Algunes solucions (via web) als exercicis de sistemes lineals per Gauss

Anem a utilitzar aquesta ferramenta web per a resoldre sistemes per Gauss:

https://matrixcalc.org/es/slu.html

Ací teniu totes les solucions als exercicis 65 a 71 de la pàgina 103 (en les formes matricial i amb incògnites, amb els enllaços, etc.). Ja vos he explicat a classe com usar la web, que té alguns avantatges respecte al que vos he contat com fer amb la calculadora científica.

[65a]
-1 1 1 5
2 -1 4 -5
1 1 -5 5

https://matrixcalc.org/en/slu.html#solve-using-Gaussian-elimination%28%7B%7B-1,1,1,5%7D,%7B2,-1,4,-5%7D,%7B1,1,-5,5%7D%7D%29

[65b]
1 -2 -1 -1
1 -1  2  2
1  2  1  3

https://matrixcalc.org/en/slu.html#solve-using-Gaussian-elimination%28%7B%7B1,-2,-1,-1%7D,%7B1,-1,2,2%7D,%7B1,2,1,3%7D%7D%29

[65c]
2 -1 -1  24
7 10  2   6
2  6  4 -10

https://matrixcalc.org/en/slu.html#solve-using-Gaussian-elimination%28%7B%7B2,-1,-1,24%7D,%7B7,10,2,6%7D,%7B2,6,4,-10%7D%7D%29

[65d]
-x+3y+z=-15
3x-y+3z=9
x+4x-2z=-4

https://matrixcalc.org/en/slu.html#solve-using-Gaussian-elimination%28%7B%7B-1,3,1,-15%7D,%7B3,-1,3,9%7D,%7B5,0,-2,-4%7D%7D%29

[65e]
3x+2y+2z=1
2x+y+z=0
3y+2z=-1

https://matrixcalc.org/en/slu.html#solve-using-Gaussian-elimination%28%7B%7B3,2,2,1%7D,%7B2,1,1,0%7D,%7B0,3,2,-1%7D%7D%29

[65f]
3x - 2y + 4z = 5
5x + 2y -2z = -1
-x +6y -10x = -11

https://matrixcalc.org/en/slu.html#solve-using-Gaussian-elimination%28%7B%7B3,-2,4,5%7D,%7B5,2,-2,-1%7D,%7B-11,6,0,-11%7D%7D%29

[66a]
Si multipliquem la tercera equació per 2, tenim aquest sistema ja sense fraccions:
x+y-z=3
2x-y-5z=0
x+6y+4z=13

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B1,1,-1,3%7D,%7B2,-1,-5,0%7D,%7B1,6,4,13%7D%7D%29


[66b]
x-y+z=1
-2x+y-z=-4
x+2y-2z=7

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B1,-1,1,1%7D,%7B-2,1,-1,-4%7D,%7B1,2,-2,7%7D%7D%29

[66c]
-x+y+z=5
2x-y-4z=-5
x+y-5z=5

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B-1,1,1,5%7D,%7B2,-1,-4,-5%7D,%7B1,1,-5,5%7D%7D%29

[66d]
x+y-5z=2
x-y-z=-4
y-2z=3

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B1,1,-5,2%7D,%7B1,-1,-1,-4%7D,%7B0,1,-2,3%7D%7D%29

[66e]
x-2z=-11
2y-z=-5
x-4y=-1

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B0,1,-2,-11%7D,%7B2,0,-1,-5%7D,%7B-4,1,0,-1%7D%7D%29

[66f]
-x+y-z=-7
3x-y-z=15
4x-2y=22

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B-1,1,-1,-7%7D,%7B3,-1,-1,15%7D,%7B4,-2,0,22%7D%7D%29

[67a]
x-y+z=3
2x+4z=2
2x-y+3z=1

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B1,-1,1,3%7D,%7B2,0,4,2%7D,%7B2,-1,3,1%7D%7D%29

[67b]
2x+y-z=0
x+3y-2z=1
x-2y+z=2

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B2,1,-1,0%7D,%7B1,3,-2,1%7D,%7B1,-2,1,2%7D%7D%29

