Ejercicio 192

CAPITULO XXV

Sistema de cuatro Ecuaciones
con cuatro incognitas
Ejercicio 192
Resolver los sistemas
  1. { x+y+z+u=4 ( 1 ) x+2y+3zu=1 ( 2 ) 3x+4y+2z+u=5 ( 3 ) x+4y+3zu=7 ( 4 ) ( 1 ) +( 2 ) x+y+z+ u =4 x+2y+3z u =1 2x+3y+4z =3 ( 5 ) ( 1 ) ( 3 ) x+y+z+ u =4 3x4y2z u =5 2x3yz =9 ( 6 ) ( 1 ) +( 4 ) x+y+z+ u =4 x+4y+3z u =7 2x+5y+4z =3 ( 7 ) ( 5 ) +( 6 ) 2x + 3y +4z =3 2x 3y z =9 3z =12 z = 3 z =4 ( 7 ) +( 6 ) 2x +5y+4z =3 2x 3yz =9 2y+3z =6 Reemplazo el valor de z 2y+3( 4 ) =6 2y+12 =6 2y =12+6 y = 2 y =3 Reemplazo los valores de z e y en ( 7 ) 2x+5(3 ) +4( 4 ) =3 2x15+16 =3 2x =31 x = 2 x =2 Reemplazo los valores de x,y,z en ( 1 ) 23+4+u =4 u =4+1 u =5 Sol{ x=2 y=3 z=4 u=5
  2. { x+y+z+u=10 ( 1 ) 2xy2z+2u=2 ( 2 ) x2y+3zu=2 ( 3 ) x+2y4z+2u=1 ( 4 ) ( 1 ) +( 2 ) x+ y +z+u =10 2x y 2z+2u =2 3xz+3u =12 ( 5 ) 2( 1 ) +( 3 ) 2x+ 2y +2z+2u =20 x 2y +3zu =2 3x+5z+u =22 ( 6 ) 2( 1 ) ( 4 ) 2x+ 2y +2z+ 2u =20 x 2y +4z 2u =1 x+6z =19 ( 7 ) ( 5 ) ( 6 ) 3x z+3u =12 3x 5zu =22 6z+2u =10 ( 8 ) ( 5 ) 3( 7 ) 3x z+3u =12 3x 18z =57 19z+3u =45 ( 9 ) 3( 8 ) 2( 9 ) 18z+ 6u =30 38z 6u =90 20z =60 z = 20 z =3 Reemplazo el valor de z en ( 8 ) 6( 3 ) +2u =10 18+2u =10 2u =10+18 u = 2 u =4 Reemplazo los valores de z en ( 7 ) x+6( 3 ) =19 x+18 =19 x =1918 x =1 Reemplazo los valores de x,u,z en ( 1 ) 1+y+3+4 =10 y =108 y =2 Sol{ x=1 y=2 z=3 u=4
  3. { x2y+z+3u=3 ( 1 ) 3x+y4z2u=7 ( 2 ) 2x+2yzu=1 ( 3 ) x+4y+2z5u=12 ( 4 ) ( 1 ) +2( 2 ) x 2y +z+3u =3 6x+ 2y 8z4u =14 7x7zu =11 ( 5 ) ( 1 ) +( 3 ) x 2y + z +3u =3 2x+ 2y z u =1 3x+2u =2 ( 6 ) 2( 1 ) +( 4 ) 2x 4y +2z+6u =6 x+ 4y +2z5u =12 3x+4z+u =6 ( 7 ) 2( 5 ) +( 6 ) 14x14z 2u =22 3x+ 2u =2 17x14z =20 ( 8 ) ( 5 ) +( 7 ) 7x7z u =11 3x+4z+ u =6 10x3z =17 ( 9 ) 10( 8 ) 17( 9 ) 170x 140z =200 170x +51z =289 89z =89 z = 89 89 z =1 Reemplazo el valor de z en ( 8 ) 17x14( 1 ) =20 17x14 =20 17x =20+14 x = 17 x =2 Reemplazo los valores de x, z en ( 7 ) 3( 2 ) +4( 1 ) +u =6 6 +4+u = 6 u =4 Reemplazo los valores de x,u,z en ( 1 ) 22y+1+3(4 ) =3 2y12 =33 2y =6+12 y = 2 y =3 Sol{ x=2 y=3 z=1 u=4
  4. { 2x3y+z+4u=0 ( 1 ) 3x+y5z3u=10 ( 2 ) 6x+2yz+u=3 ( 3 ) x+5y+4z3u=6 ( 4 ) ( 1 ) +3( 2 ) 2x 3y +z+4u =0 9x+ 3y 15z9u =30 11x14z5u =30 ( 5 ) 2( 1 ) +3( 3 ) 4x 6y +2z+8u =0 18x+ 6y 3z+3u =9 22xz+11u =9 ( 6 ) 5( 1 ) +3( 4 ) 10x 15y +5z+20u =0 3x+ 15y +12z9u =18 13x+17z+11u =18 ( 7 ) 11( 5 ) +5( 6 ) 121x154z 55u =330 110x5z+ 55u =45 231x159z =375 77x53z =125 ( 8 ) ( 6 ) ( 7 ) 22xz+ 11u =9 13x17z 11u =18 9x18z =9 x2z =1 ( 9 ) ( 8 ) 77( 9 ) 77x 53z =125 77x +154z =77 101z =202 z = 101 z =2 Reemplazo el valor de z en ( 9 ) x2(2 ) =1 x+4 =1 x =14 x =3 Reemplazo los valores de x, z en ( 6 ) 22(3 ) (2 ) +11u =9 66+2+11u =9 11u =9+64 u = 11 u =5 Reemplazo los valores de x,u,z en ( 1 ) 2(3 ) 3y2+4( 5 ) =0 63y2+20 =0 3y+12 =0 y = 3 y =4 Sol{ x=3 y=4 z=2 u=5
  5. { x+yz=4 ( 1 ) 4x+3y+2zu=9 ( 2 ) 2xy4z+u=1 ( 3 ) x+2y+3z+2u=1 ( 4 ) 3( 1 ) ( 2 ) 3x+ 3y 3z =12 4x 3y 2z+u =9 x5z+u =21 ( 5 ) ( 1 ) +( 3 ) x+ y z =4 2x y 4z+u =1 3x5z+u =5 ( 6 ) 2( 1 ) ( 4 ) 2x+ 2y 2z =8 x 2y 3z2u =1 x5z2u =7 ( 7 ) ( 5 ) ( 6 ) x 5z + u =21 3x+ 5z u =5 4x =16 x = 4 x =4 2( 6 ) +( 7 ) 6x10z+ 2u =10 x5z 2u =7 7x15z =17 ( 9 ) Reemplazo el valor de x 7( 4 ) 15z =17 2815z =17 15z =1728 z = 15 z =3 Reemplazo los valores de x, z en ( 6 ) 3( 4 ) 5( 3 ) +u =5 1215+u =5 u =5+3 u =2 Reemplazo los valores de x,u,z en ( 3 ) 2( 4 ) y4( 3 ) 2 =1 8y122 =1 y6 =1 y =1+6 y =5 Sol{ x=4 y=5 z=3 u=2
  6. { x+2y+z=4 ( 1 ) 2x+3y+4z=2 ( 2 ) 3x+y+z+u=4 ( 3 ) 6x+3yz+u=3 ( 4 ) 3( 1 ) 2( 2 ) 3x+ 6y +3z =12 4x 6y 8z =4 x5z =8 ( 5 ) ( 1 ) 2( 3 ) x+ 2y +z =4 6x 2y 2z2u =8 5xz2u =12 ( 6 ) 3( 1 ) 2( 4 ) 3x+ 6y +3z =12 12x 6y +2z2u =6 9x+5z2u =18 ( 7 ) ( 5 ) +( 7 ) x 5z =8 9x+ 5z 2u =18 10x2u =26 5x+u =13 ( 8 ) 5( 6 ) +( 7 ) 25x 5z 10u =60 9x+ 5z 2u =18 34x12u =78 17x+6u =39 ( 9 ) 6( 8 ) ( 9 ) 30x+ 6u =78 17x 6u =39 13x =39 x = 13 x =3 Reemplazo el valor de x en ( 5 ) 35z =8 5z =8+3 5z =5 z = 5 5 z =1 Reemplazo los valores de x, z en ( 6 ) 5( 3 ) 12u =12 152u =12+1 2u =11+15 u = 2 u =2 Reemplazo los valores de x,z en ( 1 ) 3+2y+1 =4 2y =44 2y =8 y = 2 y =4 Sol{ x=3 y=4 z=1 u=2
  7. { 3x+2y=2 ( 1 ) x+y+u=3 ( 2 ) 3x2yu=7 ( 3 ) 4x+5y+6z+3u=11 ( 4 ) ( 1 ) 2( 2 ) 3x+ 2y =2 2x 2y 2u =6 x2u =4 ( 5 ) ( 1 ) +( 3 ) 3x+ 2y =2 3x 2y u =7 6xu =9 ( 6 ) ( 5 ) 2( 6 ) x 2u =4 12x+ 2u =18 11x =22 x = 11 x =2 Reemplazo el valor de x en ( 5 ) 22u =4 2u =4+2 2u =6 u = 2 u =3 Reemplazo los valores de x, u en ( 2 ) 2+y 3 = 3 y =2 Reemplazo los valores de x,u,y en ( 4 ) 4(2 ) +5( 2 ) +6z+3(3 ) =11 8+10+6z9 =11 6z =11+7 z = 6 z =3 Sol{ x=2 y=2 z=3 u=3
  8. { 2x3zu=2 ( 1 ) 3y2z5u=3 ( 2 ) 4y3u=2 ( 3 ) x3y+3u=0 ( 4 ) 5( 1 ) ( 2 ) 10x15z 5u =10 3y+2z+ 5u =3 10x3y13z =7 ( 5 ) 3( 1 ) ( 3 ) 6x9z 3u =6 4y+ 3u =2 6x4y9z =4 ( 6 ) 3( 1 ) +( 4 ) 6x9z 3u =6 x3y+ 3u =0 7x3y9z =6 ( 7 ) ( 5 ) ( 7 ) 10x 3y 13z =7 7x+ 3y +9z =6 3x4z =1 ( 8 ) 3( 6 ) 4( 7 ) 18x 12y 27z =12 28x+ 12y +36z =24 10x+9z =12 ( 9 ) 10( 8 ) +3( 9 ) 30x 40z =10 30x +27z =36 13z =26 z = 13 z =2 Reemplazo el valor de z en ( 8 ) 3x4( 2 ) =1 3x8 =1 3x =1+8 x = 3 x =3 Reemplazo los valores de x, z en ( 6 ) 6( 3 ) 4y9( 2 ) =4 18 4y 18 =4 y = 4 4 y =1 Reemplazo los valores de y en ( 3 ) 4(1 ) 3u =2 43u =2 3u =2+4 u = 3 u =2 Sol{ x=3 y=1 z=2 u=2

