Ejercicio 184

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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 ) = 319208 7740 = 37 =3 y= | 7 29 5 26 | | 7 8 5 11 | = 7( 26 ) 29( 5 ) 7( 11 ) 5( 8 ) = 182145 7740 = 37 37 =1 Sol{ x=3 y=1
  2. { 3x4y=13 8x5y=5 x= | 13 4 5 5 | | 3 4 8 5 | = 13(5 ) (5 ) (4 ) 3(5 ) 8(4 ) = 6520 15+32 = 17 =5 y= | 3 13 8 5 | | 3 4 8 5 | = (5 ) ( 3 ) 13( 8 ) 3(5 ) 8(4 ) = 15104 15+32 = 17 =7 Sol{ x=5 y=7
  3. { 13x31y=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. { 15x44y=6 32y27x=1 { 15x44y=6 27x+32y=1 x= | 6 44 1 32 | | 15 44 27 32 | = 6( 32 ) (1 ) (44 ) 15( 32 ) (27 ) (44 ) = 19244 4801188 = 236 = 1 3 y= | 15 6 27 1 | | 15 44 27 32 | = (1 ) ( 15 ) (6 ) (27 ) 15( 32 ) (27 ) (44 ) = 15162 4801188 = 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 4836 = = 9 4 =2 1 4 y= | 8 0 4 3 | | 8 9 4 6 | = 8(3 ) 4( 0 ) 8( 6 ) 4( 9 ) = 24 4836 = 12 =2 Sol{ x=2 1 4 y=2
  6. { axby=1 ax+by=7 x= | 1 b 7 b | | a b a b | = b7(b ) aba(b ) = b+7b ab+ab = b 2 a b = 3 a y= | a 1 a 7 | | a b a b | = 7aa(1 ) aba(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 { 3xy22y=1 5yx33x=1 { 3x3y=1+2 5y4x=3+1 { 3x3y=3 1 3 5y4x=4 { xy=1 4x+5y=4 x= | 1 1 4 5 | | 1 1 4 5 | = 5( 1 ) 4(1 ) 5( 1 ) (4 ) (1 ) = 5+4 54 =9 y= | 1 1 4 4 | | 1 1 4 5 | = 4( 1 ) 4(1 ) 5( 1 ) (4 ) (1 ) = 4+4 54 =8 Sol{ x=9 y=8
  8. { ax+2y=2 ax 2 3y=1 ( 2 ) { ax+2y=2 ax6y=2 x= | 2 2 2 6 | | a 2 a 6 | = 2(6 ) (2 ) ( 2 ) a(6 ) 2a = 12+4 6a2a = 8 8 a = 1 a y= | a 2 a 2 | | a 2 a 6 | = a(2 ) 2a a(6 ) 2a = 2a2a 6a2a = 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 3x2y=0 x= | 48 2 0 2 | | 3 2 3 2 | = 2(48 ) ( 2 ) ( 0 ) 3(2 ) 2( 3 ) = 96 66 = 12 =8 y= | 3 48 3 0 | | 3 2 3 2 | = 3( 0 ) 3(48 ) 3(2 ) 2( 3 ) = 144 66 = 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(3a1 ) = 3 a 3 a 2 a(3a1 ) = a 2 (3a1 ) a (3a1 ) =a y= | 3 3a+1 1 2a | | 3 a 1 a 2 | = 2a( 3 ) (3a+1 ) 3 a 2 a = 6a3a1 a(3a1 ) = (3a1 ) a (3a1 ) = 1 a Sol{ x=a y= 1 a
  11. { x+2 3 y3 8 = 5 6 ( 24 ) y5 6 2x3 5 =0 ( 30 ) { 8(x+2 ) 3(y3 ) =20 5(y5 ) 6(2x3 ) =0 { 8x+163y+9=20 5y2512x+18=0 { 8x3y=2025 12x+5y=7 { 8x3y=5 12x+5y=7 x= | 5 3 7 5 | | 8 3 12 5 | = 5( 5 ) 7(3 ) 8( 5 ) (12 ) (3 ) = 25+21 4036 = 4 4 =1 y= | 8 5 12 7 | | 8 3 12 5 | = 8( 7 ) (12 ) (5 ) 8( 5 ) (12 ) (3 ) = 5660 4036 = 4 4 =1 Sol{ x=1 y=1
  12. { 3x2y=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+65m 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 ) { 34x2y3=17y+34 63y4x1=63x+105 { 34x2y17y=3+34 63y4x63x=1+105 { 34x19y=37 67x+63y=106 x= | 37 19 106 63 | | 34 19 67 63 | = 37( 63 ) (19 ) ( 106 ) 34( 63 ) (19 ) (67 ) = 2331+2014 21421273 = 869 =5 y= | 34 37 67 106 | | 34 19 67 63 | = 34( 106 ) (67 ) ( 37 ) 34( 63 ) (19 ) (67 ) = 3604+2479 21421273 = 869 =7 Sol{ x=5 y=7
  14. { x+y xy =4 (xy ) xy1 x+y+1 = 1 9 9(x+y+1 ) { x+y=4(xy ) 9(xy1 ) =x+y+1 { x+y=4x4y 9x9y9=x+y+1 { x+y4x+4y=0 9x9yxy=9+1 { 3x+5y=0 8x10y=10 1 2 { 3x+5y=0 4x5y=5 x= | 0 5 5 5 | | 3 5 4 5 | = 0(5 ) 5( 5 ) 3(5 ) 5( 4 ) = 25 1520 = 5 =5 y= | 3 0 4 5 | | 3 5 4 5 | = 0( 4 ) 3( 5 ) 3(5 ) 5( 4 ) = 15 1520 = 5 =3 Sol{ x=5 y=3
  15. { xy=2b x a+b + y ab =2 { xy=2b x(ab ) +y(a+b ) (a+b ) (ab ) =2 { xy=2b axbx+ay+by=2( a 2 b 2 ) { xy=2b (ab ) x+(a+b ) y=2( a 2 b 2 ) x= | 2b 1 2( a 2 b 2 ) a+b | | 1 1 ab a+b | = 2b(a+b ) +2( a 2 b 2 ) a+b+(ab ) = 2(a+b ) [ b +a b ] a+ b +a b = 2 (a+b )a 2 a =a+b y= | 1 2b ab 2( a 2 b 2 ) | | 1 1 ab a+b | = 2( a 2 b 2 ) 2b(ab ) a+b+(ab ) = 2(ab ) [a+ b b ] a+ b +a b = 2 (ab )a 2 a =ab Sol{ x=a+b y=ab
  16. { x+9 x9 = y+21 y+39 x+8 x8 = y+19 y+11 { (x+9 ) (y+39 ) =(y+21 ) (x9 ) (x+8 ) (y+11 ) =(y+19 ) (x8 ) { xy +39x+9y+351= xy 9y+21x189 xy +11x+8y+88= xy 8y+19x152 { 39x+9y+9y21x=351189 11x+8y+8y19x=88152 { 18x+18y=540 1 18 8x+16y=240 1 8 { x+y=30 x2y=30 x= | 30 1 30 2 | | 1 1 1 2 | = 30(2 ) ( 30 ) ( 1 ) 1(2 ) ( 1 ) ( 1 ) = 6030 21 = 3 =10 y= | 1 30 1 30 | | 1 1 1 2 | = 30( 1 ) (30 ) ( 1 ) 1(2 ) ( 1 ) ( 1 ) = 30+30 21 = 3 =20 Sol{ x=10 y=20