1.rozwiąż układy równań :
a ) 3x = 12
x - y = 1
b) 5y = 2
x + y = 2 i 2/5
c) 4y - x = 6
17y = 0
d) a = b
2a + b = 9
e) u = 4v
5u - v = 10
f) 3r = s
3 r + s = 24
2.Rozwiąż układy równań metodą podstawiania :
a) x + 3y = 3
x - y = 5
b) 2x - y = 10
x + 5y = -17
c) 3m + n = 9
2m - 5n = - 11
d) 4a - 2 = b
3a + 7b = 1,5
e) 5x = 4 - 2y
4x + y = 3
f) 6x - 3y = 9
x+ 5y = - 9,5


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2009-12-09T16:33:36+01:00
1.rozwiąż układy równań :
a ) 3x = 12
x - y = 1 |*(-3)

3x=12
-3x+3y=-3

3y=9
y=3

x-y=1
x-3=1 |+3

x=4

b) 5y = 2
x + y = 2 i 2/5

5y=2
x+y=12/5 |*(-5)

5y=2
-5x-5y=-12

-5y=-10
y=2

-5x-5y=-12
-5x-10=-12 |+10

-5y=-2
y=2/5

c) 4y - x = 6
17y = 0

0-x=6
-x=6

x=-6
y=0

d) a = b
2a + b = 9

a=b
2b+b=9

a=b
3b=9

b=3
a=3

e) u = 4v
5u - v = 10

u=4v
5*4v - v=10

u=4v
20v-v=10

u=4v
19v=10

v=10/19

u=4v
u=40/19

f) 3r = s
3 r + s = 24

3r=s
s+s=24

2s=24
s=12

3r=s
3r=12

r=4

2.Rozwiąż układy równań metodą podstawiania :
a) x + 3y = 3
x - y = 5

x=3-3y
x-y=5

x=3-3y
3-3y-y=5

-4y=2
y=-0,5

x-y=5
x+0,5=5 |-0,5

x=4,5

b) 2x - y = 10
x + 5y = -17

2x-y=10
x=-17-5y

2(-17-5y)-y=10
x=-17-5y

-34-10y-y=10
x=-17-5y

-34-11y=10
x=-17-5y

-11y=-44
y=4

2x-y=10
2x-4=10 |+4

2x=14
x=7


c) 3m + n = 9
2m - 5n = - 11

n=9-3m
2m-5(9-3m)=-11

n=9-3m
2m-45+15m=-11

n=9-3m
17m=34

m=2

n=9-3m
n=9-6

n=3

d) 4a - 2 = b
3a + 7b = 1,5

b=4a-2
3a+7(4a-2)=1,5

b=4a-2
3a+28a-14=1,5 |+14

31a=15,5
a=0,5

b=4a-2
b=2-2

b=0

e) 5x = 4 - 2y
4x + y = 3

5x=4-2y
y=3-4x

y=3-4x
5x=4-2(3-4x)

y=3-4x
5x=4-6+8x

y=3-4x
5x=-2+8x |-8x

y=3-4x
-3x=-2

x=2/3

y=3-4x
y=9/3-8/3

y=1/3


f) 6x - 3y = 9
x+ 5y = - 9,5

x=-9,5-5y
6(-9,5-5y)-3y=9

x=-9,5-5y
-57-30y-3y=9 |+57

x=-9,5-5y
-33y=66

y=-2

x=-9,5-5y
x=-9,5-10

x=-19,5

7 4 7
2009-12-09T16:49:55+01:00
Zadanie 1.
a) 3x = 12 /:3
x-y = 1
x = 4

4 - y = 1
-y = -3/ ×(-1)
y = 3
x=4
b) 5y = 2 /:5
x + y = 2 ⅖
y = ⅖
x+⅖ = 2⅖
x = 2⅖ - ⅖
x = 2

c) 4y - x = 6
17y = 0 / :17
y = 0
-x = 6/ ×(-1)
x= -6

d) a = b
2a + b = 9
2a +a =9
3a = 9/ :3
a = 3
a=b
3=3
b=3

e) u = 4v
5u - v = 10
5 × 4v - v = 10
20v - v = 10
19v = 10 / : 19
v= 10/19
u= 4 ×10/19
u= 40/19 = 2 i 2/19

zadanie 2.
a) x + 3y = 3
x - y = 5
x=3-3y
3-3y - y=5
-4y = 5-3
-4y = 2/ :(-4)
y= 2/4 = -1/2
x- (-½)=5
x= 5 - ½
x = 4½

b) 2x - y = 10
x + 5y = -17
-y-10-2x/ ×(-1)
y= -10 +2x
x+5 (-10 +2x) = -17
x- 50 +10x = -17
11x =50 - 17
11x = 33/ :11
x = 3
y = -10 +2 ×3
y= 4

c) 3m + n = 9
2m - 5n = - 11
n = 9 -3m
2m-5(9-3m)=-11
2m-45+15m=-11
17m=-11+45
17m=34/ :17
m= 2
3×2 + n = 9
6+n=9
n=9-6
n=3

d) 4a - 2 = b
3a + 7b = 1,5
3a - 7(4a-2)=1,5
3a - 28a +14 =1,5
-25a = -12,5 / :(-25)
a= 12½ : (-25)
a = ½
4 ×½ - 2 = b
b = 0

e) 5x = 4 - 2y
4x + y = 3
y=2-4x
5x= 4-2(3-4x)
5x=4-6+8x
5x-8x= -2
-3x = -2/ :(-3)
x = 2/3
4 ×2/3 +y=3
8/3+y = 3
y= 3-8/3
y= 1 i 2/3

f)6x - 3y = 9
x+ 5y = - 9,5
x=-9,5 -5y
6(-9,5-5y) - 3y =9
-57 - 30y-3y =9
-33y = 9 + 57
-33y = 66 /: (-33)
y= 2
x+ 5 × 2 = -9,5
x= -9,5 - 10
x=-19,5
4 3 4
2009-12-09T19:23:06+01:00
Zadanie 1.
a) 3x = 12 /:3
x-y = 1
x = 4

4 - y = 1
-y = -3/ ×(-1)
y = 3
x=4
b) 5y = 2 /:5
x + y = 2 ⅖
y = ⅖
x+⅖ = 2⅖
x = 2⅖ - ⅖
x = 2

c) 4y - x = 6
17y = 0 / :17
y = 0
-x = 6/ ×(-1)
x= -6

d) a = b
2a + b = 9
2a +a =9
3a = 9/ :3
a = 3
a=b
3=3
b=3

e) u = 4v
5u - v = 10
5 × 4v - v = 10
20v - v = 10
19v = 10 / : 19
v= 10/19
u= 4 ×10/19
u= 40/19 = 2 i 2/19

zadanie 2.
a) x + 3y = 3
x - y = 5
x=3-3y
3-3y - y=5
-4y = 5-3
-4y = 2/ :(-4)
y= 2/4 = -1/2
x- (-½)=5
x= 5 - ½
x = 4½

b) 2x - y = 10
x + 5y = -17
-y-10-2x/ ×(-1)
y= -10 +2x
x+5 (-10 +2x) = -17
x- 50 +10x = -17
11x =50 - 17
11x = 33/ :11
x = 3
y = -10 +2 ×3
y= 4

c) 3m + n = 9
2m - 5n = - 11
n = 9 -3m
2m-5(9-3m)=-11
2m-45+15m=-11
17m=-11+45
17m=34/ :17
m= 2
3×2 + n = 9
6+n=9
n=9-6
n=3

d) 4a - 2 = b
3a + 7b = 1,5
3a - 7(4a-2)=1,5
3a - 28a +14 =1,5
-25a = -12,5 / :(-25)
a= 12½ : (-25)
a = ½
4 ×½ - 2 = b
b = 0

e) 5x = 4 - 2y
4x + y = 3
y=2-4x
5x= 4-2(3-4x)
5x=4-6+8x
5x-8x= -2
-3x = -2/ :(-3)
x = 2/3
4 ×2/3 +y=3
8/3+y = 3
y= 3-8/3
y= 1 i 2/3

f)6x - 3y = 9
x+ 5y = - 9,5
x=-9,5 -5y
6(-9,5-5y) - 3y =9
-57 - 30y-3y =9
-33y = 9 + 57
-33y = 66 /: (-33)
y= 2
x+ 5 × 2 = -9,5
x= -9,5 - 10
x= -19,5
1 1 1