CREATED BY HOÀNG MINH THI; TRUNG ĐOÀN 1 – SƯ ĐOÀN 3 – QUÂN ĐOÀN TĂNG THIẾT GIÁP
1

CHUYÊN ĐỀ HỆ PHƯƠNG TRÌNH

BÀI TẬP GIẢI HỆ PHƯƠNG TRÌNH ĐỐI XỨNG (PHẦN 1)

Bài 1. Giải các hệ phương trình sau trên tập hợp số thực

2 2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
4,
1,
2.
2,
2,
164.
5 3,
3,
3 5.
8,
4,
5.

x

+ + =

+ + =

− =


+ =


− + = −

+ + =


+ + + =

+ + =


+ − + =

+ = −


+ =

=

1.
1,
12,
3.
3 3 7 13,
13,
5 7.
2 2 ,
14,
6.
5 5 ,
15,
1.
16,
y xy
x y xy
x y x y
xy x y
x y xy
xy x y
x xy y
x y xy
x y xy
x y xy
x y xy
x y
x y xy x y
x y xy
x y x


+ =



+ = − + +

+ − =

+ +
2 2
3,
2.
y
x y y x
=


+ =


( )
( )
( )
( )
2 2
3 3
3 3
3 3
3 3
3 3

x y
x y
x y
x y
xy x y
x y x y
xy x y
x y
x y y x
x y xy
x y
x y xy x y
x y

+ + =


+ + + =


+ =


+ =

+ =


+ =


2 2
2 2
2 2
4 2 2 4
2 2
4 2 2 4
2 2
5,
25,
2.
11,
26,
1.
4,
27,
2.
2 3 2,
28,
14.
7,
29,
21.
5,
30,
13.
5,
31,
13
x y xy
xy x y

+ + =

+ + =


+ + = +


+ =



+ + =


+ + =



+ =


− + =


− − =
+ + =
3 3
.
2 2,

2 2 2 2
2 2
2 2
30,
1,
35.
2,
2,
2.
7,
3,
2.
11,
4,
6 6
11.
18,
5,
1 1 72.
1 2 ,
6,
1 1 .
1 1
5,
7,
1 1
x y xy
x y
xy x y
x y

− =


+ + =



+ + =



+ + + =


+ + =



+ + = +


− + = −


+ + + =
+ + + =
( )
2 2
3 3
2 2

x y
x y
x x y y
x y
x y
x y
xy
xy y x
x y
xy
xy y x
x y
xy
x y
xy
xy xy
x y xy
x y xy








+ =


+ =


 

+

+ + =



+ = −

+ = +

2 2
4 4,
4,
5 3.
x y x y xy
x y xy

+ = + + −

+ = −


( )
( )
( )
( )
( )

1
1 27.
19,
x y
y x
x y
xy
x y
x y
y x
x y
x y
y x
x y
xy
x y
xy
x y
xy
x y
xy
x y

+ =

+ +


 


+ + =
 

 


 
+ + =

 

 

 

+ + =
 

 

+
( )
( )
2
2 2
3 3 2 2 3
2 2
2 2
2 2
4 ,

x y
xy
xy
xy
x y x y
x y x y
x y
x y y x
x y
x y x y xy
xy
x y

+ + =


+ + + =



 
+ + + =

 

 

+ + + =





=



CREATED BY HOÀNG MINH THI; TRUNG ĐOÀN 1 – SƯ ĐOÀN 3 – QUÂN ĐOÀN TĂNG THIẾT GIÁP
3Bài 3.
Giải các hệ phương trình sau trên tập hợp số thực

2 2
2 2
2 2
2 2
2 2
2 2
3,
1,
3 1.
4 4,
2,
5 5 8.
3 1,
3,

y x
x y
x y
+ + =


+ − = −

− − = −


+ − + = −


+ + + − =

+ = +

− + =


+ =

+ + =


+ + =


+ + =

,
2
1 1
3 1.
3 ,
9,
.
1 1 1
3,
10,
2 2
6.
1
13,
11,
1
12.
12,
12,
13.
7,
13,
x y
y x
xy x y
x y xy x y
x y xy x y
x x xy
x y x y
xy x y

+ +

+ =



+
+ + =




− − + =


− =


+ =


− − =
+ +
( )
2
2 2
175.
4 2,
14,
2 .

2 2
3 3
3,
15,
17.
18,
16,
1 1 72.
4 4 10,
17,
4.
3 3 8,
18,
3 3 4.
3
,
19,
15.
2,
20,
1.
2,
21,
x xy y
x y
x x y y
x x y y
x y y x
xy x xy y
x y


− + − =


− − =



− =

−


+ + =

+ =


+

=

+

− =
−
( )
3 3
4 4
4 4

28,
6 2.
29,
x y
x y
x y
x y
x y
x y
y x
x y xy
x y
xy
x y
xy
x y xy
x y xy
x y x y xy
x y xy
x y
xy



− + =



+ =


+ =


+ =


+ + + =


+ = −


+
=
2 2
2,
4 .
x y x y xy




+ + + =
CREATED BY HOÀNG MINH THI; TRUNG ĐOÀN 1 – SƯ ĐOÀN 3 – QUÂN ĐOÀN TĂNG THIẾT GIÁP
4

Bài 4.

