This document contains information about mathematics formulas and equations. It provides step-by-step solutions to equations and systems of equations. Various types of equations are covered, including linear, quadratic, and word problems involving one or more unknown variables. Links are provided to a book about preparing for the 10th grade mathematics exam.
3. III
Các bạn ộc giả thân mến!
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4. IV
Chúc em sẽ dễ dàng chinh phục ược ước mơ!
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5. V
MỤC LỤC
PHẦN 1: ĐẠI SỐ
PHẦN II: HÌNH HỌC
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6. VI
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7. (d)
x;
c ax
b
,x
c by
a
; y ,y
ax by c
a, b, c
a2
b2
0
a 0; b 0
a 0; b 0
a 0; b 0
(d) y abx cb
(d)
(d)
c
b
y
x
c
a
O
c
b
y
xO
y
x
c
a
O
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8. ax by c
a'x b'y c '
a, b, c, a', b', c '
a2
b2
0
a'2
b'2
0
d ax by c
a'x b'y c 'd’
d d '
x
y
O
d
d’
y
O
d
d’
y
O
d
d’
x
x
d d '
d II d '
d d '
a
a'
b
b'
a
a'
b
b'
c
c '
a
a'
b
b'
c
c '
Chú ý:
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10. 11x 55
x 5
x 5 y 12 2x y 12 2 5
x;y 5;2
y
y
54
12x 20y 372
25x 20y 515
x
13x 143 x 11
5x 4y 103 y 12
x;y 11; 12
2x 3y 5
x 3y 26
3x 21 x 7
7 3y 26 3y 19 y
19
3
x;y 7;
19
3
3x 5y 93
5x 4y 103
y BCNN 5,4 20
2
y x
y
y 12 2x
12 2x
7x 2 12 2x 31
7x 24 4x 31
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11. 7
2x y
4
2x y
74
3
2x y
2
2x y
32
1
2x y
a,
1
2x y
b
7a 4b 74
3a 2b 32
a 10, b 1
2x y
1
10
2x y 1
x
11
40
y
9
20
x;y
11
40
;
9
20
2 x2
2x y 1 0
3 x2
2x 2 y 1 7
y 1
2a b 0
3a 2b 7
a 1, b 2
x2
2x 1
y 1 2
x2
2x 1 0
y 1 4
x 1
y 3
x;y 1;3
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12. a 0
b 0
x;yx;y
x;y
m;
m;
x;yx;y m;
m;
X a, b
ax b 0
ax b 0 1
a 0
x
b
a
a 0
b 0
mx y 2m 1
4x my m 6 2
y mx 2m
4x m mx 2m m 6 m2
4 x 2m 3 m 2 3
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13. x
m2
4 0 m 2 x
2m 3 m 2
m2
4
2m 3
m 2
y
m
m 2
2m 3
m 2
;
m
m 2
m 2 y mx 2m 2x 4
x;2x 4 x
m 2 0x 4
m 2
m 2
m 2 x;y
2m 3
m 2
;
m
m 2
x;2x 4 x
m
m
x 2y m 3
2x 3y m
I
m 1
x;y x y 3
m 1
x 2y 4
2x 3y 1
2x 4y 8
2x 3y 1
x 2
y 1
x;y 2; 3
m 2 3
m 2 3 2
2m 3
2m 3 2
m
5 2
2 3 1
;
2 3
3 1
y mx
mx mx 2 m 1 x 2
2
2
2 0 m 1 0
m 0 1
m 0 1
mx y
mx y 2
x;y 2; 3
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14. x;y 2;1
x 2y m 3
2x 3y m
2x 4y 2m 6
2x 3y m
x 2y m 3
7y m 6
x
5m 9
7
y
m 6
7
x;y
5m 9
7
;
m 6
7
x y 3
5m 9
7
m 6
7
3 5m 9 m 6 21 6m 36 m
m 6 x;y x y 3
m
m
m 1 x y 2
mx y m 1
m 2
x;y 2x y 3
m
mx 2y m 1
2x my 2m 1
mx 2y m 1
2x my 2m 1
2mx 4y 2m 2
2mx m2
y 2m2
m
m2
4 y 2m2
3m 2 m 2 2m 1
2x my 2m 1
m 2
x y 2
2x y 3
x y 2
x 1
x 1
y 1
1;1
m
m
x; y m ; m
y m x
mx m x m x m .
