Forumet - Matematikhjälp sökes (Lösning av medelstort ekvationssystem igenom gauss elimination)

Matematikhjälp sökes (Lösning av medelstort ekvationssystem igenom gauss elimination)

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Uppgift

x-y-2z+2w = -1
-3x+4y+7z-12w = 2
2x-4y-3z-2w = -12
5x-y-3z-31w = -20

Lösning

x-y-2z+2w = -1
-3x+4y+7z-12w = 2
2x-4y-3z-2w = -12
5x-y-3z-31w = -20

x-y-2z+2w = -1
-3(x-y-2z+2w) = -3(-1)
-3x-(-3)y-(-3)2z+(-3)2w = 3
-3x+3y+6z-6w = 3

-3x+3y+6z-6w = 3
-3x+4y+7z-12w = 2
-3x+3y+6z-6w-(-3x+4y+7z-12w) = 3-2
-3x+3y+6z-6w+3x-4y-7z+12w = 1
-3x+3x+3y-4y+6z-7z-6w+12w = 1
-y-z+6w = 1

x-y-2z+2w = -1
2(x-y-2z+2w) = 2(-1)
2x-2y-2(2)z+2(2)w = -2
2x-2y-4z+4w = -2

2x-2y-4z+4w = -2
2x-4y-3z-2w = -12
2x-2y-4z+4w-(2x-4y-3z-2w) = -2-(-12)
2x-2y-4z+4w-2x+4y+3z+2w = 10
2x-2x-2y+4y-4z+3z+4w+2w = 10
2y-z+6w = 10

x-y-2z+2w = -1
5(x-y-2z+2w) = 5(-1)
5x-5y-5(2)z+5(2)w = -5
5x-5y-10z+10w = -5

5x-5y-10z+10w = -5
5x-y-3z-31w = -20
5x-5y-10z+10w-(5x-y-3z-31w) = -5-(-20)
5x-5y-10z+10w-5x+y+3z+31w = 15
5x-5x-5y+y-10z+3z+10w+31w = 15
-4y-7z+41w = 15

-y-z+6w = 1
2y-z+6w = 10
-4y-7z+41w = 15

-y-z+6w = 1
-2(-y-z+6w) = -2(1)
2y-(-2)z+(-2)6w = -2
2y+2z-12w = -2

2y+2z-12w = -2
2y-z+6w = 10
2y+2z-12w-(2y-z+6w) = -2-10
2y+2z-12w-2y+z-6w = -12
2y-2y+2z+z-12w-6w = -12
3z-18w = -12

3z-18w = -12
(3z-18w)/3 = -12/3
3z/3-18w/3 = -4
z-6w = -4

-y-z+6w = 1
4(-y-z+6w) = 4(1)
-4y-4z+4(6)w = 4
-4y-4z+24w = 4

-4y-4z+24w = 4
-4y-7z+41w = 15
-4y-4z+24w-(-4y-7z+41w) = 4-15
-4y-4z+24w+4y+7z-41w = -11
-4y+4y-4z+7z+24w-41w = -11
3z-17w = -11

z-6w = -4
3z-17w = -11

z-6w = -4
3(z-6w) = 3(-4)
3z-3(6)w = -12
3z-18w = -12

3z-18w = -12
3z-17w = -11
3z-18w-(3z-17w) = -12-(-11)
3z-18w-3z+17w = -1
3z-3z-18w+17w = -1
-w = -1

-w = -1
-1(-w) = -1(-1)
w = 1

z-6w = -4
w = 1
z-6(1) = -4
z = -4+6
z = 2

-y-z+6w = 1
z = 2
w = 1
-y-2+6(1) = 1
-y = 1+2-6
-y = -3

-y = -3
-1(-y) = -1(-3)
y = 3

x-y-2z+2w = -1
y = 3
z = 2
w = 1
x-3-2(2)+2(1) = -1
x = -1+3+4-2
x = 4

x = 4
y = 3
z = 2
w = 1

Resultat

x = 4
y = 3
z = 2
w = 1

Kontroll

x = 4
y = 3
z = 2
w = 1

x = 4
y = 3
z = 2
w = 1
x-y-2z+2w = 4-3-2(2)+2(1)
x-y-2z+2w = 1-4+2
x-y-2z+2w = -1

x = 4
y = 3
z = 2
w = 1
-3x+4y+7z-12w = -3(4)+4(3)+7(2)-12(1)
-3x+4y+7z-12w = -12+12+14-12
-3x+4y+7z-12w = 2

x = 4
y = 3
z = 2
w = 1
2x-4y-3z-2w = 2(4)-4(3)-3(2)-2(1)
2x-4y-3z-2w = 8-12-6-2
2x-4y-3z-2w = -12

x = 4
y = 3
z = 2
w = 1
5x-y-3z-31w = 5(4)-3-3(2)-31(1)
5x-y-3z-31w = 20-3-6-31
5x-y-3z-31w = -20

x-y-2z+2w = -1
-3x+4y+7z-12w = 2
2x-4y-3z-2w = -12
5x-y-3z-31w = -20

Resultat

x-y-2z+2w = -1
-3x+4y+7z-12w = 2
2x-4y-3z-2w = -12
5x-y-3z-31w = -20