Temat do zadania: Rozwiązywanie układów równań
/ - to będzie zastępować klamrę
\
Zadanie: Rozwiąż podane układy równań metodą podstawiania.
a)
/ x + 2y = -2
\ 2x - y = 1
b)
/x + 4y = -2
\ 3x - 8y = 10
c)
/ x - 2y = 3
\ -2x + y = 0
d)
/ 2x + y = 9
\ 5x - 4y = 3
e)
/ 3x - 2y = 5
\ 2x - y = 4
f)
/ 2x + 9y = -9
\ -3x + 3y = 8
g)
/ 2x + 3y = 4
\ 8x - 5y = -1

3

Odpowiedzi

2010-01-27T13:28:37+01:00
A) /x + 2y = -2
\ 2x - y = 1 *2
/x+2y=-2
\4x-2y=2
----------
x=0
y=-1

b)
/x + 4y = -2 *2
\ 3x - 8y = 10
/2x+8y=-4
\3x-8y=10
-------------
5x=6
x=1,2
1,2 +4y=-2
4y=3,2
y=0,8



c)
/ x - 2y = 3
\ -2x + y = 0 *2
/x-2y=3
\ -4x+2y=0
-------------
x=-1
-1-2y=3
y=-2


d)
/ 2x + y = 9 *4
\ 5x - 4y = 3
/8x+4y=36
\5x-4y=3
----------
13x=39
x=13
26+y=9
y=-17

e)
/ 3x - 2y = 5
\ 2x - y = 4 *-2
/3x-2y=5
\ -4x + 2y =-8
--------------
x=3
9-2y=5
2y=4
y=2

f)
/ 2x + 9y = -9
\ -3x + 3y = 8 *-3
/ 2x + 9y = -9
\ 9x - 9y = -24
----------------
11x=-33
x=-3
-6+9y=-9
9y=-3
y=1/3

g)
/ 2x + 3y = 4 *5
\ 8x - 5y = -1 *3
/ 10x + 15y = 20
\ 24x - 15y = -3
--------------------
34x=17
x=1/2
1+3y=4
y=1
2010-01-27T13:38:45+01:00
A)
x + 2y = -2
2x - y = 1
--------------
x = -2 -2y
2*(-2-2y) - y = 1
-4 -4y - y = 1
-5y = 5
y = -1
x = -2 -2*(-1) = -2+2 = 0
Odp. x = 0 , y = -1
b)
x + 4y = -2
3x - 8y = 10
--------------
x = -2 - 4y
3*(-2 - 4y) -8y =10
-6 -12y - 8y = 10
-20y = 16
y = -16/20 = -4/5
x = -2 -4*(-4/5) = -2 +16/5 = -10/5 + 16/5 = 6/5
Odp. x = 6/5 , y = -4/5
c)
x - 2y = 3
-2x + y = 0
--------------
x = 3 +2y
-2*(3+2y) +y = 0
-6 -4y + y = 0
-3y = 6
y = -2
x = 3+2*(-2) = 3 -4 = -1
Odp. x = - 1, y = -2
d)
2x + y = 9
5x - 4y = 3
------------------
y = 9 - 2x
5x - 4*(9-2x) = 3
5x -36 +8x = 3
13 x = 39
x = 3
y =9 - 2*3 = 9 - 6 = 3
Odp. x = 3, y = 3
e)
3x - 2y = 5
2x - y = 4
------------------
y = 2x - 4
3x -2*(2x-4) = 5
3x -4x +8 = 5
-x = 5-8 = -3
x = 3
y = 2*3 - 4 = 6 - 4 = 2
Odp. x = 3, y = 2
f)
2x + 9y = -9
-3x + 3y = 8
-------------------
3y = 3x + 8
2x + 3*(3x +8) = -9
2x +9x +24 = -9
11x = -9 -24 = -33
x = -33 :11 = -3
3y = 3*(-3) +8 = -9 +8 = -1
y = -1 :3 = -1/3
Odp. x = -3, y = -1/3
g)
2x + 3y = 4
8x - 5y = -1
--------------------
2x = 4 - 3y
x = 2 - (3/2)y
8*(2 -(3/2)y] - 5y = -1
16 - 12y - 5y = -1
-17 y = -1 - 16
-17y = -17
y = 1
x = 2 -(3/2)*1 = 4/2 - 3/2 = 1/2
Odp. x = 1/2 , y = 1
1 5 1
2010-01-27T14:27:35+01:00
A)
x + 2y = -2
2x - y = 1

x=-2-2y
2(-2-2y)-y=1
-4-4y-y=1
-5y=5
y=-1

x=-2-2(-1)
x=-2+2
x=0

b)
x + 4y = -2
3x - 8y = 10

x=-2-4y
3(-2-4y)-8y=10
-6-12y-8y=10
-20y=16
y=-0,8

x=-2-4(-0,8)
x=-2+3,2
x=1,2

c)
x - 2y = 3
-2x + y = 0

x=3+2y
-2(3+2y)+y=0
-6-4y+y=0
-3y=6
y=-2

x=3+2(-2)
x=3-4
x=-1

d)
2x + y = 9
5x - 4y = 3

y=9-2x
5x-4(9-2x)=3
5x-36+8x=3
13x=39
x=3

y=9-2*3
y=9-6
y=3

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

-y=4-2x
y=-4+2x
3x-2(-4+2x)=5
3x+8-4x=5
-x=-3
x=3

y=-4+2*3
y=-4+6
y=2

f)
2x + 9y = -9|:2
-3x + 3y = 8

3y=8+3x
2x+3(8+3x)=-9
2x+24+9x=-9
11x=-33
x=-3

3y=8+3(-3)
3y=8-9
3y=-1
y=-1/3

g)
2x + 3y = 4
8x - 5y = -1

2x=4-3y|:2
x=2-1,5y

8(2-1,5y)-5y=-1
16-12y-5y=-1
-17y=-17
y=1

x=2-1,5*1
x=2-1,5
x=0,5
1 1 1