176501487-giẢi-hỆ-phƯƠng-trinh-bẰng-phƯƠng-phap-ĐỒng-bẬc
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GII H PHNG TRNH BNG PHNG PHP NG BC
Cc bi ton v h phng trnh thng xut hin trong cc k thi i hc, Cao ng. gip cc bn hc sinh n tp tt v phn ny, bi vit ny xin nu ra mt phng php hiu qu gii quyt mt lp cc h phng trnh l phng php ng bc.
Th d 1: Gii h phng trnh
( )( )
2 2
2 2
2 3 9 1
4 5 5 2
x xy y
x xy y
+ = + =
Gii: Ly (1) nhn 5 v (2) nhn 9 ta c phng trnh ng bc
( ) ( ) ( ) ( )2 2 2 2 2 2 55 2 3 9 4 5 4 26 30 0 5 2 3 0 2 3x y
x xy y x xy y x xy y x y x yx y=
+ = + + = = =
Vi 5x y=
thay vo (1) ta c
2 2 1 218 92 2
y y y= = =
tng ng
5 22
x =
.
Vi
32yx =
thay vo (1) ta c 2 4 2y y= =
tng ng 3x =
.
Vy h phng trnh c bn nghim l
( ) ( )5 2 2 5 2 2; ; ; ; 3;2 ; 3; 2 .2 2 2 2
Th d 2: Gii h phng trnh ( )
2 2
3 3
30 (1)35 2
x y y xx y
+ =+ =
Phng trnh ny l phng trnh i xng loi mt tuy nhin chng ta cng c th gii theo phng php ng bc.Gii: Ly (1) nhn 7 v (2) nhn 6 ta c phng trnh ng bc
( ) ( ) ( ) ( ) ( )2 2 3 3 3 2 2 3 37 6 6 7 7 6 0 2 3 3 2 0 223
x y
x y y x x y x x y y x y x y x y x y x y
x y
= + = + = + = = =
Vi x y=
thay vo (2) suy ra v nghim.
Vi
32
x y=
thay vo (2) ta c 3 8 2y y= =
suy ra 3x =
.
1
-
Vi
23
x y=
thay vo (2) ta c 3 27 3y y= =
suy ra 2x =
.
Vy h c nghim l ( ) ( )3;2 , 2;3
.
Th d 3: Gii h phng trnh
( )( )
2 2
3 3
2 1 1
2 2 2
y x
x y y x
= =
Gii: T (1) v (2) ta c phng trnh ng bc( ) ( ) ( ) ( )
( )
3 3 2 2 3 2 2 3 2 2
2 2
2 2 2 2 2 5 0 3 5 0
3 5 0 3
x y y x y x x x y xy y x y x xy y
x yx xy y
= + + = + + =
=
+ + =
Vi x y=
thay vo (1) ta c 2 1 1y y= =
.
Ta c
22 2 23 113 5 0
2 4x xy y x y y + + = + +
. R rng 0x y= =
khng phi l nghim h phng trnh. Vy (3) v nghim.
Vy h cho c nghim l ( ) ( )1;1 , 1; 1
.
Th d 4: Gii h phng trnh
( )( )
2 1
5 3 2
x y x y y
x y
+ + =+ =
Gii: iu kin ca phng trnh 0x y
Phng trnh (1) ca h l phng trnh ng bc
( )2 2 2 2
22 2
2
2 02 2x+2 4 2
2
22
05 4 0
5 4 0
y xx y x y y x y y x y y x
x y y x
y xy x
yy xy
y x
+ + = = =
=
=
= =
Vi 0y =
thay vo (2) ta suy ra 9x =
(loi)
2
-
Vi 5 4 0y x =
thay vo (2) ta c
41 15
x x y= = =
(tha mn).
Vy h phng trnh c nghim l
41;5
.
Th d 5: Gii h phng trnh
2 2
5 5
3 3
3317
x xy yx yx y
+ + = += +
Gii: iu kin ca phng trnh x y
( )( ) ( ) ( )
2 22 2
5 55 5 3 3
3 3
3 3 131
7 31 27
x xy y x xy yx y
x y x yx y
+ + = + + = +
= + = + +Ly (2) nhn 3 kt hp vi (1) ta c phng trnh ng bc
( ) ( ) ( ) ( )5 5 2 2 3 3 5 4 3 2 4 421 31 10 31 31 31 10 0 3x y x xy y x y x x y x y xy y+ = + + + + + + + =.
