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DESCARGA
CAPITULO XXV

Sistema de cuatro Ecuaciones
con cuatro incognitas
Ejercicio 192
Resolver los sistemas
  1. { x+y+z+u=4 ( 1 ) x+2y+3z–u=–1 ( 2 ) 3x+4y+2z+u=–5 ( 3 ) x+4y+3z–u=–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 –3x–4y–2z– u =5 –2x–3y–z =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 –3y–z =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 2x–15+16 =–3 2x =–3–1 x =– 2 x =–2 Reemplazo los valores de x,y,z en ( 1 ) –2–3+4+u =4 u =4+1 u =5 Sol{ x=–2 y=–3 z=4 u=5
  2. { x+y+z+u=10 ( 1 ) 2x–y–2z+2u=2 ( 2 ) x–2y+3z–u=2 ( 3 ) x+2y–4z+2u=1 ( 4 ) ( 1 ) +( 2 ) x+ y +z+u =10 2x– y –2z+2u =2 3x–z+3u =12 ( 5 ) 2( 1 ) +( 3 ) 2x+ 2y +2z+2u =20 x– 2y +3z–u =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 –5z–u =–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 =19–18 x =1 Reemplazo los valores de x,u,z en ( 1 ) 1+y+3+4 =10 y =10–8 y =2 Sol{ x=1 y=2 z=3 u=4
  3. { x–2y+z+3u=–3 ( 1 ) 3x+y–4z–2u=7 ( 2 ) 2x+2y–z–u=1 ( 3 ) x+4y+2z–5u=12 ( 4 ) ( 1 ) +2( 2 ) x– 2y +z+3u =–3 6x+ 2y –8z–4u =14 7x–7z–u =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 +2z–5u =12 3x+4z+u =6 ( 7 ) 2( 5 ) +( 6 ) 14x–14z– 2u =22 3x+ 2u =–2 17x–14z =20 ( 8 ) ( 5 ) +( 7 ) 7x–7z– u =11 3x+4z+ u =6 10x–3z =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 ) 17x–14( 1 ) =20 17x–14 =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 ) 2–2y+1+3(–4 ) =–3 –2y–12 =–3–3 –2y =–6+12 y =– 2 y =–3 Sol{ x=2 y=–3 z=1 u=–4
  4. { 2x–3y+z+4u=0 ( 1 ) 3x+y–5z–3u=–10 ( 2 ) 6x+2y–z+u=–3 ( 3 ) x+5y+4z–3u=–6 ( 4 ) ( 1 ) +3( 2 ) 2x– 3y +z+4u =0 9x+ 3y –15z–9u =–30 11x–14z–5u =–30 ( 5 ) 2( 1 ) +3( 3 ) 4x– 6y +2z+8u =0 18x+ 6y –3z+3u =–9 22x–z+11u =–9 ( 6 ) 5( 1 ) +3( 4 ) 10x– 15y +5z+20u =0 3x+ 15y +12z–9u =–18 13x+17z+11u =–18 ( 7 ) 11( 5 ) +5( 6 ) 121x–154z– 55u =–330 110x–5z+ 55u =–45 231x–159z =–375 77x–53z =–125 ( 8 ) ( 6 ) –( 7 ) 22x–z+ 11u =–9 –13x–17z– 11u =18 9x–18z =9 x–2z =1 ( 9 ) ( 8 ) –77( 9 ) 77x –53z =–125 – 77x +154z =–77 101z =–202 z = – 101 z =–2 Reemplazo el valor de z en ( 9 ) x–2(–2 ) =1 x+4 =1 x =1–4 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 ) –3y–2+4( 5 ) =0 –6–3y–2+20 =0 –3y+12 =0 y = – – 3 y =4 Sol{ x=–3 y=4 z=–2 u=5
