chuyn n mi)5) Tm m ph ng trnh sau c 4 nghim phn bit: x4-2mx2-x-m2-m=0
6) Tm m h sau c nghim: ( )
=+++++
+
+
0163216168
52
4
224
2
22
mmmx x x
x
x x
7) Cho ph ng trnh: x2+(2a-6)x+a-13=0. Khi a 1 tm a nghim ln ca ph ng trnh t ln nht.
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Ph ng php iu kin cn v
Cho h ph ng trnh (hoc h bt ph ng trnh ) c tham s d i dng:
(I)( , ) 0
, x m
f x m
x D m D
=
(hoc (II)( , ) 0
, x m
f x m
x D m D
)
Trong x l bin, m l tham s; , x m D D t ng ng l cc min xc nh ca x v m. Ta cn tm iu
kin t ln tham s m h (I) (hoc h (II) ) tha mn mt tnh cht P no .Ta thc hin ph ng
php theo cc b c sau:
B c 1: (iu kin cn) Gi s h (I) (hoc (II)) tha mn tnh cht P m bi ton y/c. Da vo c th ca
tnh cht P v dng ca hm s f(x, m), ca min xc nh Dx, m ta tm c mt iu kin rng
buc no i vi m. iu kin rng buc y chnh l iu kin cn tha mn tnh cht P v gi
s n c dng . iu c ngha l: Numm m D mom th chc chn ng vi gi tr , hom
(I), (II) khng c tnh cht P.
B c 2: (iu kin ) Gi s . Ta phi tm xem trong cc gi tr y ca m, gi tr no lm cho h (I)mm
(hoc(II)) tha mn tnh cht P. Ni chung b c 2, ta ch phi xt cc h c th ( th ng l h khng
c tham s hoc nu c th h n gin i nhiu). Da vo c tr ng ca cc h y, ta vn dng cc
kin thc cn thit v l thuyt ph ng trnh, bt ph ng trnh gii chng. Kt qu ca php gii s
cho php ta loi i khi tp cc gi tr khng thch hp v m.m
Kt qu c 2 b c, ta tm c li gii ca b i ton cho.
Lp b i ton dng iu kin cn v Dng 1: H ph ng trnh c nghim duy nht.
Dng 2: H ph ng trnh c nghim vi mi gi t ca mt tham s.
Dng 3: H ph ng trnh nghim ng x D .
Dng 4 : H ph ng trnh t ng ng vi 1 PT hoc 1BPT khc..
Tnh i xng ca cc biu thc gii tch trong vic xc nh iu kin cnCc bi ton thuc mc ny th ng c du hiu nhn bit sau y:
1> Cc biu thc gii tch c tnh i xng theo mt bin, hoc 1 tp hp bin.
V d:* Bi ton c dng: F(x, y, m) = 0 ( hoc F(x, y,m) 0) trong biu thc F(x, y, m) khng thay
i gi tr khi ta i v tr ca x v y hoc khi ng thi thay x bng y, y bng x, hoc khngthay i khi thay x bng x, hoc y bng y. Da vo tnh i xng, chn, nh gi,
2> Trong cc bi ton y c yu cu tm iu kin nghim tm c c tnh duy nht. Vi nhng
loi bi ton ni trn ta th ng s dng l c sau y tm iu kin cn.
Gi s h c nghim duy nht ( c biu din bng in M). Do biu thc gii tch l i xng nn
h cn c 1 nghim m n c biu din bng im M1. im M ny i xng vi im M1 theo
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ngha i xng ca biu thc gii tch. Mt khc do tnh duy nht ca nghim nm M trng M1 . T
iu kin ny ta s tm c iu cn bi ton c nghim duy nht.