Ejercicio 191

CAPITULO XXV

Resolución y representación gráfica de un sistema de tres Ecuaciones
con tres incognitas
Para graficar este grupo de ejercicios, se utilizara el software matemático Geogebra 6, el cual es un programa multiplataforma y cada ejercicio tendrá un link para la descarga del mismo realizado en Geogebra 6.
Ejercicio 191
Resolver y representar gráficamente los sistemas:
  1. { x+2y+z=8 2x+2y+z=9 3x+3y+5z=24 x+2y+z =8 2x+2y+z =9 3x+3y+5z =24 x=0y=0 x=0y=0 x=0y=0 z=8 z=9 5z=24 z= 24 5 y=0z=0 y=0z=0 y=0z=0 x=8 2x=9 3x=24 x= 9 2 x= 3 x=8 x=0z=0 x=0z=0 x=0z=0 2y=8 2y=9 3y=24 y= 2 y= 9 2 y= 3 y=4 y=8
    punto 3d
  2. { x+y+z=5 3x+2y+z=8 2x+3y+3z=14 x+y+z =5 3x+2y+z =8 2x+3y+3z =14 x=0y=0 x=0y=0 x=0y=0 z=5 z=8 3z=14 z= 14 3 y=0z=0 y=0z=0 y=0z=0 x=5 3x=8 2x=14 x= 8 3 x= 2 x=7 x=0z=0 x=0z=0 x=0z=0 y=5 2y=8 3y=14 y= 2 y= 14 3 y=4
    imagen 3d
  3. { 2x+2y+3z=23 2x+3y+2z=20 4x+3y+2z=24 2x+2y+3z =23 2x+3y+2z =20 4x+3y+2z =24 x=0y=0 x=0y=0 x=0y=0 3z=23 2z=20 2z=24 z= 23 3 z= 2 z= 2 z=10 z=12 y=0z=0 y=0z=0 y=0z=0 2x=23 2x=20 4x=24 x= 23 2 x= 2 x= 4 x=10 x=6 x=0z=0 x=0z=0 x=0z=0 2y=23 3y=20 3y=24 y= 23 2 y= 20 3 y= 3 y=8
    planos 3d
  4. { 2x+2y+3z=24 4x+5y+2z=35 3x+2y+z=19 2x+2y+3z =24 4x+5y+2z =35 3x+2y+z =19 x=0y=0 x=0y=0 x=0y=0 3z=24 2z=35 z=19 z= 3 z= 35 2 z=8 y=0z=0 y=0z=0 y=0z=0 2x=24 4x=35 3x=19 x= 2 x= 35 4 x= 19 3 x=12 x=0z=0 x=0z=0 x=0z=0 2y=24 5y=35 2y=19 y= 2 y= 5 y= 19 2 y=12 y=7
    planos R³
  5. { 3x+4y+5z=35 2x+5y+3z=27 2x+y+z=13 3x+4y+5z =35 2x+5y+3z =27 2x+y+z =13 x=0y=0 x=0y=0 x=0y=0 5z=35 3z=27 z=13 z= 5 z= 3 z=7 z=9 y=0z=0 y=0z=0 y=0z=0 3x=35 2x=27 2x=13 x= 35 3 x= 27 2 x= 13 2 x=0z=0 x=0z=0 x=0z=0 4y=35 5y=27 y=13 y= 35 4 y= 27 5
    puntos en R³
  6. { 4x+3y+5z=42 3x+4y+3z=33 2x+5y+2z=29 4x+3y+5z =42 3x+4y+3z =33 2x+5y+2z =29 x=0y=0 x=0y=0 x=0y=0 5z=42 3z=33 2z=29 z= 42 5 z= 3 z= 29 2 z=11 y=0z=0 y=0z=0 y=0z=0 4x=42 3x=33 2x=29 x= x= 3 x= 29 2 x= 21 2 x=11 x=0z=0 x=0z=0 x=0z=0 3y=42 4y=33 5y=29 y= 3 y= 33 4 y= 29 5 y=14
    rectas en R³