  5. { x+y–z=–4 ( 1 ) 4x+3y+2z–u=9 ( 2 ) 2x–y–4z+u=–1 ( 3 ) x+2y+3z+2u=–1 ( 4 ) 3( 1 ) –( 2 ) 3x+ 3y –3z =–12 –4x– 3y –2z+u =–9 –x–5z+u =–21 ( 5 ) ( 1 ) +( 3 ) x+ y –z =–4 2x– y –4z+u =–1 3x–5z+u =–5 ( 6 ) 2( 1 ) –( 4 ) 2x+ 2y –2z =–8 –x– 2y –3z–2u =1 x–5z–2u =–7 ( 7 ) ( 5 ) –( 6 ) –x– 5z + u =–21 –3x+ 5z – u =5 –4x =–16 x = – – 4 x =4 2( 6 ) +( 7 ) 6x–10z+ 2u =–10 x–5z– 2u =–7 7x–15z =–17 ( 9 ) Reemplazo el valor de x 7( 4 ) –15z =–17 28–15z =–17 –15z =–17–28 z = – – 15 z =3 Reemplazo los valores de x, z en ( 6 ) 3( 4 ) –5( 3 ) +u =–5 12–15+u =–5 u =–5+3 u =–2 Reemplazo los valores de x,u,z en ( 3 ) 2( 4 ) –y–4( 3 ) –2 =–1 8–y–12–2 =–1 –y–6 =–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+3y–z+u=3 ( 4 ) 3( 1 ) –2( 2 ) 3x+ 6y +3z =–12 –4x– 6y –8z =4 –x–5z =–8 ( 5 ) ( 1 ) –2( 3 ) x+ 2y +z =–4 –6x– 2y –2z–2u =–8 –5x–z–2u =–12 ( 6 ) 3( 1 ) –2( 4 ) 3x+ 6y +3z =–12 –12x– 6y +2z–2u =–6 –9x+5z–2u =–18 ( 7 ) ( 5 ) +( 7 ) –x– 5z =–8 –9x+ 5z –2u =–18 –10x–2u =–26 5x+u =13 ( 8 ) 5( 6 ) +( 7 ) –25x– 5z –10u =–60 –9x+ 5z –2u =–18 –34x–12u =–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 ) –3–5z =–8 –5z =–8+3 –5z =–5 z = – 5 – 5 z =1 Reemplazo los valores de x, z en ( 6 ) –5( 3 ) –1–2u =–12 –15–2u =–12+1 –2u =–11+15 u = – 2 u =–2 Reemplazo los valores de x,z en ( 1 ) 3+2y+1 =–4 2y =–4–4 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 ) 3x–2y–u=–7 ( 3 ) 4x+5y+6z+3u=11 ( 4 ) ( 1 ) –2( 2 ) 3x+ 2y =–2 –2x– 2y –2u =6 x–2u =4 ( 5 ) ( 1 ) +( 3 ) 3x+ 2y =–2 3x– 2y –u =–7 6x–u =–9 ( 6 ) ( 5 ) –2( 6 ) x– 2u =4 –12x+ 2u =18 –11x =22 x =– 11 x =–2 Reemplazo el valor de x en ( 5 ) –2–2u =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+6z–9 =11 6z =11+7 z = 6 z =3 Sol{ x=–2 y=2 z=3 u=–3
  8. { 2x–3z–u=2 ( 1 ) 3y–2z–5u=3 ( 2 ) 4y–3u=2 ( 3 ) x–3y+3u=0 ( 4 ) 5( 1 ) –( 2 ) 10x–15z– 5u =10 –3y+2z+ 5u =–3 10x–3y–13z =7 ( 5 ) 3( 1 ) –( 3 ) 6x–9z– 3u =6 –4y+ 3u =–2 6x–4y–9z =4 ( 6 ) 3( 1 ) +( 4 ) 6x–9z– 3u =6 x–3y+ 3u =0 7x–3y–9z =6 ( 7 ) ( 5 ) –( 7 ) 10x– 3y –13z =7 –7x+ 3y +9z =–6 3x–4z =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 ) 3x–4( 2 ) =1 3x–8 =1 3x =1+8 x = 3 x =3 Reemplazo los valores de x, z en ( 6 ) 6( 3 ) –4y–9( 2 ) =4 18 –4y– 18 =4 y =– 4 4 y =–1 Reemplazo los valores de y en ( 3 ) 4(–1 ) –3u =2 –4–3u =2 –3u =2+4 u =– 3 u =–2 Sol{ x=3 y=–1 z=2 u=–2
Categories: Capítulo XXV