3> H ph ng trnh dng: t ng ng vi mt trong cc h sau y:1 21 2
( ) ( ) (1)
( ) ( ) (2)
F x F x
G x G x
= =
hoc1 2
1 1 2 2
( ) ( ) (1)
( ) ( ) ( ) ( ) (3)
F x F x
F x G x F x G x
=
= 1 1 2 2
1 1 2 2
( ) ( ) ( ) ( ) (4)
( ) ( ) ( ) ( ) (5)
F x G x F x G x
F x G x F x G x
+ = +
=
L u : iu kin ph ng trnh 1 1 2 2( ) ( ) ( ) ( )F x G x F x G x = c nghim ch l iu kin cn h (I)
(II) c nghim.
4> Gi s f(x) l mt hm lin tctn min D. ta c cc mnh sau y:
Mnh 1:H ph ng trnh :( ) f x a
x D
=
c nghim khi v ch khi m = min f(x) max f(x) =M.a
Mnh 2:Gi s D =[b,c]. Nu f(b).f(c) < 0 th ph ng trnh f(x) = 0 c t nht 1 nghim trn [b;c]
Mnh 3: a) M a l k cn v h bt PT sau c nghim( ) f x a
x D
b) m a l k cn v h bt PT ( ) f x a nghim ng vi mi x thuc D.
Mnh 4: a) m l k cn v h bt PT sau c nghim( ) f x
x D
b) M l k cn v h bt PT f(x) nghim ng vi mi x thuc D.
Mnh 5:Xt h bt ph ng trnh( ) ( )(1)
(2)
f x g x
x D
Nu max f(x) min g(x), th h trn tha mn vi mi x thuc D.
Mnh 6:Cho PT: f(x) = g(x) vi x thuc D. Gi s trn D hm f(x) lun ng bin, cn hm g(x) lun
nghch bin. Khi nu PT trn c nghim th nghim l duy nht.
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Dng 1: H chn i vi mt nA. nh ngha:H chn i vi mt n l h m khi i du n ny th h khng thay i. Nu h ny c
nghim duy nht th n ny ch c th nhn gi tr 0.
B. Ph ng php chung . Cu hi th ng t ra l:
Tm iu kin ctham s h ph ng trnh c nghim duy nht
B c 1: K cn: Gi s h c nghim (x; y) th (-x; y) cng l nghim ca h . Do h c nghim duy nht
khi x= -x x=0. Thay x= 0 vo h ta c gi tr ca tham s. chnh l k cn h c
nghim.
B c 2: K : thay gi tr ca tham s vo h, gii h. Nhn xt v kt lun.
C. B i tp: Tm k ca tham s h c nghim duy nht.
1> 2>2 2 5
cos 2
x y m
y x
+ =
+ =
4 2sin 1
( 1) cos
x y
x a y x
+ = + = +
3>2
2
sin 2 1
tan 6 2 2
ax x a y
x y
+ + = +
+ =4>
2
2 2
1 sin
tan 1
ax a y x
x y
+ =
+ =
5>2sin 1
( 1) cos x y
x a y x + =
+ = +6>
2
2 2
cos
sin 1
ax a y x
x y + = +
+ =7>
2
2 2 2
( ) xy x y z a x y z a + + =
+ + =8>
2
2 2
2
1
x x y x a
x y + = + +
+ =
9>2 2
2 2 ( 1 x y y x m
x y m
= +
+ =
)10>
2
2 2
3
5 5
x y a
3 y x x
+ + =
+ + = + + a11>
2
2
2
2 2 2
x y a
y x a
+ + =
+ + =
12> 13>
2 2 2
2
4 x y z
xyz z a
xyz z b
+ + = + =
+ =
2
2
2
3 1 1
1
1
x a y
x y a y y
+ =
+ + = + +
14>
2 2
11
0
y
y
x y b
xa
x
x
+ =
= + >
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Dng 2: H c nghim vi mi gi tr ca mt tham sDng cu hi: Thng th ng h ny c 2 tham s: Tm iu kin ca tham s th nht sao cho h c
nghim vi mi gi tr ca tham s th hai .
A- Ph ng php chung : iu kin cn: V PT ( hoc BPT ) cho ng vi x M , nn khi th 1 gi tr c th no t
tp M ta c cc gi tr ca tham s m trong nht nh cha cc gi tr cn thit. Chn mt
hoc nhiu gi tr c bit ca tham s th 2, h c dng n gin, t c suy ra k cn i
vi tham s th nht.
