# Tài liệu TÀI LIỆU LUYỆN THI ĐẠI HỌC VÀ THPT CHUYÊN; MÔN TOÁN; CHUYÊN ĐỀ HỆ PHƯƠNG TRÌNH ; BÀI TẬP GIẢI HỆ PHƯƠNG TRÌNH ĐỐI XỨNG (PHẦN 1) pot

CREATED BY HOÀNG MINH THI; XYZ1431988@GMAIL.COM 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.
2,
5,
1.
208,
6,
96.
55,
7,
24.
7,
8,
3.
2
9,
x xy y
xy x y
x y
x y
x xy y
x y xy
x y x y
xy x y
x y x y
xy x y
x y
xy
x y
xy
x y xy
x y xy
x

+ + =

+ + =

− =

+ =

− + = −

+ + =

+ + + =

+ + =

+ − + =

+ = −

+ =

=

− =

=

+ + =

+ − =

( )
( )
2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
1,
0.
2,
10,
1.
1,
11,
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
2 2
2 2
3 3
4,
17,
4,
1,
18,
7,
5,
19,
35,
2,
20,
2,
4,
21,
2,
7,
22,
2.
3 3 ,
23,
160.
5 3,
24,
2.
x y xy
xy x y x y
x y
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
3 3
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
x xy y
x xy y
x xy y
x xy y
x xy y
x y
x xy y
x x y y
x y
x x y y
x y xy
x y xy

+ + =

+ =

+ + =

− + =

+ + =

+ + =

+ + = +

+ =

+ + =

+ + =

+ =

− + =

− − =
+ + =
3 3
.
2 2,
32,
8.
x y xy
x y

+ + =

+ =

CREATED BY HOÀNG MINH THI; XYZ1431988@GMAIL.COM TRUNG ĐOÀN 1 – SƯ ĐOÀN 3 – QUÂN ĐOÀN TĂNG THIẾT GIÁP
2

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

( )
( )
( )( )
( )( )
2 2
3 3
3 3
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
x y
xy x y
x y xy
xy
x y
x y x y
xy x y
x y x y xy
x y xy xy
x y
x y
x y
x y

+ =

+ =

 − = −

− =

− =

− =

+ + =

+ + =

+ + + =

+ + =

+ + = +

− + = −

+ + + =
+ + + =
( )
2 2
3 3
2 2
4 2 2 4
4 4
6 6
2 2
2 2
9.
2,
8,
2.
10,
9,
73.
1,
10,
1.
1
13,
11,
1
12.
1
1 4,
12,
1
4.
7 5,
13,
1.
1
x y
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

+ = + + −

+ = −

( )
( )
( )
( )
( )
( )
( )
2 2
2 2
2 2
2 2
2
2 2
2
2 2
3
3 3
2
2
,
1 1 3
15,
1
1 6.
15,
16,
85.
1
1 6,
17,
1
1 18.
1
1 9,
18,
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 ,
1 1
4.
1 1
1 1 4,
20,
1 4 .
1
1 5,
21,
1
4.
10
,
3
22,
5.
20,
23,
1 1 5
.
4
3 ,
24,
1 1
x y xy
y x
x y x y
x x
y y
x y x y xy y
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

+ + =

+ + + =

 
+ + + =

 

 

+ + + =

 
+ + =

 

 

+ =

+ −

+ =

− +

+ =

+ =

+ =

+ + =
+ − 1.

