Lin e a re G le ich u n g e n
CO
PY
I n h a lt
L ö s e n d u rc h P ro b i e re n
......................................................
G le ic h u n g e n d e r Fo rm
x + a = b
E i n S c h r i tt z u r L ö s u n g
.......................................................
G le ic h u n g e n d e r Fo rm
ax = b
oder
G le ic h u n g e n d e r Fo rm a x + b = c
Z w e i S c h r i tt e z u r L ö s u n g
oder x − a = b
x
a = b
.........
.................
....................................
...................................................
G le ic h u n g e n d e r Fo rm a x + b = c x + d
Z u s a m m e n fa s s e n − U m fo r m e n
...........................
..........................................
Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
K l a m m e r n a u f l ö s e n , z u s a m m e n fa s s e n , u m fo r m e n
V e r m i s c h t e A u fg a b e n
Z a h l e n rät s e l
A l t e rs rät s e l
.............
1 − 2
3 − 4
5
6
7
8 − 10
11 − 12
13
14
15 − 16
.........................................................
17
.........................................................................
19
T e xt a u fg a b e n
.......................................................................
.....................................................................
K re u z z a h l rät s e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
G le ic h u n g e n l ö se n : G ru n d w isse n
18
20 − 2 1
22
.......................................
23
Ü b e rs i c h t : N a m e n / S e i t e n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
© 2 0 0©
1 DDeel tlto
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e rrll aag
Lin e a re G le ich u n g e n
1
L ö s e n d u rc h P ro b i e re n
G
N ame:
CO
PY
D as Lö s e n e i n e r G l e i c h u n g m it e i n e r V a ri a b l e n b e ste h t d a ri n , fü r d i e
V a ri a b l e e i n e Z a h l z u fi n d e n , d i e b e i m E i n s etze n i n d i e G l e i c h u n g z u
e i n e r w a h re n A u s s a g e fü h rt . D i e s ka n n d u rc h P ro b ie re n g e s c h e h e n .
Lö s e n d e r G l e i c h u n g 2 x + 4 = 1 0 d u rc h P ro b i e re n :
x
2x + 4 = 1 0
2
2 ·
1
3
w a h r/fa l s c h ?
2 ·
+ 4 =
6
fa l s c h , d e n n 6 = 1 0
2 ·
+ 4 = 10
wah r, den n 1 0 = 1 0
+ 4 =
8
fa l s c h , d e n n 8 = 1 0
L ö s e d i e G l e i c h u n g e n d u rc h P ro b i e re n !
Lö s u n g ?
nein
nein
ja !
P
R
O
B
IE
R
E
N
G
E
H
TÜB
E
R
S
TUD
IE
R
E
N
!
G le ic h u n g
Lös u n g
2x + 3 = 9
x = 3
3
k = 6
6
3y − 2 = 22
30 = 1 2 + 3k
1 5 − 2a = 7
v + 37 = 39
33 = 1 1 b
14 − z = 5
7t − 9 = 5
8
y = 8
4
a = 4
2
2
z = 9
9
v =
b = 3
3
t =
2
2
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
© 2 0 0©
1 DDeel tlto
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ë vo n 8
1
Lin e a re G le ich u n g e n
2
L ö s e n d u rc h P ro b i e re n
G
N ame:
CO
PY
Lö se d iese G le i c h u n g e n !
5x = 1 5
2x = 1 6
4x = 4 0
4x = 4 8
5x = 1 0 0
8x = 4 8
x =
x =
x =
x =
x =
3
8
10
x =
12
20
6
x =
15
2
17
x =
7
21
5
x =
1
16
9
x =
11
4
19
23
22
18
14
13
24
2x = 30
x · 5 = 10
x · 4 = 28
3x = 63
1 1x = 1 1
x · 3 = 48
5x = 4 5
7x = 7 7
x · 1 5 = 60
2x = 38
3x = 69
x · 3 = 66
x · 2 = 36
3x = 42
5x = 6 5
x · 3 = 72
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
3x = 5 1
x · 20 = 1 00
x =
x =
© 2 0 0©
1 DDeel tlto
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ë vo n 2 4
2
Lin e a re G le ich u n g e n
3
G l e i c h u n g e n d e r F o rm x + a = b o d e r x − a = b
G
N ame:
CO
PY
U m d ie Lö s u n g vo n G le ic h u n g e n d e r Fo rm x + 3 = 5 o d e r
x − 4 = 7 z u b est i m m e n , g e h e s o vo r:
- 3
x + 3 = 5
− 3
x + 3 − 3 = 5 − 3
+ 4
x − 4 + 4 = 7 + 4
LI = { 1 1 }
LI = { 2}
2 + 3
5
x − 4 = 7
x = 11
x = 2
x + 3
+ 4
5
S
IC
H
E
RI S
TS
IC
H
E
R
:
P
R
O
B
E
!
x − 4
5
5
11 − 4
7
5 = 5 wahr
7
7
7
7 = 7 wahr
L ö s e d i e G l e i c h u n g e n ! T ra g e d a n n d i e z u d e n E rg e b n i s s e n
g e h ö re n d e n B u c h s t a b e n re c h t s i n d i e K ä s t c h e n e i n !