K : Vi tham s va tm c s dng trong qu trnh gii hoc thay vo h gii, nhn xt v
kt lun. Ch : Trng b c tm k cn vic chn 1 hoc nhiu gi tr c bit ca tham s th 2 tm
cn st nht cc em bit ti khi nim im thun li cng vi kinh nghim tm chng.
B. B i tp:
1> Tm a h sau c nghim vi mi b:2 2
2
( 1) ( 1)
1
a y x b
a bxy x y
2+ + + =
+ + =
2>Tm b sao cho vi mi a, h ph ng trnh c nghim:2
2
(1 )
x ay b
ax b y b
+ =
+ =
3> Tm a h sau c nghim vi mi b:2 2
3 3
2 ( 1)
( 1) 1
bx a by a
a x y
+ + =
+ =
4> Cho h PT: . Bit rng h PT c nghim vi mi b. CMR: a = 0.2 2( ) aa x y x y b
y x b
+ + + =
=
5> Tm a h sau c nghim vi mi b:2 22 1 (a-1)by 1
1 0
x b x
ax bx
=
+ =
6> Tm a h :3 3
3 2 2
1( 1)
2
1
x ay a
x ax y xy
= +
+ + =
2
b
c nghim v vi mi nghim ca n u tha mn h thc: x+y=0.
7> Tm a h sau c nghim vi mi b:2 2
.2 2 ( 1)
(2 ) 1
ax bxb a
a x y
+ = +
+ =
8> Tm a h sau c nghim vi mi b:5 5
3 2
( 1) 1
( 1)bxa x y
e a by a
+ =
+ + =
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Dng 3: H da v o s nh gi mt nA - Ph ng php : gii loi h ny ta i nh gi mt n bng cch tn k i vi n ny tn ti cc
n khc. tm k th ng coi 1 n l tham s ca PT bc 2. T ta tm c gi tr
ca n ri suy ra gi tr ca cc n cn li
B. B i tp: Gii cc h sau:
1> 2>2 2 2
2 2
1
2 2 2 1
x y z
x y xy yz xz
+ + =
+ + + = 0
2 2 2
2 2
2 3
2 1
x y z xy xz zy
x y xy yz xz
+ + + =
+ + =
3> 4>
3
2 2
2 (3 )
(2 )( 2) 9 4
4
0
y x
z y y y
x z x
z
+ = + = +
+ =
3 2 2
2 2
(2 )(3 2 ) 3
3 3
6
3
2
x x z
y y x x
y z z
z
z = + = +
+ =
5>
3
2 2
3 2 (3 )
4 8
(2 )( 3) 5 16
0
y x
z y y
z x x x
z
= +
+ =
+ = +
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Vn 4: H ph ng trnh khc
A - H ph ng trnh khc khng cha tham sGii h
1) 2)2 2
2 2
3 4 1
3 2 9 8
x y x y
x y x y
+ + =
= 302 22 3
| | | | 2
+ =
+ = x xy y
x x y y3)
2
4
0
x x y
y
x xy y
+ + =
+ =
4)2 2 2
2 2 2 2
3 2 x y xy x y
x y xy x y
+ + =
+ =
5) 6)1
4
9
x xy y
y yz z
z zx x
+ + = + + = + + =
163
92
x xy
y
y xy
x
=
=
7)9
27
1 1 11
x y z
xy yz zx
x y z
+ + = + + =
+ + =
8)
3 3 3
2
2
1 2
2
x y y
x x y y
+ =
+ =
9) 10) 11) 12)2
2
4 2
2 2
x xy
y xy
+ =
+ =
3
3
6
3 3
5 5
2 ( ) 6
30 32
x y xy x y
x y xy
+ + + =
+ + =
( ) 30
( ) 120
x y xy
x y xy
= + =
2
2 2
3 4 2 6
4 4 12
x y xy
x y x y
+ + =
+ + + =
13) 14)3 3
9( ) x y x y xy
+ = + =
( 2 ) 2(2 ) 2 x y xy y x
xy xy y x
+ = + + = + 15)
2
2 33 5
x y x y x y xy
6+ + =
+ + =16)
3 3
3 32 22 2 x y y x x y y
+ = + + = + x
17)2 2
2
4
2 9 7 5
x y x y
x xy x y
+ + + =
+ + = +
B - H ph ng trnh khc cha tham s1> Cho h ph ng trnh: a. Gii h vi m=4
2( 1) ( 2
x y m
x y xy m y
+ =
+ + = + )b. Tm m h c nhiu hn 2 nghim.
