Odpowiedź:
a)
x + 3y = - 4
-x + 4y = 11
--------------------------------
x + 3y - x + 4y = - 4 + 11
x - x + 3y + 4y = 7
7y = 7
y = 7 : 7
y = 1
x + 3y = - 4
x + 3*1 = - 4
x + 3 = - 4
x= - 4 - 3
x= - 7
spr.
x + 3y = - 4
L= x + 3y = -7 + 3*1 = - 7 + 3= - 4
P = - 4
-4= -4
L=P
b)
x - 2y = 3
3x + 5y = - 2
x= 3 + 2y
3x + 5y = - 2
x= 3 + 2y
3(3 + 2y) + 5y =- 2
x= 3 + 2y
9 + 6y + 5y = - 2
x= 3 + 2y
9 + 11y = - 2
x= 3 + 2y
11y = - 2 - 9
x= 3 + 2y
11y = - 11
x= 3 + 2y
y = - 11 : 11
x= 3 + 2y
y = - 1
x = 3 + 2*(-1)
y = - 1
x= 3 - 2
y = - 1
x= 1
y = -1
spr.
x - 2y = 3
L= x - 2y = 1 - 2(-1) = 1 + 2= 3
P= 3
3=3
L=P
c)
3x + y =13
2x + 3y = - 4
y = 13 - 3x
2x + 3(13 - 3x) = - 4
y = 13 - 3x
2x + 39 - 9x = - 4
y = 13 - 3x
-7x = - 4 - 39
y = 13 - 3x
- 7x = - 43
y = 13 - 3x
x = - 43 : (-7)
y = 13 - 3x
x = 43/7
y = 13 - 3x
x= 6 i 1/7
y = 13 - 3* 6 i 1/7
x= 6 i 1/7
y = 13 - 3 * 43/7
x= 6 i 1/7
y = 13 - 129/7
x = 6 i 1/7
y = 13 - 18 i 3/7
x = 6 i 1/7
y = - 5 i 3/7
x = 6 i 1/7
spr.
2x + 3y = - 4
L= 2x + 3y = 2 * 6 i 1/7 + 3* (-5 i 3/7) = 2* 43/7 + 3*(-38/7) = 86/7 - 114/7 =
= - 28/7= - 4
P= - 4
- 4 = - 4
L=P
d)
x + 3y = 7 /*(-2)
2x + 6y = 4
-2x - 6y = - 14
2x + 6y = 4
-----------------------------
- 2x - 6y + 2x + 6y = - 14+ 4
0 ≠- 10
Układ równań sprzeczny , nie ma rozwiązania
e)
3x + 4y = 12 /*(-2)
6x - 3y = 13
- 6x - 8y = - 24
6x - 3y = 13
----------------------------------
- 6x - 8y + 6x - 3y = - 24 + 13
- 11y = - 11
y = - 11 : (-11)
y = 1
6x - 3y = 13
6x - 3 *1 = 13
6x - 3 = 13
6x = 13 + 3
6x = 16
x = 16 : 6
x = 16/6
x = 2 i 4/6
x = 2 i 2/3
spr.
3x + 4y = 12
L = 3x + 4y = 3*2 i 2/3 + 4 * 1= 3 * 8/3 + 4 = 1 * 8/1 + 4= 8 + 4= 12
P = 12
12 = 12
L=P
f)
5(x - y) = 2(2x + y) - 5
2(x -y) = y + 1
5x - 5y = 4x + 2y - 5
2x - 2y = y + 1
5x - 5y - 4x - 2y = - 5
2x - 2y - y = 1
5x - 4x - 5y - 2y = - 5
2x - 3y = 1
x - 7y = - 5
2x - 3y = 1
x = - 5 + 7y
2(-5 + 7y) - 3y = 1
x = - 5 + 7y
- 10 + 14y - 3y = 1
x = - 5 + 7y
14y - 3y = 1 + 10
x = - 5 + 7y
11y = 11
x = - 5 + 7y
y = 11 : 11
x = - 5 + 7y
y = 1
x = - 5 + 7*1
y = 1
x = - 5 + 7
y = 1
x = 2
y = 1
spr.
2(x -y) = y + 1
L= 2(x -y) = 2(2 - 1) = 2 * 1= 2
P= y + 1 = 1 + 1= 2
2=2
L=P
Szczegółowe wyjaśnienie:
