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Tài liệu TÀI LIỆU LUYỆN THI ĐẠI HỌC VÀ THPT CHUYÊN; MÔN TOÁN; CHUYÊN ĐỀ HỆ PHƯƠNG TRÌNH ; BÀI TẬP GIẢI HỆ PHƯƠNG TRÌNH ĐỒNG BẬC (PHẦN 1) doc


CREATED BY HOÀNG MINH THI; XYZ1431988@GMAIL.COM 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.
2,
4,
2 4 2 16.
3 5 4 3,
5,
9 11 8 6.
55,
6,
24.
40,
7,
x xy y
x xy y
x xy y
x xy y
x xy y
x xy y
x xy
x xy y
x xy y
y xy x
x y
xy
x
xy
y

+ + =


+ + =



+ + =


+ + =



− + = −


+ + =



− =


+ − =



− − = −


+ − =



− =

=

+ =
( )
( )
3
2 2
2 2
2 2
2 2
3 3
2 2
2 2
3 3
2 2
2 2
2
2 2
10.
2 3 ,
8,
10 .
2 2 5,
9,
3 3.
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

+ + =


+ + =



2 2
2 2
2 2
2 2
2 2
2 2
3 3 2
2 3
3 3 2 2
3 3
2 2
2 2
2 4 1 0,
15,
3 2 2 7.
2 3 9,
16,
4 5 5.
3,
17,
6 5 4 15.
3 5,
18,
3 4 7.
5,
19,
2 3.
2 2 ,
20,
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
2 2
2 2
2 2
3 3 2
3 3 3
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
x y
x y xy
y x y
x y
x x y x
y x
x y y x
x y
x y x y
x y x


− + =


+ + = +



+ + =


+ =



+ − =


+ =



+ =


+ = +



+ =


+ − =



+ =


+ = +


+
3
3 3
2
2
2
2
3 6,
2 3 5.
15,
28,
10.
3 4,
29,
4 5.
y
x y
x xy
y xy
x xy
y xy

+ =


+ =



+ =


+ =



+ =


+ =





CREATED BY HOÀNG MINH THI; XYZ1431988@GMAIL.COM 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
2
2 2
2 2
2
2
2
2
2
3 5,
1,
3 5.
3 2 1 ,
2,
2 3.
2 3,
3,
2.
3 2,
4,
2 1.
3 2,
5,
2 1.
2 1,
6,
3 2.
3 1,
7,
1.
3 2
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
2 2
2
2 2
2 2
2
2,
2 3.
3 2,
9,
4 3 .
3 ,
10,
2 3 6 1.
3 4 ,
11,
3 4 0.
2 4 2 1,
12,
4 5 3 4.
3 5 3 ,
13,
4 3 .
3
14,
x x y
x x y
y y
x y x y
x y xy
x xy y x y
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
2 2
2
2 2
2
2
4 ,
4 5 2 3.
4 7 3 4,
15,
2 3 26.
x
y y x
x yx y x y
x x y

=


= + −



+ + = + −


+ + =



(
)
( )
( )
( ) ( )
( )
( ) ( )
( )( )
( )( )
2
2
2 2
2
2
2 2
2
2
4 1,
16,
6 6.
2 4 5,
17,
3 4 3 8.
3 4 7 3 2 ,
18,
5 5 3 7 0.
2 2 2 1,
19,
6 8 1.
3 2 2 1,
20,
3 2 6.
2
3 ,
3 2
21,
1
5
x y x y
y y x y
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 2
2 2 2
2
2
3 3
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
y
y
x
x
x y x y
x y y x
x x x y
y y y x
x y
xy x y
x y x y y x
x y y x








− =


− =



− =




− =



− =


− + =



+ − + =


+ − =



+ =


+ =



+ + + =


+ =


( )
( )(
)
3 3
5 5 2 2
3 3
3 3
2 2
1,
.
7,
29,
2.
2 9 2 3 ,
30,
3.
x y
x y x y
x y
xy x y
x y x y xy
x xy y