2> Tm iu kin h c nghim :2 0
x y m
y x y
=
+ =
3> Cho h ph ng trnh:2 2
2 2
( )
3
x y a x y x y a
x y bxy
+ + = +
+ + =
a. Gii h vi a=b=1b. Xc nh tt c cc gi tr ca a, b h c nhiu hn 4 nghim khc nhau.
4> Cho h ph ng trnh: Gi s ph ng trnh c nghim vi mi b. CMR a=02 2( )a x y x y b
y x b
+ + + =
=
5> Cho h: 4 13
x y
x y m
+ =
+ =
4 Tm m h c nghim.
6> Cho h:3 3
1
( )
+ =
=
x y
x y m x ya. Gii h vi m=1
b. Tm m h c 3 cp nghim phn bit.
7> Cho h: . Tm m h c nghim.2 2 2 2
x y m
x y x
+ =
+ =
8> Cho h:3 3
3 2 2
1 ( 1)2
1
x ay a
x ax y xy
= +
+ + =
2
Vi gi tr no ca a h c nghim tho mn x+y=0.
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9 > Cho h:3 3 ( )
1
=
+ = x y m x y
x y
1. Gii h vi m=32. Tm m h c 3 cp nghim (x1;y2); (x2;y2); (x3;y3) sao cho x1; x2; x3 lp thnh CSC v c 2
gi tr x tho mn |x|>1.
10> Cho h:((1 )
2 0
= + + + + = x y a xy
xy x ya/. Gii h vi a=1/2 b/. Gii v bin lun h theo a.
11> Tm iu kin h sau c nghim:
1 2 52
22
+ + = + =
x y x y
x ya
x y
(Bin lun)
12) Tm a, b h sau c nhiu hn 5 nghim:2 2
2 2
( )
1 0
x y a x y x y
x y bxy
a + + = +
+ + =
13) Tm m h sau c nghim: 2 2 2 2 2( )( 20) (140 )( 80)
(8 20 ) (4 )
, , ,
x y x y m m
m u v m u v
x y u v N
+ + + = + =
14)Tm b h sau c nghim: 22 2 2
3 2 1
3 4 2 1
9 4 2
x y b
xy b b
x y b b
+ = + = + + + + 2
15) Tm k h c ng 4 nghim:2 2 2(2 ) 4
( 4)( 2)
x y x y
xy x y k
+ + =
=k
16) Tm m h c ng 2 nghim:1
x y m
x y m
+ =
= +
Vn 5: H ph ng trnh v tA - h ph ng trnh v t khng c tham s
Gii h:
1)
++=+
=
2
3
y x y x
y x y x2)
=++
=+++
65)(
185)(222
2222
y x y xy x
y x y xy x3)
=+
=+
xy x y y x
xy x y y x
11
2)1()1(
4)
=++
=+++
12
527
y x y x
y x y x5)
1 1
2 2 2
x y
x y y
+ =
+ = 6)
5 5
4
x y
x y
+ + + =
+ =
6
7)2 2
4
128
x y x y
x y
+ + =
+ =8)
1
1
x y
x y
+ =
+ =9)
2 2 2 2
2
4
x y x y
x y x y
+ =
+ + =
10)2
1
x y x y
x y y x
+ + =
+ =11)
3
3
31 2
31 2
x y x
y x
y
+ = +
+ = + 12)
2 2
1 1
2 2
x y x y
1 x y x y x
+ + =
+ =
13)2 2
2 2
2
2
x y x y
x y y x
+ =
+ =14)
24
2 4 2
1 8 4
2 2 2 2
x x y
x x y y
+ + + =
2 + + + = 15)
4
24
32 24 6
32 3
x x y
x y y
+ =
+ =
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16)( 3
1
y x x
x y x
+ + =
+ = +
) 317)
33
3
y x y x
x
x y x x
+ + + =
