CREATED BY HOÀNG MINH THI; TRUNG ĐOÀN 1 – SƯ ĐOÀN 4 – QUÂN ĐOÀN TĂNG THIẾT GIÁP
1

CHUYÊN ĐỀ HỆ PHƯƠNG TRÌNH

BÀI TẬP GIẢI HỆ PHƯƠNG TRÌNH ĐỒNG BẬC (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 2
2 2
3
2 3 15,
1,
2 8.
2 3 9,
2,
2 2 2.
2 4 1,
3,
3 2 2 7.

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

10,
2 3 16.
2 15,
11,
8 35.
2 3 4 1,
12,
2 1.
3 4,
13,
4 1.
14,
y
xy
x
y x y x
x x y y
x xy y
x xy y
y x
x y xy
x y xy
x y
x y xy
x y
y xy
x xy y




+ =



+ = +


= +



− =


− + =


2 2
2 2
3 2 11,
2 3 17.
x xy y
x xy y

+ + =


+ + =



3 .
x xy y
x xy y
x xy y
x xy y
x xy y
x xy y
x y xy
xy y
x y xy xy
x y
x y x y
x y x y

− + + =


+ + =



− + =


− + =



+ + =


2 2
3 3
2 2
3 3
2 2
3 3
3 2
3 2 4 ,
21,
3 3 3 5 .
5 7,
22,
3 4.
3 4 0,
23,
2 3 .
2,
24,
2 2 .
3 4,
25,
3 5 2 6 .
2,
26,
5 2 4 3 .
2
27,
x y y x
x y xy y x
x y yx

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



+ =


CREATED BY HOÀNG MINH THI; TRUNG ĐOÀN 1 – SƯ ĐOÀN 4 – QUÂN ĐOÀN TĂNG THIẾT GIÁP
2

Bài 2.
Giải các hệ phương trình sau trên tập hợp số thực
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2 2
2 2
2
3
2
3

8,
x xy y
xy y x
x x y y
x y x
x
y
y
y
x
x
x x y
y y x
x x y
y x y
y x y
x x y
x y yx
x y xy
x xy y

+ + =


+ + =



+ = +


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

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

2 3 26.
x
y y x
x yx y x y
x x y

=


= + −



+ + = + −


+ + =



(
)
( )
( )
( ) ( )
( )
( ) ( )
( )( )
( )( )
2

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

2 2
3 3 2 2
4 2 4 2
2
.
5 2
4 3 ,
22,
5 4 .
2 1 ,
23,
4 3 .
4 3 ,
24,
2 0.
3 1 0,
25,
2 3 4.
5,
26,
1.
13,
27,
468.
28,
x y
x y x y
y x y x
x
x

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

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

Bài 3.
Giải các hệ phương trình sau trên tập hợp số thực
( )
( )
( )
( )
( )
( )

2000 0,
6,
500 .
2 2
7,
x xy y x y
x xy y x y
x x y y
x y
x y y
y x
x y
x y
x xy y
y x
x y xy
x xy y
y yx x
x y x y

− + = −


+ + = −



− = +




− +
( )
( )
( )
( )
( )
2
2 2
2
3 3
2
3 3
2
2
5 5
3 3
2 2
3 2
2 2
3
3 3 3 3
3 3
3,
2 2 1.
9,
8,
7.
4,
9,

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

15,
2 .
x y xy
x y xy
x x y



− + =



+ − =


= +



( )
2 2
3 3
3 3
2 2 3
3 2
3 2
2 2
5 5
3 3
2 2

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

4 4
3 2 3
4 4
2 2
2 2
3 3
2 2
25.
6 2 56,
23,
5 49.
2 3 5,
24,
6 7.
1,
25,
4 4 .
4 8 1,
26,
2 8 2 .
1 2 4 2 ,
27,
2 .
4 2 ,
28,
1 3 1
x xy y
x xy y
x x y
y xy



+ = +



− + =


+ = +



− = +


+ − =


− = +
− = −
3 2
2 2
2 2
3 3 2 2
2 2
2 2
.
,
29,




+ + + =


+ + + =

 CREATED BY HOÀNG MINH THI; TRUNG ĐOÀN 1 – SƯ ĐOÀN 4 – QUÂN ĐOÀN TĂNG THIẾT GIÁP
4

Bài 4.
Giải các hệ phương trình sau trên tập hợp số thực
( )
( ) ( )
( )
( )
2 2
2 2
2 2
2 2
2 2
2 2
2

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

2 3 2 ,
7,
3 3 2 2 .
2 6 ,
8,
3 4 20 .
2 3 5 ,
9,
2 .
5 2 2,
10,
4 3 .
2 1,
11,
2 2 .
3 3 ,
12,
3 3.
2 1 0,
13,
x y x y xy
x y x y xy
x y x y xy
x y x y xy
x x y y
y x y x
x y xy
y x xy
y x
x y y x


− =


− = −



+ = +


− = −


+ =
( )
( )
( )
( )
( )
( )
( )
3 3
2 2
2 2
2
3 3
2 2
2 2
8 9 0.