+ =


+ = +



− =


− =



− = − +


− + =





CREATED BY HOÀNG MINH THI; XYZ1431988@GMAIL.COM 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
( )
( )
( )
( )
( )
( )
2 2
2
2 2
3 3
2 2
3 3
2 2
3
3
2 2
3 2
3 2
2
3 3 ,
1,
7 .
8 2 ,
2,
3 3 1 .
4 16,
3,
1 5 1 .
2 3 1,
4,
2 3.
5,
5,
2 5 2
0.
2
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,
2.
3 0,
10,
2 0.
31
,
7
11,
3.
3 ,
12,
3 1.
2 ,
13,
3 .
1
14,
y x x y
y x y
x y
x x y
x y
xy y x y
x xy y
x y
x y
x xy y
x xy y
x y
x y xy
x y x y
y x

=


− − =



+ =


− =



+ =


+ =



+ + − =


+ − =



+
=

+


+ + =


+ =


+ =



+ =


+ =


− =
5 5
2 2
3
,
0.
1,
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
10 10 8 8
2 2
2 2
3,
16,
2 2 .
1,
17,
2 3.
3 20,
18,
3 7.
5,
19,
11 11 .
9,
20,
6.
1,
21,
.
3 2 11,
22,
5
x y xy
x y y x
x y
x y xy y
x x y
y y x
x y
x y x y
x y
xy x y
x y
x y x y
x xy y
x xy y

+ + =


+ = +



+ =


+ + =



+ =


+ =



+ =


+ = +



+ =


+ =



+ =


+ = +


+ + =
+ +
( ) (
)
( )
2 2
2 2
3 2
3 2
3 3 2
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
x y xy
x y x y
x xy y
x y x y
x y y x y
x xy y x
x y x y
x y



=



− − =


− − =



+ =


+ =



+ − =


+ = +



− + =


+ = +



− = +


+ − =


− = +
− = −
3 2
2 2
2 2
3 3 2 2
2 2
2 2
.
,
29,
3 5.
2 4,
30,
2 5 .
4 ,
31,
2 3 7 .
x x y x y
x y xy
x xy y
x y xy x y x
x xy x y y
y xy x y x






+ = +


+ + =



+ + =


+ + + =



+ + + =


+ + + =







CREATED BY HOÀNG MINH THI; XYZ1431988@GMAIL.COM 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
2 2
2 2
2 2
2
2
2 2
2 2
2 3 ,
1,
3 6 1.
3 4 0,
2,
5 9 15 .
2 5 2 0,
3,
8 7 1.
2 ,
4,
3 2 2 7 .
2 4 ,
5,
4 5.
2 2 ,
6,
3 4.
x y xy
x y xy
x xy y
x y xy
x xy y
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 2
2 2
2 2
3 3
3 2
2 2
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
x x y x xy
y xy x
xy


− − =


− − =


+ + =


+ + =


+ =


+ =



= − −


= −



− =


− = −



+ = +


− = −


+ =
( )
( )
( )
( )
( )
( )
( )
3 3
2 2
2 2
2
3 3
2 2
2 2
8 9 0.
13,
14,
25.
2,
15,
19.
4 3 ,
16,
2.
x y x y
x y x y
x y x y
x y y
x y
xy x y
x y



− + − =



− + =


+ − =



− =


− =



− =


+ =



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 ,
18,
3 5 .
2 3 ,
19,
3 .
2 3 ,
20,
3 .
3 4 ,
21,
2 3 .
4 3 0,
22,
.
4
23,
x xy y
x y x y xy
x xy y
x y xy x
y xy x
x x y y
xy x y
x xy y y
y xy x
x y x
xy x y
x y xy x
x

= +


+ + + =



= +


+ + =



+ =


+ + =



= −


+ + =



+ =


+ =



− + =


+ = +


( )
( ) (
)
3 2 2 3
2 2
2
3
2 2
3 2 2 3
2 2
3 2 2 3
2 2
2 3 2 3
2 2
2 2 2 2
2 ,
1 3 .
4 ,
24,
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
x x y y y x
x y y x x
x y x xy y
x x y
x y xy y x
x xy y x y
x x y xy y
x y xy x



+ = + +



− = +


− = + +



= + +


+ − =



+ = +


+ + + + =



= + −


+ = +





CREATED BY HOÀNG MINH THI; XYZ1431988@GMAIL.COM 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
2 2
3 3 2 2
2 2
2
6 4 ,
1,
2 4.
6 2 3 ,
2,
3 1 .
6 3 2 ,
3,
4.
3 4 5 6 0,
4,
5.
5 3 7 ,
5,
4 3 .
6 3
6,
x x y xy y
x y x
x xy x y xy y
x y y
x x y xy y
x y xy x y
xy x y y x
x y xy x y
x y xy x y
x y x xy y
x y xy