+ + = +18)
2 2
2
1
x y x y
x y
+ + =
+ =
19)2 21
1
x y x y x y
x y
+ + = +
+ =20)
1 1
1 1
x x y
y y x
+ + + =
+ + + =21)
2
2
2 2
2 2
x y
y x
+ =
+ =
22)2 2
1 x x x y x y x y
+ + + =+ = +
y 23)2
11
1 1
x x y y
xy y x
+ + =
1+ + + =
24)
13 3
12 8
x x y y
x y y
+ + + =
+ + =
b - h ph ng trnh v t c cha tham s
1) Gii bin lun cc h: 1)
==++
a y x
a xy y x 2)
=+
=+
m x y
m y x
12
12
2) Tm tham bin cc h sau c nghim: 1)
=++=++
11
1
x y
m y x. 2)
=+=+++
a y x
a y x
3
21
3) Cho ph ng trnh: ( 021
21
=++
=++m
m x y
m y x) a, Gii ph ng trnh vi m = 9
b, Xc nh m h c nghim.
4) Tm a h sau c ng 4 nghim:2 2
1 1 7
49 4 2 1
x y
y x a x
=
+ + =
( Ph ng php chuyn n mi)5) Tm m ph ng trnh sau c 4 nghim phn bit: x4-2mx2-x-m2-m=0
6) Tm m h sau c nghim: ( )
=+++++
+
+
0163216168
52
4
224
2
22
mmmx x x
x
x x
7) Cho ph ng trnh: x2+(2a-6)x+a-13=0. Khi a 1 tm a nghim ln ca ph ng trnh t ln nht.
Dng 1: H chn i vi mt n C. B i tp: Tm k ca tham s h c nghim duy nht.
1> 2>2 2 5
cos 2
x y m
y x
+ =
+ =
4 2sin 1
( 1) cos
x y
x a y x
+ = + = +
3>2
2
sin 2 1
tan 6 2 2
ax x a y
x y
+ + = +
+ =4>
2
2 2
1 sin
tan 1
ax a y x
x y
+ =
+ =
5>2sin 1
( 1) cos
x y
x a y x
+ =
+ = +6>
2
2 2
cos
sin 1
ax a y x
x y
+ = +
+ =7>
2
2 2 2
( ) xy x y z a
x y z a
+ + =
+ + =8>
2
2 2
2
1
x x y x a
x y
+ = + +
+ =
9>2 2
2 2 ( 1 x y
y x m x y m = +
+ =) 10>
2
2 235 5
x y a3 y x x
+ + =+ + = + + a
11>2
222 2 2
x y a y x a
+ + =+ + =
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12> 13>
2 2 2
2
4 x y z
xyz z a
xyz z b
+ + = + = + =
2
2
2
3 1 1
1
1
x a y
x y a y y
+ =
+ + = + +
14>
2 2
11
0
y
y
x y b
xa
x
x
+ =
= + >
Dng 2: H c nghim vi mi gi tr ca mt tham s
B i tp:
1> Tm a h sau c nghim vi mi b:2 2
2
( 1) ( 1)
1
a y x b
a bxy x y
2+ + + =
+ + =
2>Tm b sao cho vi mi a, h ph ng trnh c nghim:2
2
(1 )
x ay b
ax b y b
+ =
+ =
3> Tm a h sau c nghim vi mi b:2 2
3 3
2 ( 1)
( 1) 1
bx a by a
a x y
+ + =
+ =
4> Cho h PT: . Bit rng h PT c nghim vi mi b. CMR: a = 0.2 2( ) aa x y x y b
y x b
+ + + =
=
5> Tm a h sau c nghim vi mi b:2 22 1 (a-1)by 1
1 0
x b x
ax bx
=
+ =
6> Tm a h :3 3
3 2 2
1( 1)
21
x ay a
x ax y xy
= +
+ + =
2
b
c nghim v vi mi nghim ca n u tha mn h thc: x+y=0.