Ejercicio 190

CAPITULO XXV

Representación gráfica en el espacio
Representación gráfica de una ecuación de prtimer grado con tres variables
Para graficar este grupo de ejercicios, se utilizara el software matemático Geogebra 6, el cual es un programa multiplataforma y cada ejercicio tendrá un link para la descarga del mismo realizado en Geogebra 6.
Ejercicio 190
Representar gráficamente las ecuaciones:
  1. 3x+6y+2z =6 x=0y=0 2z=6 z= 2 z=3 y=0z=0 3x=6 x= 3 x=2 x=0z=0 6y=6 y= 6 6 y=1
    representacion 3d ecuaciones
  2. 2x+y+4z =4 x=0y=0 4z=4 z= 4 4 z=1 y=0z=0 2x=4 x= 2 x=2 x=0z=0 y=4
    representacion 3d ecuaciones
  3. 4x+6y+3z =12 x=0y=0 3z=12 z= 3 z=4 y=0z=0 4x=12 x= 4 x=3 x=0z=0 6y=12 y= 6 y=2
    representacion 3d ecuaciones
  4. 15x+6y+5z =30 x=0y=0 5z=30 z= 5 z=6 y=0z=0 15x=30 x= 15 x=2 x=0z=0 6y=30 y= 6 y=5
    representacion 3d ecuaciones
  5. 2x+y+3z =6 x=0y=0 3z=6 z= 3 z=2 y=0z=0 2x=6 x= 2 x=3 x=0z=0 y=6
    representacion 3d ecuaciones
  6. 15x+10y+6z =30 x=0y=0 6z=30 z= 6 z=5 y=0z=0 15x=30 x= 15 x=2 x=0z=0 10y=30 y= 3 0 1 0 y=3
    representacion 3d ecuaciones
  7. 14x+10y+5z =35 x=0y=0 5z=35 z= 5 z=7 y=0z=0 14x=35 x= x= 5 2 x=0z=0 10y=35 y= y= 7 2
    representacion 3d ecuaciones
  8. 3x+y+2z =10 x=0y=0 2z=10 z= 2 z=5 y=0z=0 3x=10 x= 10 3 x=0z=0 y=10
    representacion 3d ecuaciones
  9. 4x+2y+3z =18 x=0y=0 3z=18 z= 3 z=6 y=0z=0 4x=18 x= x= 9 2 x=0z=0 2y=18 y= 2 y=9
    representacion 3d ecuaciones
  10. 15x+20y+24z =120 x=0y=0 24z=120 z= 24 z=5 y=0z=0 15x=120 x= 15 x=8 x=0z=0 20y=120 y= 20 y=6
    representacion 3d ecuaciones

Ejercicio 189

CAPITULO XXV

Representación gráfica en el espacio
Representación gráfica de puntos en el espacio

Para graficar este grupo de ejercicios, se utilizara el software matemático Geogebra 6, el cual es un programa multiplataforma y cada ejercicio tendrá un link para la descarga del mismo realizado en Geogebra 6.
Ejercicio 189
Represente gráficamente los puntos siguientes:
  1. (1,1,3)
  2. (4,2,3)
  3. (5,4,2)
  4. (3,5,6)
  5. (2,4,1)
  6. (4,3,7)
  7. (7,5,4)
  8. (3,1,6)
  9. (6,3,4)
  10. (4,0,4)
  11. (4,2,0)
  12. (5,6,0)
  13. (0,0,4)
  14. (5,0,0)
  15. (0,5,0)