[67c]
-x-y+2z=3
2x+3y+z=-2
2y+10z=8

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B-1,-1,2,3%7D,%7B2,3,1,-2%7D,%7B0,2,10,8%7D%7D%29

[67d]
x+2y-z=5
2x+y-2z=7
x-y-z=0

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B1,2,-1,5%7D,%7B2,1,-2,7%7D,%7B1,-1,-1,0%7D%7D%29

[67e]
x+5y-5z=12
-2x-3y+3z=-1
3x+y-z=0

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B1,5,-5,12%7D,%7B-2,-3,3,-1%7D,%7B3,1,-1,0%7D%7D%29

[67f]
x-5y+3z=10
2x-2y+z=-4
3x+y-z=5

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B1,-5,3,10%7D,%7B2,-2,1,-4%7D,%7B3,1,-1,5%7D%7D%29

[68a]
-x+y-z=6
y-4z=-7
-x=4

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B-1,1,-1,6%7D,%7B0,1,-4,-7%7D,%7B-1,0,0,4%7D%7D%29

[68b]
Escrivim ordenadament el sistema per a poder emprar el mètode de Gauss:
-2x-y+5z=-3
7x-5y+z=2
3x+3y+2z=2

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B-2,-1,5,-3%7D,%7B7,-5,1,2%7D,%7B3,3,2,2%7D%7D%29

[68c]

x-y+z=-4

x-z=-10

y-z=2

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B1,-1,1,-4%7D,%7B1,0,-1,-10%7D,%7B0,1,-1,2%7D%7D%29

[68d]

2x+y-2z=4

3x-2y+2z=3

-5x+3y+z=2

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B2,1,-2,4%7D,%7B3,-2,2,3%7D,%7B-5,3,1,2%7D%7D%29

[69a]
2x+4y+z=0
x-y+2z=0
-3x+2y+z=0

https://matrixcalc.org/en/slu.html#solve-using-Gaussian-elimination%28%7B%7B2,4,1,0%7D,%7B1,-1,2,0%7D,%7B-3,2,1,0%7D%7D%29


[69b]
2x-3y+z=0
x+9y-3z=0
5x+3y-z=0

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B2,-3,1,0%7D,%7B1,9,-3,0%7D,%7B5,3,-1,0%7D%7D%29


[69c]
6x+4y-12z=6
7x+6y+18z=7
-3x-2y+6z=-3

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B6,4,-12,6%7D,%7B7,6,18,7%7D,%7B-3,-2,6,-3%7D%7D%29


[69d]
3x-7y+4z=4
-10x+71y+9z=9
6x-3y-2z=10

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B3,-7,4,4%7D,%7B-10,71,9,9%7D,%7B6,-3,-2,10%7D%7D%29

[70a]
x+4y=2
2x-y+2z=1
7x =3

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B0,1,4,2%7D,%7B2,2,-1,1%7D,%7B0,7,0,3%7D%7D%29

[70b]
x+y+2z=4
-2x+y-z=3
-x+2y+z=1

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28%7B%7B1,1,2,4%7D,%7B-2,1,-1,3%7D,%7B-1,2,1,1%7D%7D%29

[71a]
Després de multiplicar cada equació per 4, 10 i 2 ens queda la següent expressió quasi matricial del sistema (que és la que podeu copiar en la web per a veure la solució):
2x-y+4z=43
20x+2y+5z=90
x-y-2z=-8
https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28{{2,-1,4,43},{20,2,5,90},{1,-1,-2,-8}}%29


[71b]
Després de desfer els parèntesis en les equacions 1a i 2a i de multiplicar per 30 la 3a obtenim la següent expressió del sistema:
-3x-3y+4z=1
2x-y-z=-5
15x+10y-6z=-1

https://matrixcalc.org/es/slu.html#solve-using-Gaussian-elimination%28{{-3,-3,4,1},{2,-1,-1,-5},{15,10,-6,-1}}%29