7> Tm a h sau c nghim vi mi b:2 2
.2 2 ( 1)
(2 ) 1
ax bxb a
a x y
+ = +
+ =
8> Tm a h sau c nghim vi mi b:5 5
3 2
( 1) 1
( 1)bxa x y
e a by a
+ =
+ + =
Dng 3: H da v o s nh gi mt nB. B i tp: Gii cc h sau:
1> 2>2 2 2
2 2
1
2 2 2 1
x y z
x y xy yz xz
+ + =
+ + + = 0
2 2 2
2 2
2 3
2 1
x y z xy xz zy
x y xy yz xz
+ + + =
+ + =
3> 4>
3
2 2
2 (3 )
(2 )( 2) 9 4
4
0
y x
z y y y
x z x
z
+ = + = +
+ =
3 2 2
2 2
(2 )(3 2 ) 3
3 3
6
3
2
x x z
y y x x
y z z
z
z = + = +
+ =
5>
3
2 2
3 2 (3 )
4 8
(2 )( 3) 5 16
0
y x
z y y
z x x x
z
= +
+ =
+ = +
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B- H mt n:1) Ph ng php nh l o: Tm m h sau c nghim: 19> Tm m h sau c nghim:
++
+
122
325
22
22
mm
y xy x
y xy x
1)2
2 2
( 1)( 2) 0 ( 1)2)
4 3 1 02 2
x x x x m m
x mx mx mx m
+ < 0
0
+
+ + > + + +
+++
+
>+++
+
0)12(
012)4
01)1(2
065)3
22
2
22
2
mm xm x
m x x
m xm x
x x
2) Ph ng php th bng bin thin:
1) Gii cc h sau:
>++
013
0123/
0
012
02
/3
2
3
3
24
x x
x xb
x x
x x
x x x
a
>++
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2. Dng h: . Trong F l mt hm s l ng gic ca x. gii h ny ta bin i
ph ng trnh u v dng tng kt hp vi ph ng trnh sau a v h ph ng trnh i s ca x v y.
==
y x
m yF xF )().(
3. Dng h:
=
=
y x
m yF xF
)()(
. Trong F l mt hm s l ng gic ca x. gii h ny ta bin i ph ng
trnh u v dng1
)()(1
)()(=
++
m yF xF
m yF xF ri bin i v tch v kt hp vi ph ng trnh sau a v h
ph
ng trnh i s ca x v
y.4. Cc tr ng hp cn li ta phi p dng cc php bin i i s hoc l ng gic thch hp ng vitng bi gii.B. Luyn tp Gii cc h sau:
c)
+=
=+
21
214
tgytgx
y x
a)
=+
=
=+
=
21
sinsin
3)
21
coscos
32
22 y x
y xb
y x
y x
d)
=
=+
2sin
sin2
y
x
y x
e)
=+
=
12533
.
y x
tgytgx f)
=+
=
3
21
coscos
y x
y x
g)
=+
=
22
1cos.cos
tgytgx
y x h)
=
=
21
sin.cos
21
cos.sin
y x
y x i)
=
=
43
cos.cos
41
sin.sin
y x
y x
k)
=+
=+
3coscos
1sinsin
y x
y x l)
=+
=+
21
cossin
0cossin
22 y x
y xvi 0++
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7/ . 8/
=++=++=++
2logloglog
2logloglog
2logloglog
16164
993
442
y x z
x z y
z y x
=+
=+
1log
433)11(
3 y x x
x x y .