Ejercicio 188

CAPITULO XXV

Ecuaciones simultáneas con tres incógnitas
Resolución por determinantes
Ejercicio 188
Resolver por determinantes:
  1. { x+y+z=11 xy+3z=13 2x+2yz=7 x = | 11 1 1 13 1 3 7 2 1 | | 1 1 1 1 1 3 2 2 1 | = | 11 1 1 13 1 3 7 2 1 11 1 1 13 1 3 | | 1 1 1 1 1 3 2 2 1 1 1 1 1 1 3 | = [(11 × 1 × 1 ) +(13 × 2 × 1 ) +(7 × 1 × 3 ) ] [(7 × 1 × 1 ) +(11 × 2 × 3 ) +(1 × 1 × 13 ) ] [(1 × 1 × 1 ) +(1 × 2 × 1 ) +(2 × 1 × 3 ) ] [(2 × 1 × 1 ) +(1 × 2 × 3 ) +(1 × 1 × 1 ) ] = [11+26+21 ] [7+6613 ] [1+2+6 ] [2+61 ] = 5846 93 = 6 = 2 y = | 1 11 1 1 13 3 2 7 1 | | 1 1 1 1 1 3 2 2 1 | = | 1 11 1 1 13 3 2 7 1 1 11 1 1 13 3 | | 1 1 1 1 1 3 2 2 1 1 1 1 1 1 3 | = [(1 × 13 × 1 ) +(1 × 7 × 1 ) +(2 × 11 × 3 ) ] [(2 × 13 × 1 ) +(1 × 7 × 3 ) +(1 × 11 × 1 ) ] [(1 × 1 × 1 ) +(1 × 2 × 1 ) +(2 × 1 × 3 ) ] [(2 × 1 × 1 ) +(1 × 2 × 3 ) +(1 × 1 × 1 ) ] = [13+7+66 ] [26+2111 ] [1+2+6 ] [2+61 ] = 6036 93 = 6 = 4 z = | 1 1 11 1 1 13 2 2 7 | | 1 1 1 1 1 3 2 2 1 | = | 1 1 11 1 1 13 2 2 7 1 1 11 1 1 13 | | 1 1 1 1 1 3 2 2 1 1 1 1 1 1 3 | = [(1 × 1 × 7 ) +(1 × 2 × 11 ) +(2 × 1 × 13 ) ] [(2 × 1 × 11 ) +(1 × 2 × 13 ) +(1 × 1 × 7 ) ] [(1 × 1 × 1 ) +(1 × 2 × 1 ) +(2 × 1 × 3 ) ] [(2 × 1 × 1 ) +(1 × 2 × 3 ) +(1 × 1 × 1 ) ] = [7+22+26 ] [22+26+7 ] [1+2+6 ] [2+61 ] = 4111 93 = 6 = 5 Sol.{ x=2 y=4 z=5
  2. { x+y+z=6 2x+yz=1 x2y+3z=6 x = | 6 1 1 1 1 1 6 2 3 | | 1 1 1 2 1 1 1 2 3 | = | 6 1 1 1 1 1 6 2 3 6 1 1 1 1 1 | | 1 1 1 2 1 1 1 2 3 1 1 1 2 1 1 | = [(6 × 1 × 3 ) +(1 × 2 × 1 ) +(6 × 1 × 1 ) ] [(6 × 1 × 1 ) +(6 × 2 × 1 ) +(1 × 1 × 3 ) ] [(1 × 1 × 3 ) +(2 × 2 × 1 ) +(1 × 1 × 1 ) ] [(1 × 1 × 1 ) +(1 × 2 × 1 ) +(2 × 1 × 3 ) ] = [18+2+6 ] [6123 ] [341 ] [1+2+6 ] = 10+21 29 = 11 11 = 1 y = | 1 6 1 2 1 1 1 6 3 | | 1 1 1 2 1 1 1 2 3 | = | 1 6 1 2 1 1 1 6 3 1 6 1 2 1 1 | | 1 1 1 2 1 1 1 2 3 1 1 1 2 1 1 | = [(1 × 1 × 3 ) +(2 × 6 × 1 ) +(1 × 6 × 1 ) ] [(1 × 1 × 1 ) +(1 × 6 × 1 ) +(2 × 6 × 3 ) ] [(1 × 1 × 3 ) +(2 × 2 × 1 ) +(1 × 1 × 1 ) ] [(1 × 1 × 1 ) +(1 × 2 × 1 ) +(2 × 1 × 3 ) ] = [312+6 ] [1+636 ] [341 ] [1+2+6 ] = 9+31 29 = 11 = 2 z = | 1 1 6 2 1 1 1 2 6 | | 1 1 1 2 1 1 1 2 3 | = | 1 1 6 2 1 1 1 2 6 1 1 6 2 1 1 | | 1 1 1 2 1 1 1 2 3 1 1 1 2 1 1 | = [(1 × 1 × 6 ) +(2 × 2 × 6 ) +(1 × 1 × 1 ) ] [(1 × 1 × 6 ) +(1 × 2 × 1 ) +(2 × 1 × 6 ) ] [(1 × 1 × 3 ) +(2 × 2 × 1 ) +(1 × 1 × 1 ) ] [(1 × 1 × 1 ) +(1 × 2 × 1 ) +(2 × 1 × 3 ) ] = [6+241 ] [6+212 ] [341 ] [1+2+6 ] = 17+16 29 = 11 = 3 Sol.{ x=1 y=2 z=3
  3. { 2x+3y+4z=3 2x+6y+8z=5 4x+9y4z=4 x = | 3 3 4 5 6 8 4 9 4 | | 2 3 4 2 6 8 4 9 4 | = | 3 3 4 5 6 8 4 9 4 3 3 4 5 6 8 | | 2 3 4 2 6 8 4 9 4 2 3 4 2 6 8 | = [(3 × 6 × 4 ) +(5 × 9 × 4 ) +(4 × 3 × 8 ) ] [(4 × 6 × 4 ) +(3 × 9 × 8 ) +(5 × 3 × 4 ) ] [(2 × 6 × 4 ) +(2 × 9 × 4 ) +(4 × 3 × 8 ) ] [(4 × 6 × 4 ) +(2 × 9 × 8 ) +(2 × 3 × 4 ) ] = [72+180+96 ] [96+21660 ] [48+72+96 ] [96+14424 ] = 204252 120216 = 48 = 1 2 y = | 2 3 4 2 5 8 4 4 4 | | 2 3 4 2 6 8 4 9 4 | = | 2 3 4 2 5 8 4 4 4 2 3 4 2 5 8 | | 2 3 4 2 6 8 4 9 4 2 3 4 2 6 8 | = [(2 × 5 × 4 ) +(2 × 4 × 4 ) +(4 × 3 × 8 ) ] [(4 × 5 × 4 ) +(2 × 4 × 8 ) +(2 × 3 × 4 ) ] [(2 × 6 × 4 ) +(2 × 9 × 4 ) +(4 × 3 × 8 ) ] [(4 × 6 × 4 ) +(2 × 9 × 8 ) +(2 × 3 × 4 ) ] = [40+32+96 ] [80+6424 ] [48+72+96 ] [96+14424 ] = 88120 120216 = 32 = 1 3 z = | 2 3 3 2 6 5 4 9 4 | | 2 3 4 2 6 8 4 9 4 | = | 2 3 3 2 6 5 4 9 4 2 3 3 2 6 5 | | 2 3 4 2 6 8 4 9 4 2 3 4 2 6 8 | = [(2 × 6 × 4 ) +(2 × 9 × 3 ) +(4 × 3 × 5 ) ] [(4 × 6 × 3 ) +(2 × 9 × 5 ) +(2 × 3 × 4 ) ] [(2 × 6 × 4 ) +(2 × 9 × 4 ) +(4 × 3 × 8 ) ] [(4 × 6 × 4 ) +(2 × 9 × 8 ) +(2 × 3 × 4 ) ] = [48+54+60 ] [72+90+24 ] [48+72+96 ] [96+14424 ] = 162186 120216 = 24 = 1 4 Sol.{ x= 1 2 y= 1 3 z= 1 4