G leich u n g
x − 1 6 = 52
70 = a + 1 8
1 3 + y = 45
Lös u n g
x = 68
32
H
68
S
y = 32
89
E
1 9 ,7
R
381
L
z = 3 7,8
39 ,9
E
52
C
c = 8 2 ,4
111
U
82 ,4
I
37 ,8
F
a =
52
84 = b − 2 7
b =
s − 99 = 2 82
s = 381
35 ,2 + z = 73
82 ,7 = w + 42 ,8
45 = c − 37 ,4
1 1 1 = p + 22
57 ,3 − t = 37 ,6
111
w = 3 9,9
p = 89
t =
1 9,7
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
68
S
52
C
32
H
111
U
381
L
F
37 ,8
F
E
39 ,9
E
I
82 ,4
I
E
89
E
R
1 9,7
R
S
C
H
U
L
© 2 0 0©
1 DDeel tlto
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ë vo n 1 0
3
Lin e a re G le ich u n g e n
4
G l e i c h u n g e n d e r F o rm x + a = b o d e r x − a = b
G
N ame:
CO
PY
Lö se d iese G le i c h u n g e n i m Ko pf !
x + 3 = 7
x + 8 = 14
3 + x = 10
2 + x = 16
x + 3 = 21
5 + x = 17
x − 3 = 2
x − 4 = 12
x − 10 = 10
x + 7 = 20
x + 5 = 13
x − 3 = 16
x − 12 = 1 1
x + 1 1 = 26
x − 6 = 5
x + 3 = 20
x − 10 = 12
1 8 + x = 21
1 1 + x = 21
x − 1 = 23
7 + x = 8
x + 7 = 28
8 + x = 10
x − 7 = 2
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
x =
4
6
7
x =
14
18
12
x =
5
16
20
x =
13
8
19
x =
23
15
11
x =
17
22
3
x =
10
24
1
x =
21
2
9
© 2 0 0©
1 DDeel tlto
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ë vo n 2 4
4
Lin e a re G le ich u n g e n
5
G l e i c h u n g e n d e r F o rm
ax = b
oder
x
a = b
G
N ame:
U m d ie Lös u n g vo n G le ic h u n g e n d e r Fo rm 6x = 1 8 o d e r
z u b est i m m e n , g e h e so vo r:
6x = 1 8
: 6
6x
=
6
: 6
18
6
· 3
x = 3
LI = { 3}
x
= 2
3
3 ·x
= 3 · 2
3
· 3
IL = { 6}
18
x
3
2
2
2
6
3
18
18
6 · 3
18
x
= 2
3
x = 6
S
IC
H
E
RI S
TS
IC
H
E
R
:
P
R
O
B
E
!
6x
CO
PY
1 8 = 1 8 wahr
2
2 = 2 wahr
L ö s e d i e G l e i c h u n g e n ! T ra g e d a n n d i e z u d e n E rg e b n i s s e n
g e h ö re n d e n B u c h s t a b e n re c h t s i n d i e K ä s t c h e n e i n !
G leic h u n g
Lös u n g
= 7
x = 35
x
5
3a = 75
30 =
c
4
240 = k · 1 2
8t = 9 6
v
11
= 7
85z = 85
d
3
= 13
32 = q · 8
4 =
b
7
D
I ELÖ
S
U
NG
S
T
E
C
KTI M
KO
F
F
E
R
!
12
R
39
A
35
K
k = 20
1 20
N
1
B
v =
28
S
1
25
O
4
S
q = 4
20
T
77
A
a =
c =
t =
z =
25
1 20
12
77
d = 39
b =
28
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
35
K
25
O
N
1 20
N
T
20
T
R
12
R
A
77
A
1
B
A
39
A
4
S
S
28
S
K
O
B
S
© 2 0 0©
1 DDeel tlto
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e rrll aag
ë vo n 1 0
5
Lin e a re G le ich u n g e n
6
E i n S c h r i tt z u r L ö s u n g
G
N ame:
Lö se d iese G le i c h u n g e n i m Ko pf !
