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Ejercicio 184

Gestor Baldor
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DESCARGA
CAPITULO XXIV

Ecuaciones simultaneas con dos incognitas
Ejercicio 184
Resolver por determinantes:
  1. { 7x+8y=29 5x+11y=26 x= | 29 8 26 11 | | 7 8 5 11 | = 29( 11 ) –8( 26 ) 7( 11 ) –5( 8 ) = 319–208 77–40 = 37 =3 y= | 7 29 5 26 | | 7 8 5 11 | = 7( 26 ) –29( 5 ) 7( 11 ) –5( 8 ) = 182–145 77–40 = 37 37 =1 Sol{ x=3 y=1
  2. { 3x–4y=13 8x–5y=–5 x= | 13 –4 –5 –5 | | 3 –4 8 –5 | = 13( –5 ) –( –5 ) ( –4 ) 3( –5 ) –8( –4 ) = –65–20 –15+32 = – 17 =–5 y= | 3 13 8 –5 | | 3 –4 8 –5 | = ( –5 ) ( 3 ) –13( 8 ) 3( –5 ) –8( –4 ) = –15–104 –15+32 =– 17 =–7 Sol{ x=–5 y=–7
  3. { 13x–31y=–326 25x+37y=146 x= | –326 –31 146 37 | | 13 –31 25 37 | = –326( 37 ) –146( –31 ) 13( 37 ) –( 25 ) ( –31 ) = –12062+4526 481+775 = – 1256 =–6 y= | 13 –326 25 146 | | 13 –31 25 37 | = 146( 13 ) –( –326 ) ( 25 ) 13( 37 ) –( 25 ) ( –31 ) = 1898+8150 481+775 = 1256 =8 Sol{ x=–6 y=8
  4. { 15x–44y=–6 32y–27x=–1 { 15x–44y=–6 –27x+32y=–1 x= | –6 –44 –1 32 | | 15 –44 –27 32 | = –6( 32 ) –( –1 ) ( –44 ) 15( 32 ) –( –27 ) ( –44 ) = –192–44 480–1188 = – 236 – = 1 3 y= | 15 –6 –27 –1 | | 15 –44 –27 32 | = ( –1 ) ( 15 ) –( –6 ) ( –27 ) 15( 32 ) –( –27 ) ( –44 ) = –15–162 480–1188 = – 177 – = 1 4 Sol{ x= 1 3 y= 1 4
  5. { 8x=–9y 2x+5+3y=3 1 2 { 8x+9y=0 2x+3y= 7 2 –5 { 8x+9y=0 2x+3y=– 3 2 2 { 8x+9y=0 4x+6y=–3 x= | 0 9 –3 6 | | 8 9 4 6 | = 6( 0 ) –9( –3 ) 8( 6 ) –4( 9 ) = 27 48–36 = = 9 4 =2 1 4 y= | 8 0 4 –3 | | 8 9 4 6 | = 8( –3 ) –4( 0 ) 8( 6 ) –4( 9 ) = –24 48–36 = 12 =2 Sol{ x=2 1 4 y=2
  6. { ax–by=–1 ax+by=7 x= | –1 –b 7 b | | a –b a b | = –b–7( –b ) ab–a( –b ) = –b+7b ab+ab = b 2 a b = 3 a y= | a –1 a 7 | | a –b a b | = 7a–a( –1 ) ab–a( –b ) = 7a+a ab+ab = a 2 a b = 4 b Sol{ x= 3 a y= 4 b
  7. { 3x–( y+2 ) =2y+1 5y–( x+3 ) =3x+1 { 3x–y–2–2y=1 5y–x–3–3x=1 { 3x–3y=1+2 5y–4x=3+1 { 3x–3y=3 1 3 5y–4x=4 { x–y=1 –4x+5y=4 x= | 1 –1 4 5 | | 1 –1 –4 5 | = 5( 1 ) –4( –1 ) 5( 1 ) –( –4 ) ( –1 ) = 5+4 5–4 =9 y= | 1 1 –4 4 | | 1 –1 –4 5 | = 4( 1 ) –4( –1 ) 5( 1 ) –( –4 ) ( –1 ) = 4+4 5–4 =8 Sol{ x=9 y=8
  8. { ax+2y=2 ax 2 –3y=–1 ( 2 ) { ax+2y=2 ax–6y=–2 x= | 2 2 –2 –6 | | a 2 a –6 | = 2( –6 ) –( –2 ) ( 2 ) a( –6 ) –2a = –12+4 –6a–2a = – 8 – 8 a = 1 a y= | a 2 a –2 | | a 2 a –6 | = a( –2 ) –2a a( –6 ) –2a = –2a–2a –6a–2a = – 4a – = 1 2 Sol{ x= 1 a y= 1 2
  9. { x 4 + y 6 =–4 ( 12 ) x 8 – y 12 =0 ( 24 ) { 3x+2y=–48 3x–2y=0 x= | –48 2 0 –2 | | 3 2 3 –2 | = –2( –48 ) –( 2 ) ( 0 ) 3( –2 ) –2( 3 ) = 96 –6–6 = – 12 =–8 y= | 3 –48 3 0 | | 3 2 3 –2 | = 3( 0 ) –3( –48 ) 3( –2 ) –2( 3 ) = 144 –6–6 = – 12 =–12 Sol{ x=–8 y=–12