  4. { 4xy+z=4 2yz+2x=2 6x+3z2y=12 { 4xy+z=4 2x+2yz=2 6x2y+3z=12 x = | 4 1 1 2 2 1 12 2 3 | | 4 1 1 2 2 1 6 2 3 | = | 4 1 1 2 2 1 12 2 3 4 1 1 2 2 1 | | 4 1 1 2 2 1 6 2 3 4 1 1 2 2 1 | = [(4 × 2 × 3 ) +(2 × 2 × 1 ) +(12 × 1 × 1 ) ] [(12 × 2 × 1 ) +(4 × 2 × 1 ) +(2 × 1 × 3 ) ] [(4 × 2 × 3 ) +(2 × 2 × 1 ) +(6 × 1 × 1 ) ] [(6 × 2 × 1 ) +(4 × 2 × 1 ) +(2 × 1 × 3 ) ] = [244+12 ] [24+86 ] [244+6 ] [12+86 ] = 3226 2614 = 6 = 1 2 y = | 4 4 1 2 2 1 6 12 3 | | 4 1 1 2 2 1 6 2 3 | = | 4 4 1 2 2 1 6 12 3 4 4 1 2 2 1 | | 4 1 1 2 2 1 6 2 3 4 1 1 2 2 1 | = [(4 × 2 × 3 ) +(2 × 12 × 1 ) +(6 × 4 × 1 ) ] [(6 × 2 × 1 ) +(4 × 12 × 1 ) +(2 × 4 × 3 ) ] [(4 × 2 × 3 ) +(2 × 2 × 1 ) +(6 × 1 × 1 ) ] [(6 × 2 × 1 ) +(4 × 2 × 1 ) +(2 × 1 × 3 ) ] = [24+2424 ] [1248+24 ] [244+6 ] [12+86 ] = 24+12 2614 = 12 = 3 z = | 4 1 4 2 2 2 6 2 12 | | 4 1 1 2 2 1 6 2 3 | = | 4 1 4 2 2 2 6 2 12 4 1 4 2 2 2 | | 4 1 1 2 2 1 6 2 3 4 1 1 2 2 1 | = [(4 × 2 × 12 ) +(2 × 2 × 4 ) +(6 × 1 × 2 ) ] [(6 × 2 × 4 ) +(4 × 2 × 2 ) +(2 × 1 × 12 ) ] [(4 × 2 × 3 ) +(2 × 2 × 1 ) +(6 × 1 × 1 ) ] [(6 × 2 × 1 ) +(4 × 2 × 1 ) +(2 × 1 × 3 ) ] = [961612 ] [481624 ] [244+6 ] [12+86 ] = 688 2614 = 12 = 5 Sol.{ x= 1 2 y=3 z=5
  5. { x+4y+5z=11 3x2y+z=5 4x+y3z=26 x = | 11 4 5 5 2 1 26 1 3 | | 1 4 5 3 2 1 4 1 3 | = | 11 4 5 5 2 1 26 1 3 11 4 5 5 2 1 | | 1 4 5 3 2 1 4 1 3 1 4 5 3 2 1 | = [(11 × 2 × 3 ) +(5 × 1 × 5 ) +(26 × 4 × 1 ) ] [(26 × 2 × 5 ) +(11 × 1 × 1 ) +(5 × 4 × 3 ) ] [(1 × 2 × 3 ) +(3 × 1 × 5 ) +(4 × 4 × 1 ) ] [(4 × 2 × 5 ) +(1 × 1 × 1 ) +(3 × 4 × 3 ) ] = [66+25104 ] [260+1160 ] [6+15+16 ] [40+136 ] = 13211 37+75 = 112 = 2 y = | 1 11 5 3 5 1 4 26 3 | | 1 4 5 3 2 1 4 1 3 | = | 1 11 5 3 5 1 4 26 3 1 11 5 3 5 1 | | 1 4 5 3 2 1 4 1 3 1 4 5 3 2 1 | = [(1 × 5 × 3 ) +(3 × 26 × 5 ) +(4 × 11 × 1 ) ] [(4 × 5 × 5 ) +(1 × 26 × 1 ) +(3 × 11 × 3 ) ] [(1 × 2 × 3 ) +(3 × 1 × 5 ) +(4 × 4 × 1 ) ] [(4 × 2 × 5 ) +(1 × 1 × 1 ) +(3 × 4 × 3 ) ] = [15390+44 ] [1002699 ] [6+15+16 ] [40+136 ] = 361+25 37+75 = 112 = 3 z = | 1 4 11 3 2 5 4 1 26 | | 1 4 5 3 2 1 4 1 3 | = | 1 4 11 3 2 5 4 1 26 1 4 11 3 2 5 | | 1 4 5 3 2 1 4 1 3 1 4 5 3 2 1 | = [(1 × 2 × 26 ) +(3 × 1 × 11 ) +(4 × 4 × 5 ) ] [(4 × 2 × 11 ) +(1 × 1 × 5 ) +(3 × 4 × 26 ) ] [(1 × 2 × 3 ) +(3 × 1 × 5 ) +(4 × 4 × 1 ) ] [(4 × 2 × 5 ) +(1 × 1 × 1 ) +(3 × 4 × 3 ) ] = [52+33+80 ] [88+5312 ] [6+15+16 ] [40+136 ] = 165+395 37+75 = 112 = 5 Sol.{ x=2 y=3 z=5
  6. { 7x+10y+4z=2 5x2y+6z=38 3x+yz=21 x = | 2 10 4 38 2 6 21 1 1 | | 7 10 4 5 2 6 3 1 1 | = | 2 10 4 38 2 6 21 1 1 2 10 4 38 2 6 | | 7 10 4 5 2 6 3 1 1 7 10 4 5 2 6 | = [(2 × 2 × 1 ) +(38 × 1 × 4 ) +(21 × 10 × 6 ) ] [(21 × 2 × 4 ) +(2 × 1 × 6 ) +(38 × 10 × 1 ) ] [(7 × 2 × 1 ) +(5 × 1 × 4 ) +(3 × 10 × 6 ) ] [(3 × 2 × 4 ) +(7 × 1 × 6 ) +(5 × 10 × 1 ) ] = [4+152+1260 ] [16812380 ] [14+20+180 ] [24+4250 ] = 1408+560 214+32 = 246 = 8 y = | 7 2 4 5 38 6 3 21 1 | | 7 10 4 5 2 6 3 1 1 | = | 7 2 4 5 38 6 3 21 1 7 2 4 5 38 6 | | 7 10 4 5 2 6 3 1 1 7 10 4 5 2 6 | = [(7 × 38 × 1 ) +(5 × 21 × 4 ) +(3 × 2 × 6 ) ] [(3 × 38 × 4 ) +(7 × 21 × 6 ) +(5 × 2 × 1 ) ] [(7 × 2 × 1 ) +(5 × 1 × 4 ) +(3 × 10 × 6 ) ] [(3 × 2 × 4 ) +(7 × 1 × 6 ) +(5 × 10 × 1 ) ] = [266+42036 ] [456+882+10 ] [14+20+180 ] [24+4250 ] = 1181348 214+32 = 246 = 5 z = | 7 10 2 5 2 38 3 1 21 | | 7 10 4 5 2 6 3 1 1 | = | 7 10 2 5 2 38 3 1 21 7 10 2 5 2 38 | | 7 10 4 5 2 6 3 1 1 7 10 4 5 2 6 | = [(7 × 2 × 21 ) +(5 × 1 × 2 ) +(3 × 10 × 38 ) ] [(3 × 2 × 2 ) +(7 × 1 × 38 ) +(5 × 10 × 21 ) ] [(7 × 2 × 1 ) +(5 × 1 × 4 ) +(3 × 10 × 6 ) ] [(3 × 2 × 4 ) +(7 × 1 × 6 ) +(5 × 10 × 1 ) ] = [29410+1140 ] [12+266+1050 ] [14+20+180 ] [24+4250 ] = 8361328 214+32 = 246 = 2 Sol.{ x=8 y=5 z=2