x + 7 = 9
x − 2 = 1
x · 3
= 27
5 + x = 15
1 0x = 1 0
x · 5
= 15
5 + x = 10
1 0x = 9 0
x =
x =
x =
x + 1 8 = 20
x =
9x = 4 5
x =
x + 1 3 = 25
x =
16 − x = 15
x =
x − 5 =
x =
1
2x = 1 6
x =
x =
x =
1 0x = 1 0 0
x =
x − 2 = 2
x =
8x = 5 6
x =
x − 4 = 2
x =
x · 7 =
x =
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
49
CO
PY
x =
2
3
8
x =
9
10
1
x =
3
5
9
x =
2
10
4
x =
5
4
11
x =
12
7
8
x =
1
6
11
x =
6
7
12
30 − x = 26
x · 5 = 55
1 2 + x = 20
22 − x = 1 1
5x = 6 0
© 2 0 0©
1 DDeel tlto
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6
7
Lin e a re G le ich u n g e n
G le ic h u n g e n d e r Fo rm a x + b = c
G
N ame:
CO
PY
D u k a n n s t G l e i c h u n g e n m i t e i n e r V a r i a b l e n d u rc h U m fo r m e n
s c h r i ttw e i s e l ö s e n .
D a z u d a rfs t d u
í a u f b e i d e n S e i t e n d e r G l e i c h u n g d i e s e l b e Z a h l a d d i e re n
í a u f b e i d e n S e i t e n d e r G l e i c h u n g d i e s e l b e Z a h l s u b t ra h i e re n
í b e i d e S e ite n d e r G l e i c h u n g m it d e rse l b e n Z a h l ( = 0 ) m u lt i p l iz i e re n
í b e i d e S e ite n d e r G l e i c h u n g d u rc h d i e se l b e Z a h l ( = 0 ) d iv i d i e re n .
Beisp iel :
− 5
3x + 5 = 1 1
− 5
3x + 5 − 5 = 1 1 − 5
: 3
3x = 6
3x
3
=
: 3
6
3
x = 2
LI = { 2 }
A u f b e id e n S e ite n
d i e s e l b e Z a h l s u b t ra h i e re n
A u f b e i d e n S e i t e n d u rc h
d i e s e l b e Z a h l d i v i d i e re n
3x + 5
S
IC
H
E
RI S
T
S
IC
H
E
R
!
3 · 2 + 5
6 + 5
11
11
11
11
11
1 1 = 1 1 wahr
L ö s e d i e G l e i c h u n g e n ! T ra g e d a n n d i e z u d e n E rg e b n i s s e n
g e h ö re n d e n B u c h s t a b e n re c h t s i n d i e K ä s t c h e n e i n !
G leich u n g
5a + 1 2 = 27
2v − 1 0 = 22
Lös u n g
a = 3
v =
16
3d + 9 = 54
d =
11g − 7 = 15
g = 2
56 = 4z − 32
z =
I ND
IE
S
E
R
S
TA
D
THA
B
E
IC
HM
E
IN
E
N
E
R
S
T
E
NF
A
LL
G
E
LÖ
S
T
!
15
22
96 = 89 + 7x
x =
6b − 8 = 34
b =
1 32 = 1 2 + 1 5y
y = 8
1
7
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
8
D
22
M
16
O
3
D
7
N
15
R
1
U
2
T
3
D
O
16
O
R
15
R
T
2
T
M
22
M
1
U
N
7
N
8
D
D
U
D
© 2 0 0©
1 DDeel tlto
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e rrll aag
ë vo n 8
7
8
Lin e a re G le ich u n g e n
Z w e i S c h r i tt e z u r L ö s u n g
Rec h n e i m Ko pf !
3x + 2 = 8
E
N ame:
2x + 6 = 30
5x + 5 = 2 0
6x + 6 = 3 0
4x + 2 0 = 1 0 0
2x + 6 = 50
3x + 1 = 40
5x + 5 = 1 1 0
1 0x + 2 0 = 8 0
7x − 9 = 40
3x − 2 = 4 0
4x − 2 = 3 0
9x − 1 = 8 0
3x − 8 = 4 0
9x + 1 0 = 1 0 0
4x + 2 = 7 0
3x − 2 = 7 0
5x + 5 = 6 0
2x + 4 = 40
2x + 1 0 = 40
2x − 8 = 30
7x + 1 5 = 50
2x − 6 = 40
2 1 x − 1 = 20
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
CO
PY
x =
2
12
3
x =
4
20
22
x =
13
21
6
x =
7
14
8
x =
9
16
10
x =
17
24
11
x =
18
15
19
x =
5
23
1
D
IE
S
ES
P
U
RI S
T
S
E
H
RH
E
IS
S. . .
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
© 2 0 0©
1 DDeel tlto
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e rrll aag
ë vo n 2 4
8
9
Lin e a re G le ich u n g e n
Z w e i S c h r i tt e z u r L ö s u n g
E
N ame:
CO
PY
Rec h n e i m Ko pf !
