PT & BPT bậc nhất

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V¤n ð« 1:
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1)Cho hai hàm s¯ y = f(x) và y = g(x) 
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2) Xét hai phß½ng trình
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3) Các phép biªn ð±i tß½ng ðß½ng
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4) Phép biªn ð±i không tß½ng ðß½ng
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V¤n ð« 2:
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no
1) Xét phß½ng trình A.x = B
;`<;p7O<L+")>?@
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2) Các phß½ng trình ðßa v« b§c nh¤t
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V¤n ð« 3:
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1)Ði«u k¸ên Ax = B có nghi®m duy nh¤t
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≠
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0A
2) Ði«u ki®n Ax = B nh§n ∀x∈  là
nghi®m
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=
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0A
3) Ði«u ki®n Ax = B có nghi®m
 •J 51<^IOQ

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4) Ði«u ki®n Ax = B vô nghi®m
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:.9g#g]<Z,:;g#gZ<][
4:<

.
9#Z,:;\#Z[<].LK5#€`
:
2x
1m2x2
2x
2x
mx3
−
−+
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−
−
^:
2x
−
9<#Z<]\:;`
c7<≤[ua4:<•Z.L<€,:<€,^:
I‚q
V¤n ð« 4:
k
•J1")>?@



=+
=+
222
111
cybxa

cybxa
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22
11
ba
ba
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,
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.
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22
11
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bc
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.
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Q'+
r"AB9#;
D
D
x
-O;
D
D
y
:
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#

≠`Lw
O
≠`71LU
51<
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#
;w
O
;`71 LUAB
51<C
,
#]4
,
O;
,
5"
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tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt
,au.vece"w5Ae5055Ox</L/L-x8"/L/L
vvb[a`y
.
  
:



=+
=+
1,1y7x2
8,7y24x27

4:



=−
=+
21y5x13
23y9x7
:



=−
=−
8y9x7
3y4x5
^:



=−
=+
11y5/3x2/5
16y3/2x4/3
_:






=+
=−
3y2x5
1yx3
8:





=−−
−=++
22y)12(x2
12yx)12(
4.2.1



=−
=−
154v7u3
50v9u5
/~O5C51L+AIO
?51<E1








=
−
+
+
=
−
−
+
308
2y
7
3x
3
100
2y
9
3x
5

c7I;.ƒ-L;,`#;Z,byuab-O;\,u.`
4.3.5C5L+451'I!pF51 51<@<
135„#-O%='!"%B5LK5<r-
4
:



+=+
+=+

5m2myx2
1my2mx
 4:



=+++
=−+
2y)1m(x)2m(
5y)2m(mx
:



−=−+
−=+−
m1yx)2m(
1m3y2x)1m(
^:



=+
+=−+
2myx2
1my)1m(mx
_:




=−+−
=+−+
my)4m(x)1m2(
4y)2m(x)4m(
8:





=−
=−
2
2
bybx
ayax
:



=+
+=+
ab2aybx
babyax
22
:






=−
−=−
b4ybbx
babyax
2
2
V¤n ð« 5:
mzjkk
{jk0-|jk0-{|cjk0
1) H® có nghi®m duy nh¤tF5w≠`
2) H® vô nghi®m :
 …w-5C5w;`<
O<L+15")>?@F$
I@LU51<
3) H® vô s¯ nghi®m :
…w-5C5w;`<
O<L+15")>?@5B
I@ LUAB51<
5"
5.1.1



+=+
=+
1m2ymx
m3myx
9<<AB:
:&<%P1 51<^IOQp56

13'5'M5„$51<%='!"%B5
LK5<
4:@<<∈%P51<E1'+$AB
IO
c7:<≠±,-9#9#Z.:;9OZ,:9OZ[:4:Z[-
Z.-Z,-`
5.2.1



+=+
=+
1mmyx
m2ymx
9<'+<AB:
:&<%P1LU51<p
4:&ABIO<%P1 51<^IO
Q'+ABIOp…$51<% p
c7:<;Z,4:<;`79,e`:-<;Z.79[e.:
5.3.&<%P1 &<^IOQ
:



=−+++
=−++
0m31y)3m(mx
0m4y8x)1m(
c7<≠,-<≠[
4:








−=+−
=++
)1m(2
y
2
x
2
)2m(
m
y
1
m
x
2
)1m(
c7<≠`-.-.±
b
-Z,±
3
5.4.@<%52IF51%P1 LUAB51<
tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt
,au.vece"w5Ae5055Ox</L/L-x8"/L/L
vvb[a`y

[
  
:



+=++
+=+−
m3y2x)6m(
m1myx4
c7<;Z.
4:



−=+=+
−=+++
1by)ba(2x)a5(
aby)ba(x)a1(

c7;[-4;aL;Z4;Z\±
17
5.5.@<%52IF51-4%P1LU51<



−=+
+=+
baaybx
babyax

c7;±4≠`
V¤n ð« 6:
†0‡†
ˆj‰oŠi
1)иnh nghîa7≥4Z4≥`
‹$?)Y*"^)>7
.
≥`-gg≥`-
A
≥`-…E5ABH^QI/≥`-
Œ$4@")>
.
]
.
]
.
≥`
2) Tính ch¤t 7
:…=7
‹=5L6H<=AB7
≥]F≥]F
‹=LLKL6E54Q%‡3H
52I7
≥n≥w]≥]w
9Gq7FU%)*?TL6LKL6:
4:…7
‹5L6E4Q%‡3H
<=AB^)>%)*4Q%‡3H52I
-AB<@4Q%‡3%Œ52I
‹≥LK/€`

A
1
≤
B
1
‹≥€`

≥

/n
n
A
≥
n
B
:…4•(I7
≥n≥≥
5"
6.1.wH%&Ž3<57
:∀-4-∈7
.
]4
.
]
.
≥4]4]
4:∀-4∈74]]4≤,]
.
]4
.

/
:∀-4--^-_∈7
 
.
]4
.
]
.
]^
.
]_
.
≥94]]^]_:
^:∀-4-∈7
 
.
9,]4
.
:]4
.
9,]
.
:]
.
9,]
.
:≥b4
6.2.3<5
:∀-4∈
]

7
2
a
b
b
a
≥+
4:∀-4∈7
2
22
2
ba
2
ba






+
≥
+
:]4€`7
3
33
2
ba
2
ba







+
≥
+
^:∀≠`7
2
≥+


,

6.3.∀-4-#-O∈/3<54Q%‡
3
I5AU"AF57g/#]4/Og≤
)yx)(ba(
2222
++
T% AIO?7a/A5],.A≤,[
6.4 3<5$4Q%‡3?4)A_"
:6I≥4-#≥O7.9/#]4/O:≥9]4:9#]
O:
4:6I≥4-#≤O7.9/#]4/O:≤9]4:9#]
O:
:6I≥4≥-#≥O≥•7
[9/#]4/O]/•:≥9]4]:9#]O]•:

^:6I≥4≥-#≤O≤•7
[9/#]4/O]/•:≤9]4]:9#]O]•:
_:T% AIO?7?<D∆'IU'IU
 
 5:]4]≥
3
π
9]4]:
 55:
≤
++
cba
hhh
S.18
]4]
6.5.]4;,/•$<5
:
.
]4
.
≥
2
1
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[
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[
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4
1

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\
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\
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8
1
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.
]9Z4:
.
≥,]`4:%PqZ4≥Z
2
1
9
.
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.
:
:9
.
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.
:
.
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.
Z4
.
:

.
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4
1
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6.6.-4-'+4M<5$
:/4/≥9]4Z:9Z4]:9Z]4]:
4:
cba
3
ac
1
cb
1
ba
1
++
>
+
+
+
+
+
:
[
]4
[
]
[
≥[//4/

w7:≥g4Zg
.
≥
.
Z94Z:
.
-…
4:]4]€]4-…=:9///:/≥`
6.7.3<5
:6Igg≤,-g4g≤,@g]4g≤g4],g
4:6I-4-<€`L+
b
a
≤,@
b
a
≤
mb
ma
+
+
tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt
,au.vece"w5Ae5055Ox</L/L-x8"/L/L
vvb[a`y
\
  
:6I-4-<€`L+
b
a
≥,@

b
a
≥
mb
ma
+
+
w7:4@5L64:L@≤4:L@≥4
6.8.3<5
:6I≥4≥,@
b.a1
2
b1
1
a1
1
22
+
≥
+
+
+
4:6I#-O∈L+#]O€`@
yxyx
21
2
41
1
41
1

+
+
≥
+
+
+
w7:WIO%‘%)L29Z4:
.
94Z,:≥`4:
)>’
6.9.-4--^-#-O-•€`/3<5
:,•
ac
c
cb
b
ba
a
+
+
+
+
+
•.
4:.•
bad
ad
adc
dc
dcb

cb
cba
ba
++
+
+
++
+
+
++
+
+
++
+
•[
:6I
z
c
y
b
x
a
≤≤
@
z
c
z
c
y
b

x
a
x
a
≤++≤
w7:u9]4]:•u9]4:4:9]4:u9]4
]]^::•#9u•:L+;#9u#:=L6
F6WIC
6.10.3<5
:6I-4--^€`L+
22
db
cdab
d
c
b
a
+
+
<<
4

@5
4:6I-4--^€`@
db
1
ca
1
d
1

c
1
b
1
a
1
111
++
+
≤
+
+
+
:6I€4€`L+<€-<-∈@ 

nn
nn
mm
mm
ba
ba
ba
ba
+
−
>
+
−
^:6I
3

4
b1
1
a1
1
≥
+
+
+
@/4≤
4
1
w7:WIO%U^Z4≥`9%:4:WIO%‘
9^Z4:
.
≥`9%::WI5%‘
<Z
€4
<Z

9%:
V¤n ð« 7:
†0‡†
ˆQ%‡3IO
1) B¤t ðÆng thÑc Cauchy cho 2 s¯7
  -4≥`7
 ]4≥.
ab
4≤
2

2
ba






+
 wQI%‡3#CO?F5;4
2) B¤t ðÆng thÑc Cauchy cho n s¯ :
 AB
,
-
.
-////-

≥`

,
]
.
]////]

≥
n
n21
a....aa

,


.
/////

≤
n
n21
n
a.......aa






++
 wQI%‡3#CO?F5
,
;
.
;///;


/
3) H® quä :
‹6I5AB-4 ŒFU%Œ@…
%MlF5;47/4≤
2
2
ba







+
‹6I5AB-4 …FU%Œ@Œ
%MF5;47]4≥.
b.a
5"
7.1.wH4Q%‡3IO5AB
3<57
:-4€`7
a
b
b
a
+
≥.
4:-4≥`79]4:9,]4:≥\4
:-4€`7






+







+
a
b
1
.b
a
1
≥\
^:-4€`7






+






+
a
b

1.
b
a
1
≥\
_:-4€`7






+






+
b
b
1
a
a
1
≥\
8:-4-€`7
8c
c

1
b
b
1
a
a
1
≥






+






+






+
:-4-€`7

81
a
c
1
c
b
1
b
a
≥






+






+







+
:-4-€`7
8a
c
1
c
b
1
.b
a
1
≥






+






+







+
5:-4€`7
1k
kk
2
a
b
1
b
a
1
+
≥






++






+
eF∈

7.2.-4≥,/3<5
:
2
ab
1b.a
≤−
 4:
b.a1ab1b.a
≤−+−
:
cbacabcb.a
++≤++
7.3.-4-≥`/cN^“4Q%‡3
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3
c.b.a
7.4.-4--^€`/3<5
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c
1
b
1
a
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3
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c.b.a
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c.b.a
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