  7. { 4x+7y+5z=2 6x+3y+7z=6 xy+9z=21 x = | 2 7 5 6 3 7 21 1 9 | | 4 7 5 6 3 7 1 1 9 | = | 2 7 5 6 3 7 21 1 9 2 7 5 6 3 7 | | 4 7 5 6 3 7 1 1 9 4 7 5 6 3 7 | = [(2 × 3 × 9 ) +(6 × 1 × 5 ) +(21 × 7 × 7 ) ] [(21 × 3 × 5 ) +(2 × 1 × 7 ) +(6 × 7 × 9 ) ] [(4 × 3 × 9 ) +(6 × 1 × 5 ) +(1 × 7 × 7 ) ] [(1 × 3 × 5 ) +(4 × 1 × 7 ) +(6 × 7 × 9 ) ] = [54301029 ] [315+14+378 ] [10830+49 ] [1528+378 ] = 111377 127365 = 238 = 5 y = | 4 2 5 6 6 7 1 21 9 | | 4 7 5 6 3 7 1 1 9 | = | 4 2 5 6 6 7 1 21 9 4 2 5 6 6 7 | | 4 7 5 6 3 7 1 1 9 4 7 5 6 3 7 | = [(4 × 6 × 9 ) +(6 × 21 × 5 ) +(1 × 2 × 7 ) ] [(1 × 6 × 5 ) +(4 × 21 × 7 ) +(6 × 2 × 9 ) ] [(4 × 3 × 9 ) +(6 × 1 × 5 ) +(1 × 7 × 7 ) ] [(1 × 3 × 5 ) +(4 × 1 × 7 ) +(6 × 7 × 9 ) ] = [21663014 ] [30588108 ] [10830+49 ] [1528+378 ] = 452+666 127365 = 238 238 = 1 z = | 4 7 2 6 3 6 1 1 21 | | 4 7 5 6 3 7 1 1 9 | = | 4 7 2 6 3 6 1 1 21 4 7 2 6 3 6 | | 4 7 5 6 3 7 1 1 9 4 7 5 6 3 7 | = [(4 × 3 × 21 ) +(6 × 1 × 2 ) +(1 × 7 × 6 ) ] [(1 × 3 × 2 ) +(4 × 1 × 6 ) +(6 × 7 × 21 ) ] [(4 × 3 × 9 ) +(6 × 1 × 5 ) +(1 × 7 × 7 ) ] [(1 × 3 × 5 ) +(4 × 1 × 7 ) +(6 × 7 × 9 ) ] = [252+12+42 ] [624882 ] [10830+49 ] [1528+378 ] = 198+912 127365 = 238 = 3 Sol.{ x=5 y=1 z=3
  8. { 3x5y+2z=22 2xy+6z=32 8x+3y5z=33 x = | 22 5 2 32 1 6 33 3 5 | | 3 5 2 2 1 6 8 3 5 | = | 22 5 2 32 1 6 33 3 5 22 5 2 32 1 6 | | 3 5 2 2 1 6 8 3 5 3 5 2 2 1 6 | = [(22 × 1 × 5 ) +(33 × 3 × 2 ) +(33 × 5 × 6 ) ] [(33 × 1 × 2 ) +(22 × 3 × 6 ) +(32 × 5 × 5 ) ] [(3 × 1 × 5 ) +(2 × 3 × 2 ) +(8 × 5 × 6 ) ] [(8 × 1 × 2 ) +(3 × 3 × 6 ) +(2 × 5 × 5 ) ] = [110+192+990 ] [66396+800 ] [15+12240 ] [16+54+50 ] = 1072470 21388 = 301 = 2 y = | 3 22 2 2 32 6 8 33 5 | | 3 5 2 2 1 6 8 3 5 | = | 3 22 2 2 32 6 8 33 5 3 22 2 2 32 6 | | 3 5 2 2 1 6 8 3 5 3 5 2 2 1 6 | = [(3 × 32 × 5 ) +(2 × 33 × 2 ) +(8 × 22 × 6 ) ] [(8 × 32 × 2 ) +(3 × 33 × 6 ) +(2 × 22 × 5 ) ] [(3 × 1 × 5 ) +(2 × 3 × 2 ) +(8 × 5 × 6 ) ] [(8 × 1 × 2 ) +(3 × 3 × 6 ) +(2 × 5 × 5 ) ] = [4801321056 ] [512594+220 ] [15+12240 ] [16+54+50 ] = 1668138 21388 = 301 = 6 z = | 3 5 22 2 1 32 8 3 33 | | 3 5 2 2 1 6 8 3 5 | = | 3 5 22 2 1 32 8 3 33 3 5 22 2 1 32 | | 3 5 2 2 1 6 8 3 5 3 5 2 2 1 6 | = [(3 × 1 × 33 ) +(2 × 3 × 22 ) +(8 × 5 × 32 ) ] [(8 × 1 × 22 ) +(3 × 3 × 32 ) +(2 × 5 × 33 ) ] [(3 × 1 × 5 ) +(2 × 3 × 2 ) +(8 × 5 × 6 ) ] [(8 × 1 × 2 ) +(3 × 3 × 6 ) +(2 × 5 × 5 ) ] = [991321280 ] [176+288+330 ] [15+12240 ] [16+54+50 ] = 1313795 21388 = 301 = 7 Sol.{ x=2 y=6 z=7
  9. { x+y+z=3 x+2y=6 2x+3y=6 x = | 3 1 1 6 2 0 6 3 0 | | 1 1 1 1 2 0 2 3 0 | = | 3 1 1 6 2 0 6 3 0 3 1 1 6 2 0 | | 1 1 1 1 2 0 2 3 0 1 1 1 1 2 0 | = [(3 × 2 × 0 ) +(6 × 3 × 1 ) +(6 × 1 × 0 ) ] [(6 × 2 × 1 ) +(3 × 3 × 0 ) +(6 × 1 × 0 ) ] [(1 × 2 × 0 ) +(1 × 3 × 1 ) +(2 × 1 × 0 ) ] [(2 × 2 × 1 ) +(1 × 3 × 0 ) +(1 × 1 × 0 ) ] = [0+18+0 ] [120+0 ] [0+30 ] [4+0+0 ] = 1812 34 = 6 1 = 6 y = | 1 3 1 1 6 0 2 6 0 | | 1 1 1 1 2 0 2 3 0 | = | 1 3 1 1 6 0 2 6 0 1 3 1 1 6 0 | | 1 1 1 1 2 0 2 3 0 1 1 1 1 2 0 | = [(1 × 6 × 0 ) +(1 × 6 × 1 ) +(2 × 3 × 0 ) ] [(2 × 6 × 1 ) +(1 × 6 × 0 ) +(1 × 3 × 0 ) ] [(1 × 2 × 0 ) +(1 × 3 × 1 ) +(2 × 1 × 0 ) ] [(2 × 2 × 1 ) +(1 × 3 × 0 ) +(1 × 1 × 0 ) ] = [0+6+0 ] [120+0 ] [0+30 ] [4+0+0 ] = 612 34 = 6 1 = 6 z = | 1 1 3 1 2 6 2 3 6 | | 1 1 1 1 2 0 2 3 0 | = | 1 1 3 1 2 6 2 3 6 1 1 3 1 2 6 | | 1 1 1 1 2 0 2 3 0 1 1 1 1 2 0 | = [(1 × 2 × 6 ) +(1 × 3 × 3 ) +(2 × 1 × 6 ) ] [(2 × 2 × 3 ) +(1 × 3 × 6 ) +(1 × 1 × 6 ) ] [(1 × 2 × 0 ) +(1 × 3 × 1 ) +(2 × 1 × 0 ) ] [(2 × 2 × 1 ) +(1 × 3 × 0 ) +(1 × 1 × 0 ) ] = [12+9+12 ] [12+18+6 ] [0+30 ] [4+0+0 ] = 3336 34 = 3 1 = 3 Sol.{ x=6 y=6 z=3