− =


− =



− =


+ =



3 2 3
2 2
3 2 3
2 2
3 2 3
2 2
2 3 3
2 2
3 2 3
2 2
2 3 3
2 2
2 ,
17,
2 3 7 .
16 2 ,

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

2 3 .
5 3 ,
25,
3 2 3.
3 5 0,
26,
7 8 18 2 .
3 5 ,
27,
3 5 .
3 2 ,
28,
9
x y xy y
x y xy
x y x y
x x y
x x y xy y
xy x y
y xy x y x
xy x y x
xy y x y x
x y x y xy
x x y y y x
x

= + +


+ + =

− = +
( ) (
)
( )
(
)
( )
2
2 2
2 2
3 2
3
2 2 3 3
2 2
3 2 2 3
2 2
20 12 13 4 .
3 2 2 ,
29,
4 1 2 2 1 .
6 4 ,
30,
2 2.
5 2 6 ,
31,
2 2 8.
6 5 3 2 ,
32,
3 .
y x y x



+ + + + =



= + −


+ = +


CREATED BY HOÀNG MINH THI; TRUNG ĐOÀN 1 – SƯ ĐOÀN 4 – QUÂN ĐOÀN TĂNG THIẾT GIÁP
5

Bài 5. Giải các hệ phương trình sau trên tập hợp số thực

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

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

4 6.
,
9,
3 6.
10 4 5 ,
10,
1 2 3 .
10 5 4 ,
11,
y x
x y
x y y x xy
x y xy x
x xy x y y
x y x y
x y x y
x y x x y
x x y xy y
x x y y
x x y xy y
x y x y x y

+ =


+ =



+ = +

3 2 2 3
2 2
3 3 2 2
2 2
3 3 2 2
2 2
3 2 2 3
3 2 2 2
6.
8 6 ,
12,
4 .
9 7 ,
13,
3 .
7 4 2 ,
14,
4 .
3 2 2 ,
15,
2 5 .
2 5 4 ,
16,
3 .
x x y xy y
x x y y y
y x y xy x
x x y xy
y x x y xy
x y x y xy

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

3 3 2 2
2
3 2 2
2 2
3 3 2 2
2 2
2 2 3 3
2
3 2 3 2
2 2
3 2 2 3
2
,
17,

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

27,
3 4 5 12 .
x x y
x y x y xy
x y y
x y x y xy
x x y y
x y x y xy
x x y y
x y x y xy
x x y y xy
x y x y xy
x y xy x



+ = +



+ = +


+ =



− = −



2
2 2 3
2 2
2 2 3 3
2 2
3 7 5 ,
28,
2 3 4 10 .
5 6 ,
29,
3 .
3 7 10 ,
30,
2 4.
4 2 3 ,
31,
3 .
5 8 4 ,
32,
3 5 .
xy x y x y
x x y y xy
x y xy
x x xy y
x xy y
x y x
xy x x y y
x x y xy
x y xy x y
y xy y x

− = −


+ + =



CREATED BY HOÀNG MINH THI; TRUNG ĐOÀN 1 – SƯ ĐOÀN 4 – QUÂN ĐOÀN TĂNG THIẾT GIÁP
6

Bài 6.
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

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

5 4 ,
3 2 4 5 11 .
2 3 2 1 2 4 2 0,
7,
2 2 3 2 1.
3 1 2 1 0,
8,
3 2 .
2 3 8 6 4 ,
9,
2 2 3 1.
5 7 1 5 ,
10,
4 3 1.
x y x y
x y x y xy
x x y y y
x x y
x x y y
x y xy x
x x xy x y y y
x yx y x
x y y x x y
x xy xy

+ = −


+ = + +


(
)
2 2
2
2 2
2
2
2 2
2 2
2
2
2 2
2 2
8 9 2 1 9 ,
11,
2 4 .
7 8 1 2 8 ,
12,
3 4.
1 2 2 1 ,
13,
2 4 2 5 2 .
2 1 3 ,
14,
3 5 .
3 2 ,
15,
2.
16,
x y xy x y




+ + + = + + +


+ + =



− + = −


+ =


2 2
2 2
2 2 0,
1.
x xy y y
x y

+ − − − =


+ =




3 7 11 .
2 2 3 1 2 0,
20,
2 4 4 1 2 2 0.
2 3 2 2
21,
y y x x x
y y x x x
x y x y y
x y y x y
x xy y x x y
x x y xy
x y x y x y
x x y y x y
x y x y x

+ − + − =


+ − = + −



+ + + + =


+ + + + =




2 1
3 4 .
22,
2
4 1,
2
3 4 ,
23,
4 3 2 .
2 3 3 3 ,
24,
2 3 5 .
2 2 ,
25,
2 4 .
2 3 2
26,
y
x y
x y
x y
x y x y x y
x y
x
x y
x y xy x y
x y x y
xy y x x y
x xy y y x
x y xy x


+ + = +



+ = +


+ = −


− = −
3
2 2
3 3 2
2 2
4 4
3 3 2
4 4 2 2
2 2
3 3 2 2
2 2
4 4
3 3 2
,
2 1.
2 3 3 ,
27,
3 3.
,

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