R rng 0x y= =
khng phi l nghim h phng trnh. t x ty=
thay vo (3) ta c:
( )( ) ( )
5 5 4 3 5 4 3
4 3 24 3 2
10 31 31 31 10 0 10 31 31 31 10 0
1 01 10 21 10 21 10 0
10 21 10 21 10 0
y t t t t t t t t
tt t t t t
t t t t
+ + + + = + + + + =
+ = + + + + + =
+ + + + =
Vi 1 0 1t t+ = =
hay 0x y x y= + =
(loi).
Vi ( )4 3 210 21 10 21 10 0 3t t t t+ + + + =
. V 0t =
khng phi l nghim ca phng trnh
(3) chia hai v phng trnh cho 2t ta c:
22
1 110 21 10 0t tt t
+ + + + =
,
t
2 2 2 22 2
1 1 12; u 2 2u t u t t ut t t
= + = + + + =
. Khi (3) tr thnh
2
2510 21 10 0
52
uu u
u
=
+ = =
3
(loi)
-
Vi
52
u =
ta c
22
1 5 2 5 2 0 122
tt t t
t t
= + = + + = =
Vi 2t =
ta c 2x y=
th vo (1) ta c 2 23 3 1 1y y y= = =
tng ng 2x =m
.
Vi
12
t =
ta c 2y x=
th vo (1) ta c 2 23 3 1 1x x x= = =
tng ng 2y =m
.
Vy h cho c bn nghim l ( ) ( ) ( ) ( )1; 2 , 1;2 , 2; 1 , 2;1 .
Th d 6: Gii h phng trnh
3 4
2 2 3
7 2 9
x y yx y xy y
=+ + =
Gii:
( ) ( )( ) ( )
3 33 4
2 2 3 2
7 17 2 9 9 2
y x yx y yx y xy y y x y
= =
+ + = + =
T h suy ra .y 0; y, y 0x x >
. Ly phng trnh (1) ly tha ba, phng trnh (2) ly tha bn. Ly hai phng
trnh thu c chia cho nhau ta thu c phng trnh ng bc:
( )( )
33 3 3 3
8 44
79
y x y
y x y
=
+
.
t x ty=
ta c phng trnh:
( )( ) ( )
33 3
8 4
1 7 391
t
t
=
+
. T phng trnh ny suy ra 1t >.
Xt
( ) ( )( )33
8
1; t 1.
1
tf t
t
= >+
( ) ( ) ( ) ( ) ( )( )( ) ( ) ( )
( )( ) ( ) ( )
( )
2 3 28 7 72 3 3 3 3 2 3
8 8
2 73 3 2
8
9 1 1 8 1 1 1 1 9 9 8 8f'
1 1
1 1 9 80 1
1
t t t t t t t t t tt
t t
t t t tt
t
+ + + + += =
+ +
+ + += > >
+
4
-
Vy f(t) ng bin vi mi 1t >. Nhn thy
2t = l nghim ca (3). Vy
2t = l
nghim duy nht. Vi 2t =
ta c 2x y=
th vo (1) ta c 4 1 1y y= =
(v 0y >
)
suy ra 2x =
.
Vy h c nghim l ( )2;1
.
Bi tp t lmGii cc h phng trnh sau
Bi 1:
3 3 2
4 4
14 4x y xyx y x y
+ =+ = +
.
Bi 2: ( )
3 3
2 2
8 2
3 3 1
x x y y
x y
= + = +
.
Bi 3:
2 0
1 4 1 2
x y xy
x y
=
+ =
Bi 4:
3 2 2 36 9 4 0
2
x x y xy y
x y x y
+ = + + =
Bi 5:
4 4
6 6
11
x yx y
+ =+ =
Bi 6:
5 5
9 9 4 4
1x yx y x y
+ =+ = +
Bi 7:
( ) ( )( ) ( )
2 2
2 2
13
25
x y x y
x y x y
+ =+ =
Bi 8:
2 2
2 2 2
x xy y 3(x y)x xy y 7(x y)
+ = + + =
Tc gi
5
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L Xun ThngGV THPT Triu Sn 4, Triu Sn, Thanh Ha
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