=

CREATED BY HOÀNG MINH THI; XYZ1431988@GMAIL.COM TRUNG ĐOÀN 1 – SƯ ĐOÀN 3 – QUÂN ĐOÀN TĂNG THIẾT GIÁP
3

Bài 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,
3 3 5.
1,
4,
2.
7,
5,
13.
3 ,
6,
1 1
1.
2,
7,
1 1
4.
8,
x y xy
x y xy
x y xy
x y x y
x y x y xy
x y xy
x y xy
x y
x xy y
x xy y
x y x y xy
xy
x y
x y
y x
x y
x y
+ + =

+ − = −

− − = −

+ − + = −

+ + + − =

+ = +

− + =

+ =

+ + =

+ + =

+ + =

+ = +

+ =

+ + + =

( ) ( )
( )
( )
( )
( )
( )
2 2
2 2
2 2 2 2
2 2 2 2
2 2
2 2
2 2
2 2
2
1
,
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
x y
xy
xy xy
x y
xy
xy y x
x y xy
x y xy
x y x y
x y x

+ =

+ +

= + +

+ + =

+ − =

+ + =

+ +

+ =

+
+ + =

− − + =

− =

+ =

− − =
+ +
( )
2
2 2
175.
4 2,
14,
2 .
y
x y xy
x y xy

=

+ − =

= −

( ) ( )
( )( )
( ) ( )
( )( )
( )
( )
( )
2 2
4 4
2 2
2 2
2 2
2 2
2 2
3 3
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
xy x y
x y
x y
x y x y
x y
x y
x y
xy x y
x y

+ + =

+ =

+ + + =

+ + =

+ + + =

+ + =

− + − =

− − =

− =

+ + =

+ =

+

=

+

− =

( )
3 3
4 4
4 4
9 9
2 2
2 2
4 4
2 2
2 2
2 2
2
6.
1,
22,
1.
1,
23,
1.
12,
24,
1
.
3
82,
25,
3.
2,
26,
1.
2 ,
27,
2 .
7 7 16 ,
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; XYZ1431988@GMAIL.COM TRUNG ĐOÀN 1 – SƯ ĐOÀN 3 – QUÂN ĐOÀN TĂNG THIẾT GIÁP
4

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

( ) ( )
( )( )
( )
3 3
2
3 3
3 3
2 2
2 2
2 2
4 4
4 2 2 4
2 2
3 3 3 3
5,
1,
1 1 35.
4,
2,
.
1 1
5,
3,
1 1
22.
1 1
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.
3 3 9,
9,
3,
3 5 ,
10,
4 3 .
4 4 10,
11,
2 4.
2 9 9 ,
12,
3
3.
2 3 2 ,
13,
2 3
x y xy
x y
y x
x y xy
x y xy x y
xy x y
xy y x y
y xy x
x
x xy y
y
x
x
y
x y xy y x y
x
x y
y
xy y y x
x y

+ + =

+ =

+ + =

+ + + + =

+ + =

+ + =

− =

+ + + =

+ =

− + + =

− + =

− = −
− 2 .x y

=

( )
( )
( )
3 3
2 2
3 3
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
x y xy
x y x y xy
x y xy
x y
x y xy x y
x y
x y
x y
x y
xy x y
xy x y xy
x y xy
x y x y
x

+ + =

+ + =

+ + + + =

+ + =

+ =

+ + + =

+ + + =

+ + + =

+ =

+ − =

+ = −

+ = +

( )
2 2
2
2
2 2
2 2
2 2
2
2
2
4 2 2
2
2
2,
2 .
3 1 ,
21,
3 4 .
4 2,
22,
1,
3
2 2,
23,
3 9
6 4 3.
2 2 ,
24,
2 4 4 2 .
3 2,
25,
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

+ + =

= +

−
+ =

CREATED BY HOÀNG MINH THI; XYZ1431988@GMAIL.COM TRUNG ĐOÀN 1 – SƯ ĐOÀN 3 – QUÂN ĐOÀN TĂNG THIẾT GIÁP
5

Bài 5.
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 1,
1,
2 2 2 3.
2 2 1,
2,
2 3 2 4 .
1,
3,
2 3.
5 2 2 26,
4,
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 ,
10,
2 4 4 6.
2 2 2 1,
11,
9 16 4 .
2
12,
x y xy y xy y
xy y y
x y xy
x y y xy
x y xy
x y x xy y
x y xy
x y x xy
x y x xy
x x xy y
x y

