# Bai tap hpt on thi vao 10

BÀI TẬP CHUYÊN ĐỀ

HỆ PHƯƠNG TRÌNH BẬC NHẤT HAI ẨN
Họ và tên: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bài 1. (TS vào 10 tp Đà Nẵng năm học 2015 - 2016)

3

−y =6

2x
Giải hệ phương trình
1

 + 2y = −4
x

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Bài 2. (TS vào 10 tp Hà Nội năm học 2015 - 2016)
Giải hệ phương trình

2 x + y + x +1 = 4
x + y − 3 x + 1 = −5

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 3. (TS vào 10 tỉnh Khánh Hòa năm học 2015 - 2016)
Không dùng máy tính cầm tay, giải hệ phương trình:

4 x −3 y = 4
2 x+

y =2

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 4. (TS vào 10 tỉnh Phú Thọ năm học 2015 - 2016)
Cho hệ phương trình:

(m − 2)x − 3y = −5
x +my = 3

(1) (với m là tham số)

1. Giải hệ phương trình (1) với m = 1.
2. Chứng minh hệ phương trình (1) có nghiệm duy nhất với mọi m . Tìm nghiệm duy nhất đó
theo m .
Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 5. (TS vào 10 tỉnh Đắc Lắc năm học 2014 - 2015)
Cho hệ phương trình:

2x − a y = 5b − 1
. Tìm a, b biết hệ có nghiệm
bx − 4y = 5

x =1
y =2

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 6. (TS vào 10 tỉnh Tây Ninh năm học 2014 - 2015)

2

 +y =3
Giải hệ phương trình: x

 1 − 2y = 4

x

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 7. (TS vào 10 tp Hà Nội năm học 2014 - 2015)
4
1
+
=5
x + y y −1
Giải hệ phương trình:
1
2

= −1
x + y y −1

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 8. (TS vào 10 tỉnh Hải Dương năm học 2014 - 2015)
Cho hệ phương trình:

x + y = 3m + 2
(tham số m ). Tìm m để hệ đã cho có nghiệm (x; y) thỏa
3x − 2y = 11 − m

mãn x 2 − y 2 đạt giá trị lớn nhất.
Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 9. (TS vào 10 tỉnh Cà Mau năm học 2014 - 2015)
Giải hệ phương trình:

8x − y = 6
x 2 − y = −6

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 10. (TS vào 10 tỉnh Nam Định năm học 2014 - 2015)
Giải hệ phương trình

x(y + 2) + y = 6
x + 2y − 3 = 0

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 11. (TS vào 10 tỉnh Hưng Yên năm học 2014 - 2015)
Giải hệ phương trình

x 2 + 2y 2 − 3x y − 2x + 4y = 0
x2 − 5

2

= 2x − 2y + 5

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 12. (TS vào 10 tỉnh Thừa Thiên Huế năm học 2013 - 2014)
(x + y)2 − 6(x + y) − 7 = 0
Giải hệ phương trình
x − y −3 = 0

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 13. (TS vào 10 tp Hà Nội năm học 2013 - 2014)
Giải hệ phương trình

3(x + 1) + 2(x + 2y) = 4
4(x + 1) − (x + 2y) = 9

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 14. (TS vào 10 tp Hải Phòng năm học 2013 - 2014)
Cho hệ phương trình

x + 2y = m + 3
(I )
2x − 3y = m

1. Giải hệ phương trình (I ) khi m = 1.
2. Tìm m để hệ phương trình (I ) có nghiệm duy nhất (x, y) thỏa mãn: x + y = −3.
Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 15. (TS vào 10 tỉnh Lào Cai năm học 2013 - 2014)
Cho hệ phương trình

(m − 1)x + y = 2
(m là tham số)
mx + y = m + 1

1. Giải hệ phương trình khi m = 2.
2. Chứng minh rằng với mọi giá trị của m thì hệ phương trình luôn có nghiệm duy nhất (x, y)
thỏa mãn: 2x + y ≤ 3.
Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 16. (TS vào 10 tỉnh Nam Định năm học 2013 - 2014)

x +2
2

 x +1 + y −2 = 6
Giải hệ phương trình
1
 5

=3
x +1 y −2

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 17. (TS vào 10 tỉnh Quảng Ngãi năm học 2013 - 2014)
Giải hệ phương trình

x + y = xy −1
x + 2y = x y + 1

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 18. (TS vào 10 tp Hà Nội năm học 2012 - 2013)

2 1

x + y =2
Giải hệ phương trình:
6 2

 − =1
x y

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 19. (TS vào 10 tỉnh Vĩnh Phúc năm học 2012 - 2013)
Cho hệ phương trình:

2x + a y = −4
ax − 3y = 5

1. Giải hệ phương trình với a = 1.
2. Tìm a để hệ phương trình có nghiệm duy nhất.
Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 20. (TS vào 10 tỉnh Hải Dương năm học 2012 - 2013)
Cho hệ phương trình

3x + y = 2m + 9
có nghiệm (x; y). Tìm m để biểu thức (x y + x − 1) đạt giá
x+y =5

trị lớn nhất.
Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 21. (TS vào 10 tỉnh Ninh Thuận năm học 2012 - 2013)
Xác định các giá trị của m để hệ phương trình sau vô nghiệm:

(m + 2)x + (m + 1)y = 3
(m là tham
x + 3y = 4

số)
Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 22. (TS vào 10 tỉnh Thừa Thiên Huế năm học 2012 - 2013)