  10. { 3x2y=1 4x+z=28 x+2y+3z=43 x = | 1 2 0 28 0 1 43 2 3 | | 3 2 0 4 0 1 1 2 3 | = | 1 2 0 28 0 1 43 2 3 1 2 0 28 0 1 | | 3 2 0 4 0 1 1 2 3 3 2 0 4 0 1 | = [(1 × 0 × 3 ) +(28 × 2 × 0 ) +(43 × 2 × 1 ) ] [(43 × 0 × 0 ) +(1 × 2 × 1 ) +(28 × 2 × 3 ) ] [(3 × 0 × 3 ) +(4 × 2 × 0 ) +(1 × 2 × 1 ) ] [(1 × 0 × 0 ) +(3 × 2 × 1 ) +(4 × 2 × 3 ) ] = [0+0+86 ] [02+168 ] [0+02 ] [0+624 ] = 86166 2+18 = 16 = 5 y = | 3 1 0 4 28 1 1 43 3 | | 3 2 0 4 0 1 1 2 3 | = | 3 1 0 4 28 1 1 43 3 3 1 0 4 28 1 | | 3 2 0 4 0 1 1 2 3 3 2 0 4 0 1 | = [(3 × 28 × 3 ) +(4 × 43 × 0 ) +(1 × 1 × 1 ) ] [(1 × 28 × 0 ) +(3 × 43 × 1 ) +(4 × 1 × 3 ) ] [(3 × 0 × 3 ) +(4 × 2 × 0 ) +(1 × 2 × 1 ) ] [(1 × 0 × 0 ) +(3 × 2 × 1 ) +(4 × 2 × 3 ) ] = [252+01 ] [012912 ] [0+02 ] [0+624 ] = 253+141 2+18 = 16 = 7 z = | 3 2 1 4 0 28 1 2 43 | | 3 2 0 4 0 1 1 2 3 | = | 3 2 1 4 0 28 1 2 43 3 2 1 4 0 28 | | 3 2 0 4 0 1 1 2 3 3 2 0 4 0 1 | = [(3 × 0 × 43 ) +(4 × 2 × 1 ) +(1 × 2 × 28 ) ] [(1 × 0 × 1 ) +(3 × 2 × 28 ) +(4 × 2 × 43 ) ] [(3 × 0 × 3 ) +(4 × 2 × 0 ) +(1 × 2 × 1 ) ] [(1 × 0 × 0 ) +(3 × 2 × 1 ) +(4 × 2 × 3 ) ] = [08+56 ] [0168+344 ] [0+02 ] [0+624 ] = 48176 2+18 = 16 = 8 Sol.{ x=5 y=7 z=8
  11. { x 3 y 4 + z 4 =1 x 6 + y 2 z=1 x 2 y 8 z 2 =0 x = | 1 1 4 1 4 1 1 2 1 0 1 8 1 2 | | 1 3 1 4 1 4 1 6 1 2 1 1 2 1 8 1 2 | = | 1 1 4 1 4 1 1 2 1 0 1 8 1 2 1 1 4 1 4 1 1 2 1 | | 1 3 1 4 1 4 1 6 1 2 1 1 2 1 8 1 2 1 3 1 4 1 4 1 6 1 2 1 | = [(1 × 1 2 × 1 2 ) +(1 × 1 8 × 1 4 ) +(0 × 1 4 × 1 ) ] [(0 × 1 2 × 1 4 ) +(1 × 1 8 × 1 ) +(1 × 1 4 × 1 2 ) ] [( 1 3 × 1 2 × 1 2 ) +( 1 6 × 1 8 × 1 4 ) +( 1 2 × 1 4 × 1 ) ] [( 1 2 × 1 2 × 1 4 ) +( 1 3 × 1 8 × 1 ) +( 1 6 × 1 4 × 1 2 ) ] = [ 1 4 1 32 +0 ] [0+ 1 8 + 1 8 ] [ 1 12 1 192 + 1 8 ] [ 1 16 + 1 24 + 1 48 ] = 9 32 1 4 7 192 1 8 = 17 32 17 = 6 y = | 1 3 1 1 4 1 6 1 1 1 2 0 1 2 | | 1 3 1 4 1 4 1 6 1 2 1 1 2 1 8 1 2 | = | 1 3 1 1 4 1 6 1 1 1 2 0 1 2 1 3 1 1 4 1 6 1 1 | | 1 3 1 4 1 4 1 6 1 2 1 1 2 1 8 1 2 1 3 1 4 1 4 1 6 1 2 1 | = [( 1 3 × 1 × 1 2 ) +( 1 6 × 0 × 1 4 ) +( 1 2 × 1 × 1 ) ] [( 1 2 × 1 × 1 4 ) +( 1 3 × 0 × 1 ) +( 1 6 × 1 × 1 2 ) ] [( 1 3 × 1 2 × 1 2 ) +( 1 6 × 1 8 × 1 4 ) +( 1 2 × 1 4 × 1 ) ] [( 1 2 × 1 2 × 1 4 ) +( 1 3 × 1 8 × 1 ) +( 1 6 × 1 4 × 1 2 ) ] = [ 1 6 +0 1 2 ] [ 1 8 +0 1 12 ] [ 1 12 1 192 + 1 8 ] [ 1 16 + 1 24 + 1 48 ] = 2 3 1 24 7 192 1 8 = 17 24 17 = 8 z = | 1 3 1 4 1 1 6 1 2 1 1 2 1 8 0 | | 1 3 1 4 1 4 1 6 1 2 1 1 2 1 8 1 2 | = | 1 3 1 4 1 1 6 1 2 1 1 2 1 8 0 1 3 1 4 1 1 6 1 2 1 | | 1 3 1 4 1 4 1 6 1 2 1 1 2 1 8 1 2 1 3 1 4 1 4 1 6 1 2 1 | = [( 1 3 × 1 2 × 0 ) +( 1 6 × 1 8 × 1 ) +( 1 2 × 1 4 × 1 ) ] [( 1 2 × 1 2 × 1 ) +( 1 3 × 1 8 × 1 ) +( 1 6 × 1 4 × 0 ) ] [( 1 3 × 1 2 × 1 2 ) +( 1 6 × 1 8 × 1 4 ) +( 1 2 × 1 4 × 1 ) ] [( 1 2 × 1 2 × 1 4 ) +( 1 3 × 1 8 × 1 ) +( 1 6 × 1 4 × 1 2 ) ] = [0+ 1 48 1 8 ] [ 1 4 1 24 +0 ] [ 1 12 1 192 + 1 8 ] [ 1 16 + 1 24 + 1 48 ] = 7 48 5 24 7 192 1 8 = 17 48 17 = 4 Sol.{ x=6 y=8 z=4
  12. { x 3 +y=2z+3 xy=1 x+z= y 4 +11 { x 3 +y2z=3 xy=1 x y 4 +z=11 x = | 3 1 2 1 1 0 11 1 4 1 | | 1 3 1 2 1 1 0 1 1 4 1 | = | 3 1 2 1 1 0 11 1 4 1 3 1 2 1 1 0 | | 1 3 1 2 1 1 0 1 1 4 1 1 3 1 2 1 1 0 | = [(3 × 1 × 1 ) +(1 × 1 4 × 2 ) +(11 × 1 × 0 ) ] [(11 × 1 × 2 ) +(3 × 1 4 × 0 ) +(1 × 1 × 1 ) ] [( 1 3 × 1 × 1 ) +(1 × 1 4 × 2 ) +(1 × 1 × 0 ) ] [(1 × 1 × 2 ) +( 1 3 × 1 4 × 0 ) +(1 × 1 × 1 ) ] = [3+ 1 2 +0 ] [22+0+1 ] [ 1 3 + 1 2 +0 ] [2+0+1 ] = 5 2 23 1 6 3 = 2 17 = 9 y = | 1 3 3 2 1 1 0 1 11 1 | | 1 3 1 2 1 1 0 1 1 4 1 | = | 1 3 3 2 1 1 0 1 11 1 1 3 3 2 1 1 0 | | 1 3 1 2 1 1 0 1 1 4 1 1 3 1 2 1 1 0 | = [( 1 3 × 1 × 1 ) +(1 × 11 × 2 ) +(1 × 3 × 0 ) ] [(1 × 1 × 2 ) +( 1 3 × 11 × 0 ) +(1 × 3 × 1 ) ] [( 1 3 × 1 × 1 ) +(1 × 1 4 × 2 ) +(1 × 1 × 0 ) ] [(1 × 1 × 2 ) +( 1 3 × 1 4 × 0 ) +(1 × 1 × 1 ) ] = [ 1 3 22+0 ] [2+0+3 ] [ 1 3 + 1 2 +0 ] [2+0+1 ] = 65 3 1 1 6 3 = 3 17 = 8 z = | 1 3 1 3 1 1 1 1 1 4 11 | | 1 3 1 2 1 1 0 1 1 4 1 | = | 1 3 1 3 1 1 1 1 1 4 11 1 3 1 3 1 1 1 | | 1 3 1 2 1 1 0 1 1 4 1 1 3 1 2 1 1 0 | = [( 1 3 × 1 × 11 ) +(1 × 1 4 × 3 ) +(1 × 1 × 1 ) ] [(1 × 1 × 3 ) +( 1 3 × 1 4 × 1 ) +(1 × 1 × 11 ) ] [( 1 3 × 1 × 1 ) +(1 × 1 4 × 2 ) +(1 × 1 × 0 ) ] [(1 × 1 × 2 ) +( 1 3 × 1 4 × 0 ) +(1 × 1 × 1 ) ] = [ 11 3 3 4 +1 ] [3 1 12 +11 ] [ 1 3 + 1 2 +0 ] [2+0+1 ] = 41 12 95 12 1 6 3 = 17 6 = 4 Sol.{ x=9 y=8 z=4

