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
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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.
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; 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; 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; 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; 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; 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
+ + = +
+ + =