2x + 1 = 2 1
6x − 2 = 4
2x − 4 = 20
3x − 1 = 4 1
5x + 2 = 82
7x − 2 = 1 9
9x − 7 = 2 9
8x + 1 2 = 1 0 0
3x − 1 = 50
2x + 2 = 40
1 1 x + 4 = 59
4x − 2 = 5 0
6x + 3 = 4 5
3x − 8 = 6 1
5x + 9 = 9 9
1 0x + 9 = 9 9
4x − 3 = 8 1
9x + 2 = 2 0
7x + 3 = 45
7x − 7 = 49
3x + 4 = 7 0
1 0x + 3 = 2 0 3
3x + 3 = 7 5
x =
x =
x =
x =
x =
9x + 5 = 1 4 0
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
x =
KO
MB
IN
IE
R
ED
E
RTÄ
T
E
RI S
T
G
E
W
IC
H
TH
E
B
E
R
!
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
x =
10
1
12
x =
14
16
3
x =
4
11
17
x =
19
5
13
x =
7
23
18
x =
15
9
21
x =
2
6
8
x =
22
20
24
© 2 0 0©
1 DDeel tlto
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e rrll aag
ë vo n 2 4
9
Lin e a re G le ich u n g e n
10
Zw e i S c h ritte z u r Lö s u n g
K
N ame:
CO
PY
Rec h n e i m Ko pf !
x +
1
2
= 1
x =
4x =
x =
x −
1
2
x − 2 ,5 = 7 ,5
x =
0 , 5x = 5 , 5
x =
3x +
1
2
= 36 ,5
x =
x −
1
4
=
x =
2
x = 2
3
x =
=
2
3
3x =
6
5
x =
x =
1
2
x − 1
x =
1
x − 1 = 0
7
2x − 2 ,5 = 2 7 ,5
x =
1
1
x + 2 = 6
2
H
IE
RS
T
E
H
E
ND
IE
LÖ
S
U
NG
E
N
.
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
1
1
4
x =
1
8
2
5
4
1
2
x =
10
5
2
x =
11
7
9
12
15
17
16
14
18
3
4
6
13
3
8
19
= 0 ,5
1
3 x + 0 ,2 5 = 5 1 4
x =
2
x − 1 = 3
9
x + 1 ,5 = 7 2
x =
0 ,5
0 , 5x + 5 , 5 = 1 0
x =
x =
x =
1
x = 2
2
x + 2,1 = 7,1
x =
3
= 1
4
x +
1
x + 3 = 10
2
0 ,2 5x + 1 = 5
x =
1
3
x =
2x + 2
1
1
= 28 3
3
x =
1
x + 2 = 3
19
x =
© 2 0 0©
1 DDeel tlto
o VVe
e rrll aag
ë vo n 2 4
10
11
Lin e a re G le ich u n g e n
G le ic h u n g d e r Fo rm a x + b = c x + d
G
N ame:
CO
PY
D u k a n n s t G l e i c h u n g e n m i t e i n e r V a r i a b l e n d u rc h U m fo r m e n
s c h r i tt w e i s e l ö s e n .
D a z u d a rfs t d u
í a u f b e i d e n S e i t e n d e r G l e i c h u n g d i e s e l b e Z a h l a d d i e re n
í a u f b e i d e n S e i t e n d e r G l e i c h u n g d i e s e l b e Z a h l s u b t ra h i e re n
í b e i d e S e ite n d e r G l e i c h u n g m it d e rs e l b e n Z a h l ( = 0 ) m u lt i p l iz i e re n
í b e i d e S e ite n d e r G l e i c h u n g d u rc h d i e s e l b e Z a h l ( = 0 ) d iv i d i e re n .
B e i G le ic h u n g e n d e r Fo rm 5x + 7 = 3x + 1 1 g e h e so vo r:
− 3x
− 7
: 2
5x + 7 = 3x + 1 1
2x + 7 = 1 1
− 3x
Va ri a b le n a u f e i n e S e ite
: 2
x b e re c h n e n
− 7
2x = 4
x = 2
Z a h l e n a u f d i e a n d e re S e i t e
IL = { 2 }
5x + 7
S
IC
H
E
RI S
TS
IC
H
E
R
:
P
R
O
B
E
!
5 · 2 + 7
10 + 7
17
3x + 1 1
3 · 2 + 11
6 + 11
17
1 7 = 1 7 wahr
L ö s e d i e G l e i c h u n g e n ! T ra g e d a n n d i e z u d e n E rg e b n i s s e n
g e h ö re n d e n B u c h s t a b e n re c h t s i n d i e K ä s t c h e n e i n !
G leic h u n g
Lös u n g
D
U
R
C
HB
L
IC
K
IS
TA
LLE
S
!