9/
=+
=+
3loglog23
10
33
3/23/13/13/2
y x
y x y x. 10/
+=
=+
)log)(log2(
2
22
22
x y xyee
y x y x
11/ . 12/
=++=++++
+
+
1)4(log)5(log6)12(log)22(log2
21
221
x y x x y x xy
y x
y x
+=+
+=+
23
loglog3log
32
loglog12log
222
333
x y y x
y y x x .
B i 3: Gii cc b i c tham bin:
1/ Cho h ph ng trnh:
=++
=+++
4)(log).(log
4)(log)(log
bxaybyax
bxaybyax
y x
y x
a/ Gii h khi a=3; b=5. b/ Gii bin lun h khi a>0; b>0.
2/ Cho h ph ng trnh:
=++
=+++
4)sincos(log).sincos(log
4)sincos(log)sincos(log
a xa ya ya x
a xa ya ya x
y x
y x
a/ Gii h khi a= /4. b/ Gii v bin lun vi 0
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Gii:
2> Cho h ph ng trnh:2 2
6 x y
x y m
+ =
+ =. Xc nh m h:
a. V nghim. B. C nghim duy nht. C. C 2 nghim phn bit.
Gii:
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3> Cho h ph ng trnh:3 3
1( )
x y x y m x y
+ =
= .
a. Gii h vi m= 3. b. Xc nh gi tr ca m h c ng mt nghim.
Gii:
.
4> Cho h ph ng trnh:2 2
1 (1)
(2)
x y
x y m
+ =
+ =.
a. Gii h vi m= 1/2. b. Xc nh gi tr ca m h c nghim.
Gii:
.
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5> Cho h ph ng trnh:2
2 2
( ) 16
2(1 )
x y
x y m
+ =
+ = +.
a. Gii h vi m= 12. b. Xc nh gi tr ca m h v nghim.
Gii:
6> Cho h ph ng trnh: .2 2 2 6
x y m
x y m
+ =
+ = +
a. Tm iu kin ca m h c nghim.
b. Tm gi tr ln nht , nh nht ca biu thc F= xy+2(x+y).Gii:
7> Cho h ph ng trnh:2
2 2
( ) 4
2(1 )
x y
x y m
+ = + = + .
Xc nh gi tr ca m h c ng 2 nghim.
Gii:
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8> Tm m h sau c nghim duy nht:2 2
4 x y x y xy m
+ =
+ =
Gii:
9> Cho h ph ng trnh:2 2
1
( ) 1
x y2 xy x y m
+ =
+ =
a. Gii h vi m=3. b. CMR: h c nghim vi mi gi tr ca m.
Gii:
10>Cho h ph ng trnh: .2
0
2 2 3
x y m
y x m
=
+ = 0
2
a. Gii h vi m=1.b. Tm m h c 2 cp nghim phn bit (x1; y1) v (x2; y2) tha mn: 2 2 21 1 2 2 x y x y+ = + .
Gii:
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11> Cho h ph ng trnh: .2 2 0
0
x y x
x my m
+ =
+ =
a. Gii h khi m=1b. Bin lun s nghim ca h theo m.
c. Tm m h cho c 2 nghim phn bit .
d. Gi (x1; y1) v (x2; y2) l cc nghim ca h cho. CMR: 2 22 1 2 1( ) ( ) x x y y 1 + .
e. Khi h c 2 nghim phn bit. Tm m A= 22 1 2 1( ) (2) x x y y + t GTLN.
Gii:
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B i tp t luyn:
12> Cho h ph ng trnh: . Tm m h c 2 nghim tha mn:2 2 1 0
4
x y mx my m
x y
+ + + =
+ =
2 22 1 2 1( ) ( ) x x y y + =4.
13> Cho h ph ng trnh: . Tm m h c nghim duy nht.2 29 16 144 x y
x y m
=
=
14> Cho h ph ng trnh: . Gii m=1, tm m h c nghim duy nht.2 29 4 3
2 5
x y
x my
+ =
+ =6
15> Cho h ph ng trnh:3 3 x y m
x y m
+ =
+ =. Gii h vi m=4, CMR: m h c nghim.