4,
1 1 4 .
78,
5,
97.
481,
6,
37.
1
7,
xy x y
x y
x
x y
y
x xy y
x y
x y
x y
x y
x y
x y
x y xy
x y xy
x y
x x y y
x xy y
x y x y
+ + =


+ =


+ =



+ + =


+ + =


+ + =
2 2
2 2
2
2
2
2 2
2
2
2 2 3 4 2 2
2
2
7,
5.
2,
8,
3.

x y xy y x y
x
x y
y
xy y y x
x y


+ + =


+ =



+ + =


+ + + + =

+ + =


+ + =


− =




2 2
3 3
3 3
3 3
2 2
2 2
3 3
2 2
3,
14,
3.
5,
15,
2 4.
2,
16,
3 8.
1 1
4,
17,
1 1
4.
2,
18,
3 3 5 2.
3 1,
19,
.
20,
x y xy



+ + + =



+ + + =




+ + + =



+ =


+ − =



+ = −


+ = +


( )
2 2

1
26,
x y
y
y
x y y
xy y y
x y y
x y xy
x y xy x y
x
x y
y
x y x
x x y x y
y
x y x
x y y x
x y x
xy y
x y
+

− + =



− =




+ = +

+ =

+ = +
5 ,
1 5 .
2 1,
27,
1 3
1.
x
xy y x
xy y
y
x
x


+ + =

= +


−
+ =




3 2 11.
2
2 1 0,
5,
4
4 4 1 0.
4 1,
6,
2 4 2.
x xy y
x x y xy
x xy y
x x y xy
x y
x x y
x xy y
x x y x y
x y
x
x xy y
x
x y
x x y

+ + =


+ + + =




+ =


+ = +


( )
( )
2 2 3 4 2
2
2 2
2
2 2
2 2
2 2
2
2 2
2
2 2
2 1 2 ,
7,
2 3 .
5 2 2,
8,
3 9 2 2 7.
2 5 2 1 ,
9,
4 2 2 5.
2 2 1 ,




+ = −


+ + + = +



+ = +


− + = +



+ = + +


+ = + +


+ =
( )
2 2
2
2 2
2 2
2 2 2 2 2

+ + =

= +


+ = +


+ = + +


+ + = +



+ = +

+ = +


− + + = −


+ + =



( )
( )
( )

y xy x
x y y
x y x xy
x y x y
y x y x
x y xy y
x xy y
x x y xy x y xy y
x y x x x y
x y y x x
x y
+ = +


+ = +


+ + = +


+ = + +



+ = + +


+ − = +



1 1 4 ,
2 3 5 2.
4 2 4 ,
23,
1 3 .
2 2 ,
24,
2 4 4 .
4 5 4,
25,
3 2 5 2 .
5 7 4 3 ,
26,
2 2 3 11.
1
27,
x y y y
xy y x x
x x y x y x y xy
x y xy
xy x x y
x y xy x y x
x y xy x y
xy x y
x y x y
x xy x y
x y y

+ + + =


+ +
( )
( )
( )( )
( ) ( )
( )
( )
( )
( )
( )
2 2 3 4 2 2
2
2 2
2
2
2 2 2
2
2
2
2
2
2 0,
2 3 .
2 3 1 14,
28,
3 9.
2 ,
29,
2 .
1 2 ,




= + +


+ = +



− + + = +


+ − = +



− + = +


= − +



− = +


+ = +



1 5 2 .
2 1 2 ,
2,
2 .
2 ,
3,
1 1 .
1 3 1,
4,
4 .
1 2 ,
5,
2 1.
1 2 ,
6,
1 2
x y y y xy
xy y y
x xy x
x xy x y
x y y x x x y
x x xy
x y x x xy y
x y x y x y
x xy y x
xy x x
xy y
x y

+ + = +

− = −


= +
+ =
( )
( )
( )
( )
2
2 2
2
2 2
2
2 2
2
2 2
2
2
4 3 2 2
2 2
2 .
2 4 2,
7,
4 2 1.
1 2 ,
8,
1 2 3 .
3 8 4 3,
9,


+ − =


+ = + +



− + =


− + = +



+ = +


+ − =



+ + + − = +


+ − = +


− + =
( )

3 1 3 .
2 2 7 ,
16,
3 1 1 8 3.
y
x xy y x
x y y x y
x x y y
x xy y y
x x y y
x xy y y
y x x y
y x y y
x x y x y
 +


− + = +



+ + + =


+ + − =



+ + = +


2 2
2
3 2 1 ,
17,
3 4 4 3 2 .
3 2 3 2,
18,
2 2 5.
2 2 1,
19,
2 2 1.
1 2,
20,
1.
3,
21,
2 3 3 7.
x x x y y x y x xy
x x y x y x
x xy y x y xy
x xy y
xy x y y
x xy y x y
x y x y y
xy x x y
x xy y
x xy y

+ + + + + = + + +


( )
2
2
2
2
2
2 2
2 2
2 2
2
2
2
3 2 2
2
1 2
5 4 3,
22,
2 1 1 2 .
1 1,
23,
2 4.
2 3 7,
24,
2 4 4 5 0.
2 3 4 ,
25,
3
2 .
2
2

−

− + =



+ = + +


+ − =


+ = +



+ = +


− − + =



+ − + =


− −
=

−

2
29,
2 1
.
1
5 1,
30,
4 3
1 .
x y
x y
x y
y x y x
y x y x x y
x y
y x
y
y x
x
y x
xy x y
x x
y




+ −

+ + =