D
16
T
10
D
12
E
12
E
a = 4
E
T
5
E
4
T
10
D
5
E
1 1 z + 44 = 3z + 1 00
z =
E
4
T
7
K
1 9g − 5 = 1 3g + 91
g =
K
2
I
16
T
21 k + 1 1 = 6k + 41
k =
T
7
K
2
I
V
3
V
3
V
1 2x + 1 7 = 6x + 7 7
x =
9d − 32 = 2d + 52
d =
1 0a + 2 = 8a + 1 0
8y − 1 7 = 4y + 3
42t − 7 = 9t + 92
y =
10
12
5
7
16
2
t = 3
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
I
© 2 0 0©
1 DDeel tlto
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11
12
Lin e a re G le ich u n g e n
G le ic h u n g d e r Fo rm a x + b = c x + d
E
N ame:
CO
PY
Lö se d iese G le i c h u n g e n i m Ko pf!
x − 7 = 6
x + 5 = 11
26 = 1 6 + x
x =
x =
x =
2 4x = 1 2 0
5x = 6 0
4x = 7 2
x =
x =
1 2x + 3 = 1 5
x =
2x = x + 8
x =
7 x = 3x + 6 8
x =
9x − 7 = 3 x + 1 1
x =
1 6 + 7x = 44
x =
5x + 5 = 3x + 3 5
x =
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
x =
5x − 5 = 4 0
x =
3x − 2x + 6 = 20
x =
4x + 2 1 = 7 x
x =
3x + 2 x = 1 6 + 4x
x =
9x + 2 − 5x = 1 0
x =
6x − 5 = 4x + 1 7
x =
© 2 0 0©
1 DDeel tlto
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e rrll aag
6
13
10
12
5
18
1
9
8
14
17
7
3
16
4
2
15
11
ë vo n 1 8
12
Z u s a m m e n fasse n − U m fo rm e n
13
Lin e a re G le ich u n g e n
Z u s a m m e n fa s s e n − U m fo r m e n
G
N ame:
CO
PY
Z u e rs t a u f b e i d e n S e i t e n d e r G l e i c h u n g s o w e i t w i e m ö g l i c h
z u s a m m e n fa s s e n . D a n n d i e V a r i a b l e d u rc h U m fo r m e n b e re c h n e n :
8x + 1 2 − 2 x − 3 = 4 1 − 1 0x + 4 + 7x
6x + 9 = 4 5 − 3 x
9x + 9 = 4 5
9x = 3 6
S
IC
H
E
RI S
T
S
IC
H
E
R
:
P
R
O
B
E
!
| + 3x
|
|
x = 4
9
: 9
LI = { 4}
Z u s a m m e n fasse n
Va ri a b le n a u f e i n e S e ite
Z a h le n a u f d ie a n d e re S e ite
x b e re c h n e n
8x + 1 2 − 2x − 3
8 ·4 + 12 − 2 ·4 − 3
32 + 1 2 − 8 − 3
33
4 1 − 1 0x + 4 + 7x
41 − 1 0 · 4 + 4 + 7 · 4
41 − 40 + 4 + 28
33
33 = 33 wahr
L ö s e d i e G l e i c h u n g e n ! T ra g e d a n n d i e z u d e n E rg e b n i s s e n
g e h ö re n d e n B u c h s t a b e n re c h t s i n d i e K ä s t c h e n e i n !
G le ic h u n g
Lö s u n g
5 + 8y = 6 5 + 1 2
y = 9
3b + 1 1 + 2b = 4b + 1 3
38 + 1 6 m − 6 − 1 3 m = 85 − 1 4
7p + 1 3 − 2 − 5p = 23
88 = 3z + 1 5 − 1 1 + 4z
5 − 6x − 3 6 + 9x = 1 5 − 1 1 x − 2 1 + 9 x
D
E
RW
E
GZ
U
RLÖ
S
U
NG
IS
TN
IC
H
TM
E
H
RW
E
I T. . .
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
6
E
9
R
5
N
2
U
D
2
U
13
D
9
R
6
E
R
13
D
12
R
12
R
5
N
R
2
U
p = 6
E
b =
m = 13
z =
x =
12
5
N
© 2 0 0©
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13
Lin e a re G le ich u n g e n
14
Test
N ame:
E
CO
PY
L ö s e d u rc h P ro b i e re n !
G leich u ng
Lös u n g
1 9 = 4x + 1 1
x =
21 − 3m = 6
m =
23 = z − 1 1 ,4
p
5
5
Lö se i m Ko pf!
G le ic h u n g
2
2
Lösu ng
E
R
S
T
R
E
C
HN
E
N
,
D
A
NN
KO
N
TR
O
LL
IE
R
E
N
!
34 ,4
z = 3 4 ,4
32
4a = 56
p = 32
a =
v + 49 = 67
v =
18
8
= 4
14
14
18
L ö s e d u rc h U m fo r m e n !
G le ic h u n g
Lös u n g
1 2k + 4 = 8k + 20
k = 4
5b − 1 9 = 1 6
1 5w − 9 = 3w + 2 7
7 7 = 5 + 6y
b =
4
7
7
3
w = 3
y =
12
12
E rs t z u s a m m e n fa s s e n , d a n n u m fo r m e n !