y m
x y m m m 4m m
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15. m2
4 0 m 2
m 2
y
m 2 2m 1
m2
4
2m 1
m 2
2
3
m 2
x
m 1
m 2
1
3
m 2
x, y m 2 3 1; 1;3; 3
m 2 1, 3 m 1; 3;1; 5
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16. 2 x y 5 3xy
2 x y 5 3xy 3xy 2x 2y 5
y 3x 2
2
3
3x 2 5
4
3
3x 2 3y 2 19
x, y 3x 2 1; 3y 2 1
19 1 19 19 1
3x 2 1
3y 2 19
I
3x 2 19
3y 2 1
II
1;7 7;1
S 17 ; 7 1
x4
4x3
6x2
4x y2
x4
4x3
6x2
4x y2
x4
4x3
6x2
4x 1 y2
1
x 1
4
y2
1
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17. x 1
2
y x 1
2
y 1
x 1
2
y 1
x 1
2
y 1
x 1
2
y 1
x 1
2
y 1
T x y
1 y 1 y
1 y 1 y
y 0 x 1 1
x 0
x 2
S 0;0 ; 2;0
m m
a m a b m b m
a b b c a c
ab c b c 1 a c
a m b n ab mn
a b a c b c 1 a bc
xy x 2y 3
xy x 2y 3 y x 2 x 3
x 2
y
x 3
x 2
y 1
1
x 2
85
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18. y x 2
x 2 1 x 2 1 x 1 x 3
S 1; 2 ; 3;0
3
x y 1 2 y 1 1
x 2 y 1
2x2
4x 19 3y2
2x2
4x 2 21 3y2
2 x 1
2
3 7 y2
*
3 7 y2
2 7 y2
2 y
2 x 1
2
18
x 1 3 x1
2 x2
4
S 2;1 ; 2; 1 ; 4 1 ; 4; 1
9x 2 y2
y
22 2
3 7 2 1 0 7 y2
y 3 y y 1
y 3
y 3
y 1 y y 1
86
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19. x k k 1 y 3k 1
S x k k 1 y 3k 1| k
x2
y2
2018
Cách 1:
Cách 2:
x y x y 2018
x y x y 2x x y x y
x y x y 2018 x y x y
x y x y
x2
y2
x2
y2
0;1;3
x2
y2
x y 8
x2
y2
x y 8 4x2
4y2
4x 4y 32
4x2
4x 1 4y2
4y 1 34
2x 1
2
2y 1
2
32
52
87
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20. x, y
2;3 3;2 1; 2 2; 1 2; 2 2;2 1;3 3; 1
2
52
2x 1 3
2y 1 5
2x 1 5
2y 1 3
-
x y 2 xy x y 0
x y
a2
b2
c2
d2
ac bd
2
x2
1 x2
y2
4x2
y
x y 1
x y 1
S
ad bc
-
22 2 2 2 2 2
x y z a b c ax by cz .
x, y
x2
1 x2
y2
4x2
y
x2
1 2x
x 1
x2
y2
2xy
x y
88
a b c
x y z
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21. x y 1
2
3 x2
y2
1
x y 1
2
12
12
12
x2
y2
12
3 x2
y2
1
x
1
y
1
1
1
x y 1
x y 1
n a2
n2
a 1
2
a
a2
n2
a 2
2
a n n a 1
x4
x2
1 y2
X
x2
1
2
y2
x2
0 x2
1
2
y2
y2
x4
x2
1 0 y2
x4
x2
2
y2
x2
1
2
y2
x2
1
2
x2
0 x 0
S 0;1 ; 0; 1
x0
;y0
;z0
; f x;y;z;
x0
;y0
;z0
; x0
x0
y0
z0
x1
;y1
;z1
;
x0
;y0
;z0
; x0
x1
;y1
;z1
;
x1
x0
x0
;y0
;z0
f x; y; z;...
n a 1 .