  10. { 3x+ay=3a+1 x a +ay=2 a { 3x+ay=3a+1 x+ a 2 y=2a x= | 3a+1 a 2a a 2 | | 3 a 1 a 2 | = a 2 ( 3a+1 ) –2a( a ) 3 a 2 –a = 3 a 3 + a 2 –2 a 2 a( 3a–1 ) = 3 a 3 – a 2 a( 3a–1 ) = a 2 ( 3a–1 ) a ( 3a–1 ) =a y= | 3 3a+1 1 2a | | 3 a 1 a 2 | = 2a( 3 ) –( 3a+1 ) 3 a 2 –a = 6a–3a–1 a( 3a–1 ) = ( 3a–1 ) a ( 3a–1 ) = 1 a Sol{ x=a y= 1 a
  11. { x+2 3 – y–3 8 = 5 6 ( 24 ) y–5 6 – 2x–3 5 =0 ( 30 ) { 8( x+2 ) –3( y–3 ) =20 5( y–5 ) –6( 2x–3 ) =0 { 8x+16–3y+9=20 5y–25–12x+18=0 { 8x–3y=20–25 –12x+5y=7 { 8x–3y=–5 –12x+5y=7 x= | –5 –3 7 5 | | 8 –3 –12 5 | = –5( 5 ) –7( –3 ) 8( 5 ) –( –12 ) ( –3 ) = –25+21 40–36 = – 4 4 =–1 y= | 8 –5 –12 7 | | 8 –3 –12 5 | = 8( 7 ) –( –12 ) ( –5 ) 8( 5 ) –( –12 ) ( –3 ) = 56–60 40–36 = – 4 4 =–1 Sol{ x=–1 y=–1
  12. { 3x–2y=5 mx+4y=2( m+1 ) x= | 5 –2 2( m+1 ) 4 | | 3 –2 m 4 | = 5( 4 ) –( –4 ) ( m+1 ) 3( 4 ) –( –2 ) m = 20+4m+4 12+2m = 4m+24 2( m+6 ) = ( m+6 ) 2 ( m+6 ) =2 y= | 3 5 m 2( m+1 ) | | 3 –2 m 4 | = 6( m+1 ) –5m 3( 4 ) –( –2 ) m = 6m+6–5m 12+2m = m+6 2 ( m+6 ) = 1 2 Sol{ x=2 y= 1 2
  13. { 2x– 2y+3 17 =y+2 ( 17 ) 3y– 4x+1 21 =3x+5 ( 21 ) { 34x–2y–3=17y+34 63y–4x–1=63x+105 { 34x–2y–17y=3+34 63y–4x–63x=1+105 { 34x–19y=37 –67x+63y=106 x= | 37 –19 106 63 | | 34 –19 –67 63 | = 37( 63 ) –( –19 ) ( 106 ) 34( 63 ) –( –19 ) ( –67 ) = 2331+2014 2142–1273 = 869 =5 y= | 34 37 –67 106 | | 34 –19 –67 63 | = 34( 106 ) –( –67 ) ( 37 ) 34( 63 ) –( –19 ) ( –67 ) = 3604+2479 2142–1273 = 869 =7 Sol{ x=5 y=7
  14. { x+y x–y =4 ( x–y ) x–y–1 x+y+1 = 1 9 9( x+y+1 ) { x+y=4( x–y ) 9( x–y–1 ) =x+y+1 { x+y=4x–4y 9x–9y–9=x+y+1 { x+y–4x+4y=0 9x–9y–x–y=9+1 { –3x+5y=0 8x–10y=10 1 2 { –3x+5y=0 4x–5y=5 x= | 0 5 5 –5 | | –3 5 4 –5 | = 0( –5 ) –5( 5 ) –3( –5 ) –5( 4 ) = –25 15–20 = – – 5 =5 y= | –3 0 4 5 | | –3 5 4 –5 | = 0( 4 ) –3( 5 ) –3( –5 ) –5( 4 ) = –15 15–20 = – – 5 =3 Sol{ x=5 y=3
  15. { x–y=2b x a+b + y a–b =2 { x–y=2b x( a–b ) +y( a+b ) ( a+b ) ( a–b ) =2 { x–y=2b ax–bx+ay+by=2( a 2 – b 2 ) { x–y=2b ( a–b ) x+( a+b ) y=2( a 2 – b 2 ) x= | 2b –1 2( a 2 – b 2 ) a+b | | 1 –1 a–b a+b | = 2b( a+b ) +2( a 2 – b 2 ) a+b+( a–b ) = 2( a+b ) [ b +a– b ] a+ b +a– b = 2 ( a+b ) a 2 a =a+b y= | 1 2b a–b 2( a 2 – b 2 ) | | 1 –1 a–b a+b | = 2( a 2 – b 2 ) –2b( a–b ) a+b+( a–b ) = 2( a–b ) [ a+ b – b ] a+ b +a– b = 2 ( a–b ) a 2 a =a–b Sol{ x=a+b y=a–b
  16. { x+9 x–9 = y+21 y+39 x+8 x–8 = y+19 y+11 { ( x+9 ) ( y+39 ) =( y+21 ) ( x–9 ) ( x+8 ) ( y+11 ) =( y+19 ) ( x–8 ) { xy +39x+9y+351= xy –9y+21x–189 xy +11x+8y+88= xy –8y+19x–152 { 39x+9y+9y–21x=–351–189 11x+8y+8y–19x=–88–152 { 18x+18y=–540 1 18 –8x+16y=–240 – 1 8 { x+y=–30 x–2y=30 x= | –30 1 30 –2 | | 1 1 1 –2 | = –30( –2 ) –( 30 ) ( 1 ) 1( –2 ) –( 1 ) ( 1 ) = 60–30 –2–1 = – 3 =–10 y= | 1 –30 1 30 | | 1 1 1 –2 | = 30( 1 ) –( –30 ) ( 1 ) 1( –2 ) –( 1 ) ( 1 ) = 30+30 –2–1 = – 3 =–20 Sol{ x=–10 y=–20
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