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DESCARGA
CAPITULO XXV

Ecuaciones simultáneas con tres incógnitas
Resolución por determinantes
Ejercicio 188
Resolver por determinantes:
  1. { x+y+z=11 x–y+3z=13 2x+2y–z=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+66–13 ] [1+2+6 ] –[–2+6–1 ] = 58–46 9–3 = 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+21–11 ] [1+2+6 ] –[–2+6–1 ] = 60–36 9–3 = 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+6–1 ] = 41–11 9–3 = 6 = 5 Sol.{ x=2 y=4 z=5
  2. { x+y+z=–6 2x+y–z=–1 x–2y+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 ] –[–6–12–3 ] [3–4–1 ] –[1+2+6 ] = –10+21 –2–9 = – 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 ) ] = [–3–12+6 ] –[–1+6–36 ] [3–4–1 ] –[1+2+6 ] = –9+31 –2–9 = – 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+24–1 ] –[–6+2–12 ] [3–4–1 ] –[1+2+6 ] = 17+16 –2–9 = – 11 = –3 Sol.{ x=–1 y=–2 z=–3
  3. { 2x+3y+4z=3 2x+6y+8z=5 4x+9y–4z=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+216–60 ] [–48+72+96 ] –[96+144–24 ] = 204–252 120–216 = – 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+64–24 ] [–48+72+96 ] –[96+144–24 ] = 88–120 120–216 = – 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+144–24 ] = 162–186 120–216 = – 24 – = 1 4 Sol.{ x= 1 2 y= 1 3 z= 1 4
  4. { 4x–y+z=4 2y–z+2x=2 ↔ 6x+3z–2y=12 { 4x–y+z=4 2x+2y–z=2 6x–2y+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 ) ] = [24–4+12 ] –[24+8–6 ] [24–4+6 ] –[12+8–6 ] = 32–26 26–14 = 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+24–24 ] –[12–48+24 ] [24–4+6 ] –[12+8–6 ] = 24+12 26–14 = 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 ) ] = [96–16–12 ] –[48–16–24 ] [24–4+6 ] –[12+8–6 ] = 68–8 26–14 = 12 = 5 Sol.{ x= 1 2 y=3 z=5
  5. { x+4y+5z=11 3x–2y+z=5 4x+y–3z=–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+25–104 ] –[260+11–60 ] [6+15+16 ] –[–40+1–36 ] = –13–211 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 ) ] = [–15–390+44 ] –[100–26–99 ] [6+15+16 ] –[–40+1–36 ] = –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+5–312 ] [6+15+16 ] –[–40+1–36 ] = 165+395 37+75 = 112 = 5 Sol.{ x=–2 y=–3 z=5
  6. { 7x+10y+4z=–2 5x–2y+6z=38 3x+y–z=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 ] –[–168–12–380 ] [14+20+180 ] –[–24+42–50 ] = 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+420–36 ] –[456+882+10 ] [14+20+180 ] –[–24+42–50 ] = 118–1348 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 ) ] = [–294–10+1140 ] –[12+266+1050 ] [14+20+180 ] –[–24+42–50 ] = 836–1328 214+32 = – 246 = –2 Sol.{ x=8 y=–5 z=–2
  7. { 4x+7y+5z=–2 6x+3y+7z=6 x–y+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 ) ] = [–54–30–1029 ] –[–315+14+378 ] [108–30+49 ] –[15–28+378 ] = –1113–77 127–365 = – – 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 ) ] = [216–630–14 ] –[30–588–108 ] [108–30+49 ] –[15–28+378 ] = –452+666 127–365 = 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 ] –[–6–24–882 ] [108–30+49 ] –[15–28+378 ] = –198+912 127–365 = – 238 = –3 Sol.{ x=5 y=–1 z=–3
  8. { 3x–5y+2z=–22 2x–y+6z=32 8x+3y–5z=–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 ] –[66–396+800 ] [15+12–240 ] –[–16+54+50 ] = 1072–470 –213–88 = – 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 ) ] = [–480–132–1056 ] –[512–594+220 ] [15+12–240 ] –[–16+54+50 ] = –1668–138 –213–88 = – – 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 ) ] = [99–132–1280 ] –[176+288+330 ] [15+12–240 ] –[–16+54+50 ] = –1313–795 –213–88 = – – 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 ] –[12–0+0 ] [0+3–0 ] –[4+0+0 ] = 18–12 3–4 = 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 ] –[12–0+0 ] [0+3–0 ] –[4+0+0 ] = 6–12 3–4 = –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+3–0 ] –[4+0+0 ] = 33–36 3–4 = –3 –1 = 3 Sol.{ x=–6 y=6 z=3
  10. { 3x–2y=–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 ] –[0–2+168 ] [0+0–2 ] –[0+6–24 ] = 86–166 –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+0–1 ] –[0–129–12 ] [0+0–2 ] –[0+6–24 ] = –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 ) ] = [0–8+56 ] –[0–168+344 ] [0+0–2 ] –[0+6–24 ] = 48–176 –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 x–y=1 ↔ x+z= y 4 +11 { x 3 +y–2z=3 x–y=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
Categories: Capítulo XXV