89
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22. k
k
k
x2
5y2
0
x0
;y0
x0
2
5y0
2
0 x0
x0
5x1
x1
25x1
2
5y0
2
0 5x1
2
y0
2
0
y0
y0
5y1
y1
5x1
2
25y1
2
0 x1
2
5y1
2
0
x0
;y0
x0
5
;
y0
5
xk
5
;
yk
5
x0
y0
5k
x0
y0
0
x y 0
Cách 1:
Cách 2:
0;0
; 0;0x y
0 0;x y 0 0x y
0 0;x y 2 2
0 05 5x y 0 5x
0 1 15x x x 2 2 2 2
1 0 0 125 5 0 5 5x y y x 0 5y
0 1 15y y y 2 2 2 2
1 1 1 15 25 0 5 0x y x y
1 1;x y
0 0
1 1 0 0
5 5
x y
x y x y 0x y
0x y
90
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23. x, y
x, y
x, y
12x2
26xy 15y2
461
x, y
x;y
5x2
8y2
2041
x;y
2x2
5y2
41 2xy.
x2
5y2
4xy 4x 4y 3
x;y
x3
y3
6xy
7x2
3y2
71
x;y
x2019
y2019
y1346
y673
x3
xy 2 x y.
x y z3 3 3
2 4
xy 2x 3y 1
x4
x2
y2
y 10
x;y x2
2y2
1
x y
y 2 x2
1 y .
.
.
2
16 x3
y3
15xy 371
x2
y 3 y x2
3
2
x x 1 x 2 x 3 y2
91
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24. xy 2x 3y 1 0
x y 2 3 y 2 5
x 3 y 2 5
x, y x 3 y 2
5 1 5 1 5
x 3 1
y 2 5
;
x 3 5
y 2 1
;
x 3 5
y 2 1
;
x 3 1
y 2 5
x;y
4;7 ; 8;3
x4
x2
y2
y 10 0
x2
2
2
1
2
x2 1
4
y2
2
1
2
y
1
4
10 0
x2 1
2
2
y
1
2
2
10
x2 1
2
y
1
2
x2 1
2
y
1
2
10
x2
y 1 x2
y 10
x2
y 1 x2
y 2x2
1
x2
0 x 2x2
1 0
10 10 1 10
2 5
x;y
x
y x
x2
3 x 3 x 3 y 2
x2
x2
1 3 2y2
3
y 3
x 3 y 2
6
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25. A
B
C D
180
180
o
o
A C
B D
A
B
C D
1
1
1 1A B
A, B
CD
A
B
C D
O
OA OB OC OD
A
B
C
Dx
A BCx
BCx C
A
B
C
D
90o
CBD CAD
A, B
CD.
A
B
C
D90o
A C
cbit
cbit
cùng
cách
u
m
t
i
m
haigóc
ibù
nhau
góc trong = góc ngoài t i
nh
i di n
hai
nh
cùng
nhìn
m
t c
nh
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26. I ABCD A, B, C, D
I IA IB IC ID I A,B, C,
D. ABCD I IA.
ABCD
ABCD
ABCD
DC ABCD
ABCD ABCD
ABCD A
C C ABCD
ABCD
ABCD
180o
A C 180o
B D
A C B D
DAC DBC
E BD .BAC EAD
( . . . (1
AD DE
DAE CAB AD BC AC DE
AC CB
KL
ABCD (O)
. . . .AC BD AB DC AD BC
D
A
B
OE
C
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27. ABC A AD, BE H O là
AHE.