G le ic h u n g
4c + 9 − 7 + 5c = 3c + 50
1 4 + 7q = 39 + 5q - 1 3
Lös u n g
8
c = 8
6
q = 6
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
© 2 0 0©
1 DDeel tlto
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14
Lin e a re G le ich u n g e n
15
K l a m m e rn a u fl ö s e n , z u s a m m e n fas s e n , u m fo rm e n
G
N ame:
CO
PY
K o m m e n i n e i n e r G l e i c h u n g K l a m m e ra u s d r ü c k e m i t V a r i a b l e n v o r ,
s o s i n d d i e K l a m m e r n a u fz u l ö s e n . D a n n w i rd d i e V a r i a b l e d u rc h
Z u s a m m e n fa s s e n u n d U m fo r m e n b e re c h n e t .
1 1 − (1 0 − 2 x) = 5
1 1 − 1 0 + 2x = 5
K la m m e rn a ufl öse n
1 + 2x = 5
Z u sa m m e nfasse n
2x = 4
LI = { 2}
5
S
IC
H
E
RI S
T
S
IC
H
E
R
:
P
R
O
B
E
!
5 = 5 wahr
4x − 3 6 = 2 0
4x = 5 6
x = 14
LI = { 1 4}
20
4 (x − 8 ) − 4
4 (14 − 8) − 4
4 · 6 − 4
24 − 4
20
5
5
5
5
1 1 − (1 0 − 2 · 2)
1 1 − (1 0 − 4)
11 − 6
5
4x − 3 2 − 4 = 2 0
x a u f e i n e S e ite ,
Za h le n a uf d ie a n d e re S e ite
x = 2
1 1 − ( 1 0 − 2 x)
4 (x − 8 ) − 4 = 20
20 = 20 wahr
20
20
20
20
L ö s e d i e G l e i c h u n g e n ! T ra g e d a n n d i e z u d e n E rg e b n i s s e n
g e h ö re n d e n B u c h s t a b e n re c h t s i n d i e K ä s t c h e n e i n !
Lös u n g
G leic h u n g
8 x − 3 (x + 5 )
1 4 + (1 2 y + 6)
=
=
20
44
3 (w − 7 ) + 2 w = 1 9
6 a − (9 − 5 a) = 2 4
7
S
8
N
7
S
7
S
2
I
w = 8
N
5
N
8
N
G
18
E
3
G
3
G
18
E
N
2
I
5
N
x =
y =
a = 3
5 ( b − 8 ) + 3 (4 − b ) = 8
b =
4 0 + (3 m − 1 8 ) = 3 7
m =
LÖ
S
U
NGD
E
R
G
LE
IC
H
U
NG
?
2
18
5
I
E
G
LE
IC
H
U
NG
E
N
S
I NDM
IR
G
LE
IC
HG
Ü
LT
IG
!
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
© 2 0 0©
1 DDeel tlto
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15
16
Lin e a re G le ich u n g e n
L i n e a re G l e i c h u n g e n m i t K l a m m e r n
E
N ame:
CO
PY
L ö s e s c h r i ft l i c h o d e r i m K o p f !
3 · (x + 1 ) = 6
7 · (x + 4) = 4 2
4 · (x − 3) = 3 2
6 · (x − 4 ) = 2 4
5 · (x + 3) = x + 5 1
8 · (x + 5 ) = x + 3 3
x =
x =
x =
x =
x =
x =
1 0 · (x − 1 1 ) = 1 6 + x
2 x − 5 = 7 · (x − 5)
x =
x =
2 · (x + 3 ) = 2 0
9 · (x + 2 ) = 4 5
x =
5 + 6 · (x + 1 ) = 3 5
x =
x =
2 · (x − 3 ) + 1 2
= x + 4
x =
1 0 · (x − 4) = 8 · (x − 2)
4 · (x + 7) = 1 2 · (x − 1 )
8 · (2 x − 5 ) = 1 0 · (x + 2)
4 · (x − 2) = 5 · (x − 1 )
x =
x =
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
x =
x =
© 2 0 0©
1 DDeel tlto
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1
2
11
8
9
−1
14
6
7
3
4
−2
12
5
10
−3
ë vo n 2 6
16
17
Lin e a re G le ich u n g e n
Lö se n vo n G l e i c h u n g e n − Ve rm is c h te A u fg a b e n
E
N ame:
CO
PY
L ö s e d i e G l e i c h u n g e n ! T ra g e d a n n d i e z u d e n E rg e b n i s s e n
g e h ö re n d e n B u c h s t a b e n u n t e n i n d i e K ä s t c h e n e i n !