2 x + y = 5 x − y
20
20
Giải hệ phương trình:

+
=7
x+y x−y

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 23. (TS vào 10 tỉnh Lào Cai năm học 2012 - 2013)
1. Giải hệ phương trình

3x − 2y = 1
.
−x + 3y = 2

2. Tìm m để hệ phương trình

2x − y = m − 1
có nghiệm (x; y) thỏa mãn điều kiện x + y > 1.
3x + y = 4m + 1

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 24. (TS vào 10 tỉnh Phú Thọ năm học 2012 - 2013)
Giải hệ phương trình

x 2 − 5y 2 − 8y = 3
(2x + 4y − 1)

2x − y − 1 = (4x − 2y − 3)

x + 2y

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 25. (TS vào 10 tỉnh Thanh Hóa năm học 2007 - 2008)
2
1
+
=1
x +2 y +2
Giải hệ phương trình:
8
5

=1
x +2 y +2

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 26. (TS vào 10 tỉnh Thanh Hóa năm học 2006 - 2007)
Giải hệ phương trình:

5(3x + y) = 3y + 4
3 − x = 4(2x + y) + 2

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 27. (TS vào 10 tỉnh Thanh Hóa năm học 2004 - 2005)
Giải hệ phương trình:

2(x − y) + 3y = 1
3x + 2(x − y) = 7

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 28. (TS vào 10 tỉnh Thanh Hóa năm học 2003 - 2004)

 x + y = −1
Giải hệ phương trình: 1 2
 − =2
x y

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 29. (TS vào 10 tỉnh Thanh Hóa năm học 2001 - 2002)
Cho hệ phương trình:

x+y =1
.
mx + y = 2m

1. Giải hệ phương trình với m = 2.
2. Xác định m để hệ phương trình có một nghiệm? Vô nghiệm? Vô số nghiệm?
Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 30. (*)
Giải hệ phương trình:

x + y +xy = 5
x2 + y 2 + x y = 7

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 31. (*)
Giải hệ phương trình:

x 2 − 5x y − 2y 2 = 3
y 2 + 4x y + 4 = 0

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 32. (*)
Giải hệ phương trình:

x 2 − 3x y + 2y 2 = 0
2x 2 − 3x y + 5 = 0

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 33. (*)
Giải hệ phương trình:

x 2 + y 2 = 25
x y = 12

13

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 34. (*)
Giải hệ phương trình:

x + y + 3x y = −3
xy +1 = 0

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 35. (*)
Giải hệ phương trình:

x 2 − 2x = y
y 2 − 2y = x

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 36. (*)
Giải hệ phương trình:

x 2 − 3x = 2y
y 2 − 3y = 2x

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 37. (*)
Giải hệ phương trình:

x2 + y 2 = 1
x2 − x = y 2 − y

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 38. (*)
Giải hệ phương trình:

x 2 − y 2 = 16
x+y =8

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 39. (*)
Giải hệ phương trình:

x + y +xy = 5
x2 + y 2 + x y = 7

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 40. (*)
2x + y = 3a − 5
x−y =2
Gọi nghiệm của hệ là (x, y), tìm giá trị của a để x 2 + y 2 đạt giá trị nhỏ nhất.

Cho hệ phương trình:

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 41. (*)
a 2 x − y = −7
Cho hệ phương trình:
2x + y = 1

1. Giải hệ phương trình khi a = 1
2. Gọi nghiệm của hệ phương trình là (x, y). Tìm các giá trị của a để x + y = 2
Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 42. (*)
Giải hệ phương trình:

x 2 − y 2 = 16
x+y =8

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 43. (*)
Giải hệ phương trình:

x 2 − 3x y + 2y 2 = 0
2x 2 − 3x y + 5 = 0

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 44. (*)
Giải hệ phương trình:

x − y = −3
x y = 10

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 45. (*)
Giải hệ phương trình:

3x = 2y = 6z
x + y + z = 18

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 46. (*)
Cho hệ phương trình:

(a + 1)x + y = 4
ax + y = 2a

1. Giải hệ khi a = 1.
2. Chứng minh rằng với mọi giá trị của a , hệ luôn có nghiệm duy nhất (x; y) sao cho x + y ≥ 2.
Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 47. (*)
Cho hệ phương trình:

mx − y = −m
(1 − m 2 )x + 2m y = 1 + m 2

1. Chứng tỏ phương trình có nghiệm với mọi giá trị của m .
2. Gọi (x; y) là nghiệm của hệ phương trình, chứng minh với mọi giá trị của m ta luôn có:
x 2 + y 2 = 1.
Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 48. (*)
Cho hệ phương trình:

x + ay = 2
ax − 2y = 1

1. Giải hệ phương trình trên.
2. Tìm số nguyên a lớn nhất để hệ phương trình có nghiệm (x; y) thỏa mãn x y < 0.
Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 49. (*)
Giải hệ phương trình:

x 3 − y(1 + 2) = 1
x(1 − 2) + y 3 = 1

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 50. (*)
Giải hệ phương trình:

|x − 1||y − 5| = 1
y = 5 + |x − 1|

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 51. (*)
Giải hệ phương trình:

|x − 1| + y = 0
x + 3y − 3 = 0

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 52. (*)
Giải hệ phương trình:

2|x| −

y −1 = 3

|x| + 5

y −1 = 7

21

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Bài 53. (*)
Giải hệ phương trình:

x4 + y 4 = 1
x3 + y 3 = x2 + y 2

Lời giải: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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