= + +


+ + =



+ = + +


= + −



= + +


+ + + =



+ + − =


+ + + + =



+ = +


+ = + −



( )
( )
( )
2 3 3
2 2
2 3 3 2
2
3 3
3 3 2
3
3 3
3 2 2 3
3 3
3 2 2 3
2 2 3 3
2 5 ,
2.
6 4 3 ,
7,
2 5.
7 3 ,
8,
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 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
x y x y xy
x x y xy y
x x y xy y
x xy xy y



=



= + +


+ + + =



= + +


+ + =



− = +


+ + + =



= − +


+ + = −



= − +


+ + =



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,
2 3.
7 9 2 ,
18,
3 1.
4 2 ,
19,
3 4 8 .
4 2 ,
20,
2 4 .
3 7 8 2 ,
21,
1 4 .
5 3 0,
22,
2
x y x y xy
x y
x x y xy
x y xy
x y xy x y
x xy y y
x y xy x y
x y y
x xy y x y
x x y y
x xy xy y
y

+ = +


+ =



= −


+ = −



− = +


+ + =



+ = −


+ + =



+ = +


+ + + =


− + + =
+
2
3 3 2 2
2 2
3 3 2 2
2 2
3 3 2 2
2 3
3 3 2 2
2 2
3 3 2 2
2 2
1 2 2 .
10 2 9 ,
23,
3 4 .
3 7 5 ,
24,
3 .
4 3 6 ,
25,
.
3 5 ,
26,
4 .
6 4 ,
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



+ = +



+ = +


+ =



− = −


+ + =



− = −


+ = +



+ = −


+ + + =



− = +


+ + =

( )
( )
3 3
2 2
2 2
2 2
2 2
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; XYZ1431988@GMAIL.COM 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
2
3 3 10 ,
1,
2 8.
3 3 9 10 3 ,
2,
1
3 0.
3
2 2 2 4,
3,
3 5 2.
3 3 9 5 1,
4,
2 5 .
3 4 5 8 ,
5,
6 1 3 7 .
2 2 2 2
6,
x y xy
x x y
x y x y x y
x y
x y
x x xy y
x x xy y
x y xy y x
xy y x y
x y xy x y
y x xy x
x y

+ =


+ + =



+ − − = −


+ + =




+ + + =


+ + + =



+ + = + −


+ = +



+ + + = +


− + + =


− +
( )
( )
( )
( ) ( )( )
( )
( ) ( )
( )
( )
2
2 2
2
2
2
2
2
2
2
2 2
2 2
2
2
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
x x xy y
x y y x xy
xy x y x
x x y y x
x xy x x y
x y x xy x x y
x xy x xy
x xy y x y
x y
x


+ − + + =


+ + =



+ + + = +


+ + = +



+ = + − +


+ + = +



+ + + = + + +


+ + =



− + = −


+ =


2 2
2 2
2 2 0,
1.
x xy y y
x y

+ − − − =


+ =



(
)
(
)
( )
( )
( )
( )
( )( )
( ) ( )( )
( )( )
( ) ( )
2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
2
3 1 1 4,
17,
2 1 3 1 2 .
3 1 2 4,
18,
3 2 4 1 4.
3 4 1 ,
19,
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
2 2
2 2
2 2
2 2
2 2
2 2
3 3 2
3 3 2
3
4 0,
2 3
3 5.
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
x y xy y
x y y 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.
,
28,
2 4.
2 3 ,
29,
3 3.
2 2 5 ,
30,
3 2 2.
5 3 ,
31,
2 2.
2
32,
x y
x y x y xy
x y xy
x y x y
x y xy
x y x xy y
x yx y
x y xy x y x y
x xy y
x y x y
x y xy



− =



+ = + +


+ − =



+ = +


+ + =



+ = − +


+ − =



+ + + = +


+ − =



+ = −


− + =


3 2 2
2 2
2 2 3 ,
2 2 5.
x y x xy x y
x xy y

+ + = +


+ + =




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