G leich u n g
Lös u n g
1 1 6 = x + 32
x = 84
B
84
B
13
E
6v + 4 − v = 2 5 + 3v − 7
v =
E
7
R
1 1 w − 9 = 5w + 63
w =
12
R
N
12
N
4
H
14
A
14
A
10
R
z = 0
R
D
8
D
61
I
d = 5
I
N
5
N
54
E
n =
E
R
11
R
7y − 4 = 8 7
y =
a =
3a = 42
z =
1 0 z − 5 (z − 6 ) = 8 0
2 1 + 1 5z = 1 4 1
b − 24 = 37
5 − 8d − 1 4 + 1 1 d = 6d + 49 + 7 − 1 6d
27n − 1 97 = 7n + 23
7
m = 4
3 5 − (2 9 − 4 m ) = 2 2
q
= 6
9
13
H
10
b = 61
q =
54
11
D
A
SI S
TJAE
IN
D
IC
KE
RH
U
ND
!
7
61
11
13
4
5
84
10
12
54
8
14
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
© 2 0 0©
1 DDeel tlto
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ë vo n 1 2
17
Lin e a re G le ich u n g e n
18
Z a h l e n rä t s e l
N ame:
E
W i rd e i n e Z a h l m i t 3 m u l t i p l i z i e rt u n d d a n n d a z u 2 1 a d d i e rt , s o
e rh ä lt m a n 5 7 . D ie g es u c h te Z a h l h e i ßt
ë
.
12
W i rd v o m D o p p e l t e n e i n e r Z a h l 1 3 s u b t ra h i e rt , s o e r h ä l t m a n 3 5 .
D ie g es u c h te Z a h l h e i ßt
ë
.
24
W i rd e i n e Z a h l d u rc h 4 d i v i d i e rt u n d d a n n d a z u 1 0 a d d i e rt , s o
e rh ä lt m a n 1 5 . D ie g es u c h te Z a h l h e i ßt
ë
.
20
D i e D i ffe re n z a u s d e m F ü n ffa c h e n e i n e r Z a h l u n d d e r Z a h l 9 i s t 2 6 .
D ie g es u c h te Z a h l h e i ßt
ë
.
7
CO
PY
12
24
20
7
A d d i e rt m a n 1 7 z u e i n e r Z a h l , s o e r h ä l t m a n d a s s e l b e , a l s w e n n
m a n v o m D re i fa c h e n d i e s e r Z a h l 5 s u b t ra h i e rt . D i e g e s u c h t e Z a h l
heißt
11
ë
.
11
O
BI C
HV
O
ND
E
M
E
TW
A
SLE
R
N
E
N
KA
NN
... ?
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
© 2 0 0©
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18
Lin e a re G le ich u n g e n
19
A l t e rs rä t s e l
K
N ame:
V at e r u n d M u tt e r s i n d z u s a m m e n 8 1 J a h re a l t . D i e M u tt e r i s t 5 J a h re
j ü n g e r a l s d e r V at e r . D a n n i s t d e r V at e r
J a h re a l t .
ë
4 3 u n d d i e M u tt e r
ë
38
CO
PY
43
V at e r i s t d o p p e l t s o a l t w i e s e i n e T o c h t e r . B e i d e z u s a m m e n s i n d
7 8 J a h re a l t . D i e T o c h t e r i s t
2 6 u n d d e r V at e r
ë
5 2 J a h re a l t .
ë
38
52
26
P e t e rs G ro ß m u tt e r i s t 6 2 J a h re a l t . P e t e r ü b e r l e g t : „ W ä re i c h f ü n f m a l
s o a l t w i e i c h b i n , s o w ä re i c h 3 J a h re ä l t e r a l s m e i n e G ro ß m u tt e r . “
P ete r ist
13
ë
1 3 J a h re a l t .
M u tt e r i s t d re i m a l s o a l t w i e i h r S o h n . D e r A l t e rs u n t e rs c h i e d b e t rä g t
2 4 J a h re . D e r S o h n i s t
ë
1 2 u n d d i e M u tt e r
ë
3 6 J a h re a l t .
12
36
J u l i a , C h r i s t i n e u n d M a r i e s i n d z u s a m m e n 3 3 J a h re a l t . C h r i s t i n e i s t
d re i J a h re ä l t e r a l s J u l i a , J u l i a i s t d o p p e l t s o a l t w i e M a r i e . D a n n i s t
J u lia
1 2 , C h rist i n e
ë
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A
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ë
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
6 J a h re a l t .
ë
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6
19
20
Lin e a re G le ich u n g e n
T e x t a u fg a b e n
N ame:
K
CO
PY
E i n e z w e i t ä g i g e A u s s t e l l u n g w u rd e v o n i n s g e s a m t 5 1 6 0 P e rs o n e n
b e s u c h t . A m z w e i t e n T a g w a re n e s 4 9 2 w e n i g e r a l s a m e rs t e n T a g .