Ejercicio 187

CAPITULO XXV

Hallar el valor de una determinante de tercer orden
Ejercicio 187
Hallar el valor de las siguientes determinantes:
  1. | 1 2 1 1 3 4 1 0 2 | | 1 2 1 1 3 4 1 0 2 1 2 1 1 3 4 | = [(1 × 3 × 2 ) +(1 × 0 × 1 ) +(1 × 2 × 4 ) ] [(1 × 3 × 1 ) +(4 × 0 × 1 ) +(2 × 2 × 1 ) ] = [6+0+8 ] [3+0+4 ] = 147 = 7
  2. | 1 2 2 1 3 3 1 4 5 | | 1 2 2 1 3 3 1 4 5 1 2 2 1 3 3 | = [(1 × 3 × 5 ) +(1 × 4 × 2 ) +(1 × 2 × 3 ) ] [(2 × 3 × 1 ) +(3 × 4 × 1 ) +(5 × 2 × 1 ) ] = [1586 ] [6+12+10 ] = 2916 = 45
  3. | 3 4 1 2 3 0 1 2 7 | | 3 4 1 2 3 0 1 2 7 3 4 1 2 3 0 | = [(3 × 3 × 7 ) +(2 × 2 × 1 ) +(1 × 4 × 0 ) ] [(1 × 3 × 1 ) +(3 × 2 × 0 ) +(2 × 4 × 7 ) ] = [63+4+0 ] [3+0+56 ] = 6753 = 14
  4. | 2 5 1 3 4 3 6 2 4 | | 2 5 1 3 4 3 6 2 4 2 5 1 3 4 3 | = [(2 × 4 × 4 ) +(3 × 2 × 1 ) +(6 × 5 × 3 ) ] [(1 × 4 × 6 ) +(2 × 2 × 3 ) +(3 × 5 × 4 ) ] = [326+90 ] [24+12+60 ] = 5296 = 44
  5. | 5 1 6 2 5 3 3 4 2 | | 5 1 6 2 5 3 3 4 2 5 1 6 2 5 3 | = [(5 × 5 × 2 ) +(2 × 4 × 6 ) +(3 × 1 × 3 ) ] [(3 × 5 × 6 ) +(5 × 4 × 3 ) +(2 × 1 × 2 ) ] = [50+489 ] [90+60+4 ] = 89+26 = 115
  6. | 4 1 5 3 2 6 12 3 2 | | 4 1 5 3 2 6 12 3 2 4 1 5 3 2 6 | = [(4 × 2 × 2 ) +(3 × 3 × 5 ) +(12 × 1 × 6 ) ] [(12 × 2 × 5 ) +(4 × 3 × 6 ) +(3 × 1 × 2 ) ] = [16+4572 ] [12072+6 ] = 1154 = 65
  7. | 5 2 8 3 7 3 4 0 1 | | 5 2 8 3 7 3 4 0 1 5 2 8 3 7 3 | = [(5 × 7 × 1 ) +(3 × 0 × 8 ) +(4 × 2 × 3 ) ] [(4 × 7 × 8 ) +(5 × 0 × 3 ) +(3 × 2 × 1 ) ] = [35+0+24 ] [224+0+6 ] = 59230 = 171
  8. | 3 2 5 1 3 4 3 2 5 | | 3 2 5 1 3 4 3 2 5 3 2 5 1 3 4 | = [(3 × 3 × 5 ) +(1 × 2 × 5 ) +(3 × 2 × 4 ) ] [(3 × 3 × 5 ) +(3 × 2 × 4 ) +(1 × 2 × 5 ) ] = [4510+24 ] [45+2410 ] = 31+31 = 0
  9. | 5 2 3 6 1 2 3 4 5 | | 5 2 3 6 1 2 3 4 5 5 2 3 6 1 2 | = [(5 × 1 × 5 ) +(6 × 4 × 3 ) +(3 × 2 × 2 ) ] [(3 × 1 × 3 ) +(5 × 4 × 2 ) +(6 × 2 × 5 ) ] = [25+72+12 ] [9+40+60 ] = 109109 = 0
  10. | 12 5 10 8 6 9 7 4 2 | | 12 5 10 8 6 9 7 4 2 12 5 10 8 6 9 | = [(12 × 6 × 2 ) +(8 × 4 × 10 ) +(7 × 5 × 9 ) ] [(7 × 6 × 10 ) +(12 × 4 × 9 ) +(8 × 5 × 2 ) ] = [144+320+315 ] [420+43280 ] = 779+68 = 847
  11. | 9 3 4 7 5 3 4 6 1 | | 9 3 4 7 5 3 4 6 1 9 3 4 7 5 3 | = [(9 × 5 × 1 ) +(7 × 6 × 4 ) +(4 × 3 × 3 ) ] [(4 × 5 × 4 ) +(9 × 6 × 3 ) +(7 × 3 × 1 ) ] = [4516836 ] [80+162+21 ] = 159263 = 422
  12. | 11 5 7 12 3 8 13 1 9 | | 11 5 7 12 3 8 13 1 9 11 5 7 12 3 8 | = [(11 × 3 × 9 ) +(12 × 1 × 7 ) +(13 × 5 × 8 ) ] [(13 × 3 × 7 ) +(11 × 1 × 8 ) +(12 × 5 × 9 ) ] = [29784+520 ] [273+88+540 ] = 733355 = 378