+ + + + =

+ + =

+ + =

+ + + =

+ = −

+ + + = +

+ = +

− + = +

+ = + +

+ = + +

+ =
( )
2 2
2
2 2
2 2
2 2 2 2 2
2 ,
8.
4 6 3 ,
13,
4 2 4 .
10 10 5 18 9,
14,
2 4 1 .
4 4 2 ,
15,
3 4 2 .
2 2 1,
16,
4
1 .
x y
x y xy
xy y x
x y x y
xy x x y
x y x y
x y x y
xy x y
x y xy x x y
y
x y
x

+

+ + =

= +

+ = +

+ = + +

+ + = +

+ = +

+ = +

− + + = −

+ + =

( )
( )
( )
( )
( )
2 2
2
2 2
2
2 2
2
2 2 2 2 2 2
4 4 3 2 2 2
2 2
2
2
2 3 9,
17,
2 1.
4 2 4 13,
18,
2 2 1 1.
3 1 6 2 2,
19,
5 4 2 1.
3 6 ,
20,
2 2 .
2 2 ,
21,
2 1 .
1 1
22,
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
+ = +

+ = +

+ + = +

+ = + +

+ = + +

+ − = +

+ =

+ − + = +

+ − + =

+ = + +

+ +
( )( )
( )
( )
( )
( )
2
2
2
2 2 2 2 2
2
2
2 2 2 2 2
2 2
2 2
2
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 ,
30,
1 2 1 .
2 1 1 3 ,
31,
2 2 3 .
1 3,
32,
1 3 .
y
x y xy y x xy x y y
x x y x
x x y
x xy y x
x xy y x
y xy x y x
x y y y
y x x y
x x y xy
y x y
y x y x
− =

+ + + + + =

 + − =

+ + =

= + +

+ = +

− + + = +

+ − = +

− + = +

= − +

− = +

+ = +

CREATED BY HOÀNG MINH THI; XYZ1431988@GMAIL.COM TRUNG ĐOÀN 1 – SƯ ĐOÀN 3 – QUÂN ĐOÀN TĂNG THIẾT GIÁP
6

Bài 6.
Giải các hệ phương trình sau trên tập hợp số thực
( )
( ) ( )
( )
( )
( )
( )
2 2 4 2 3
2
2
2 2
2
2 2
2 2 2 2
2
2
4 2 2 2 2 2
2
2
2 2
2 2
1 4 2 4 ,
1,
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,
5 10 2 2 7.
3 9 5,
10,
3 3 3.
2 1 1,
11,
5 4 5 .
2 3 2
12,
xy y
x xy y
y x x
xy x x
x x y x
x xy y
x xy x y
xy x y
xy x y
x x y y xy x
xy x y x y
x x y

+

+ + =

+ + =

+ − =

+ = + +

− + =

− + = +

+ = +

+ − =

+ + + − = +

+ − = +

− + =
( )
( )
( )
( )
( )
( )
( )
( )
( )
( ) ( )
2 2
2
2
2
2
2 2
2
2
2
2
2
7,
2 4 4 2 1.
1 4 ,
13,
1 2 .
2 3 3 ,
14,
2 3 1 .
3 5 1,
15,
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
 +

− + = +

+ + + =

+ + − =

+ + = +

+ − − =

+ = + +

+ − − =

+ + + =

− + + + = −

(
)
( )
4 3 2 2 2
3 2 2
2 2
2
2
2 2
2 2
2
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
4,
2
26,
2 .
3
2
2 1
27,
xy
x x
y
x y y x x
x y
x y y x
x y y
x y x y
x xy x y
x y
x
x
x y
x y
y
x x y x y
x y
y x
y

− + =

+ = + +

+ − =

+ = +

+ = +

− − + =

+ − + =

− −
=

+ −
+ = +

+ = + +

+
= − +

( )( )
( )
( )
( )
2
2
2 2
2
5,
2 3
2 .
2 2
1 2 1 1,
28,
2 2 2 1.
2
2 1,
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

+ −

+ + =

− −

+ − − = +

− + + = + +

+ − =

− −

=

− +

+ = +

− +

=

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