A l s o k a m e n a m e rs t e n T a g
ë
2 3 3 4 Besu c her.
2 82 6
2 8 2 6 u n d a m zw e ite n Tag
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Te i l n e h m e r m u ss n o c h
31
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M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
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21
Lin e a re G le ich u n g e n
22
K re u z z a h l rä t s e l
N ame:
1
NAD
A
NNE
B
E
NM
IT
KÖ
P
F
C
H
E
N
...
4
1
2
3
8
1
3
W a a g e re c h t
1:
4x = 68
4:
115
3:
6:
8:
1 0:
1 1 b − 45 = 9b − 3
g
= 3
1:
2y + 5 = 87
5:
1 03
3:
3c = 750
z
= 4
7:
5p − 1 3 − 2 p = 35
9:
7q − 47 = 3q + 9
8:
6d = 78
4
5
1
2
10
1
2
0
9
1
4
1
7
1
6
2
1
7
4
2
3
1
2
4
5
1
0
2
3
1
4
ë
12
12
ë
21
21
ë
101
1 01
1 42
m =
ë
123
1 23
c =
ë
250
250
ë
16
16
z =
p =
d =
q =
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
2
17
ë
142
y =
1
1
345
v =
1
6
ë
345
a =
2:
6
3
4
g =
b =
8a − 6 − 2 a = 3a + 42 0
97 = m − 26
5
2
ë
17
w =
S e n k re c h t
7
CO
PY
x =
1 37 = w + 36
7v − 5 = 79
E
ë
41
41
ë
41 2
41 2
13
ë
13
ë
14
14
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22
23
Lin e a re G le ich u n g e n
G le ic h u n g e n löse n : G ru n d w isse n
U m fo r m e n
CO
PY
N ame:
D u k a n n s t G l e i c h u n g e n m i t e i n e r V a r i a b l e n d u rc h U m fo r m e n
s c h r i tt w e i s e l ö s e n .
D a z u d a rfs t d u
í a u f b e i d e n S e i t e n d e r G l e i c h u n g d i e s e l b e Z a h l a d d i e re n
í a u f b e i d e n S e i t e n d e r G l e i c h u n g d i e s e l b e Z a h l s u b t ra h i e re n
í b e i d e S e ite n d e r G l e i c h u n g m it d e rs e l b e n Z a h l ( = 0 ) m u lt i p l iz i e re n
í b e i d e S e ite n d e r G l e i c h u n g d u rc h d i e s e l b e Z a h l ( = 0 ) d iv i d i e re n .
Beisp iel :
− 5
3x + 5 = 1 1
− 5
3x + 5 − 5 = 1 1 − 5
: 3
3x = 6
3x
3
=
6
3
x = 2
A u f b e i d e n S e ite n
d i e s e l b e Z a h l s u b t ra h i e re n
: 3
IL = { 2 }
A u f b e i d e n S e i t e n d u rc h
d i e s e l b e Z a h l d i v i d i e re n
S
IC
H
E
RI S
T
S
IC
H
E
R
!
3x + 5
3 · 2 + 5
6 + 5
11
11
11
11
11
1 1 = 1 1 wahr
K la m m e rn
K o m m e n i n e i n e r G l e i c h u n g K l a m m e ra u s d r ü c k e m i t V a r i a b l e n v o r ,
s o s i n d d i e K l a m m e r n a u fz u l ö s e n . D a n n w i rd d i e V a r i a b l e d u rc h
Z u s a m m e n fa s s e n u n d U m fo r m e n b e re c h n e t .
1 1 − (1 0 − 2 x) = 5
1 1 − 1 0 + 2x = 5
K la m m e rn a uflöse n
1 + 2x = 5
Z u sa m m e nfasse n
2x = 4
x a u f e i n e S e ite ,
Za h le n a u f d ie a n d e re S e ite
x = 2
LI = { 2}
1 1 − ( 1 0 − 2 x)
1 1 − (1 0 − 2 · 2)
1 1 − (1 0 − 4)
11 − 6
5
5 = 5 wahr
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
5
5
5
5
5
S
IC
H
E
RI S
TS
IC
H
E
R
:
P
R
O
B
E
!
4 (x − 8 ) − 4 = 20
4x − 3 2 − 4 = 2 0
4x − 3 6 = 2 0
4x = 5 6
x = 14
LI = { 1 4}
4 (x − 8) − 4
4 (14 − 8) − 4
4 · 6 − 4
24 − 4
20
20 = 20 wahr
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20
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20
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Lin e a re G le ich u n g e n
Ü b e rs i c h t
N a m e n / S e ite n
K lasse :
1
S c h u lj a h r :
2
3
4
5
M a g i s t e r h e ft M 2 0 : L i n e a re G l e i c h u n g e n
6
7
8
CO
PY
9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 20 2 1 22 23
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