CEHD
AEDB
CEHD
( .BAE CAD
. . (2
BE AB
CD AC
BE AC CD AB
. . . .AD BC ABCD AC DE EB AC
. . .AD BC ABCD AC DB
90o
CEH
AD
CEHD.
CEHD
BE
AD
I AB, AC M N
AM AN
E D AB
E D AB.
A, E, D, B
AEDB
180 .o
CEH CDH
,CEH CDH
90 .o
AD BC BDA
ABC I
AB, AC M N P Q CI, BI
MN
INQC
BPQC
CNQ ANM
180
2 2
o
A B C
.IBC ICB CIQ
90 .o
BE AC BEA
.A B C
A
E
O
H
DB C1
2
1
1
3
A
N Q
C
I
B
P M
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28. INQC
AC I N
CDNM CD
BCMP BC
POMN
IMPB
BPQC
BC.
ANCP
O AC.
INQC I N QC
.INC IQC
IN AC 90 .o
INC
90 (1o
IQC
90 .o
IMB IPB
90o
BPC BQC
ABCD O M, N, P
C BD, AD, AB NMOP
90CPA CNA
2 .PON PCN
180 .o
PCN NAP
180o
PCN NAP ABC AD II BC
2 (1PON ABC
180o
PMN PMB NMD
90 90 .o o
NMD NCD CDN ABC
90
180 90 90 2 (2
o
o o o
PMB PCB ABC
PCB ABC ABC ABC
PON PMN
O O’ M, N M O
(O’ B M O’ O A P M N.
MAPB
( . .AMN MBN
D
N
A P
M
B
C
O
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29. MN NP KMP K,
KM KP.
KA KB KM KP.
AMBP
K M OO’.
OMO’K
OK, O’K MA, MB
' , ' .OM IIO K O M IIOK
(1KA KB KM
AMBP
(c. .c
AN MN AN NP
MN BN NP BN
ANP PNB
NAP NPB
MAP MAN NAP
PMB MPB MBP180 .
A O AB, AC
ABCD O. M, N, P, Q
O O’ AB CD
D B C D OD AB, AC
E F. Khi D BC AEOF
AD O O’ E F
ABCD
O. AD
O H BC D O
BC M MO AB, AC E F.
B MO AD P
BHPD
O EF.
O AB, BC, CD, AD MNPQ
(AB AC)ABC
2
. .MD MB MC
O
A
K
N
P
B
O’
M
M
N
O O’
B
P
A
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30. ABC AB < AC BD CE H
I BC
K
BEI
CDI .K I
.BDK CEK
DE BC M M, H, K
BKMD
O O’ A, B (O, O’
AB B O O’ C,
D A CD O O’ M, N (M, N
A CM, DN E P Q
MN BC BD
AE CD.
BCED
EPQ
ABC O AB < AC M
BC (M B C AM O D A.
MCD AC E C
MBD AB F B.
BECF
ECD, FBD E, M, F
O R M O
MA, MB O A, B AB C
(C A B I, K MA, MC KA
O D.
BCDM
E MD O N KE.
KE O F ANFI là
2 2 2
.KO KM R
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31. AB O B
B D OE
90o
EBO
90 (o
EDO
EBOD
ODCF
AEOF
AEOF
AMOQ
AMOQ
MNPQ
M OO’
AB O O’
OB(1BEO BDO
(2OFC BDO
.OFC BEO
180 .o
AMO AQO
1 1M A OQ
2 1.M B
1 1.QMN A B
1 1 1 1 180 .o
QMN QPN A B C D
.MH AB
ABO’O
Mà M OO’,
AB.
MC MD.
AB CD là I.
'
OA AB
O B AB
'OA II O B AB
MH IIOA AB
H
.HM AB MA MB
1 2O O
(c. .cAOM COM MA MC .MA MB MC MD
ABCD
D
O
C
F
A
E
B
M
B’
N
C
P
D
Q
A
1
1
1
1
11
2
2
O
C
D
A
H
B
O’
Q
F
M
E
P
O
1
2
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