Ejercicio 185

CAPITULO XXIV

Ecuaciones simultaneas con dos 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 185
Resolver gráficamente:
  1. { xy=1 x+y=7 x=0 { y=1 y=7 y=0 { x=1 x=7 Sol{ x=4 y=3
    interseccion de lineas
  2. { x2y=10 2x+3y=8 x=0 { y=5 y= 8 3 y=0 { x=10 x=4 Sol{ x=2 y=4
    punto entre dos rectas
  3. { 5x3y=0 7xy=16 x=0 { y=0 y=16 y=0 { x=0 x= 16 7 Sol{ x=3 y=5
    ejericio 3
  4. { 3x=4y 5x6y=38 x=0 { y=0 y= 19 3 y=0 { x=0 x= 38 5 Sol{ x=4 y=3
    4
  5. { 3x+4y=15 2x+y=5 x=0 { y= 15 4 y=5 y=0 { x=5 x= 5 2 Sol{ x=1 y=3
    5
  6. { 5x+2y=16 4x+3y=10 x=0 { y=8 y= 10 3 y=0 { x= 16 5 x= 5 2 Sol{ x=4 y=2
    6
  7. { x+8=y+2 y4=x+2 { xy=6 x+y=6 x=0 { y=6 y=6 y=0 { x=6 x=6 Son ecuaciones equivalentes
  8. { 3x 5 + y 4 =2 x5y=25 { 12x+5y=40 x5y=25 x=0 { y=8 y=5 y=0 { x= 10 3 x=25 Sol{ x=5 y=4
    8
  9. { x 2 y 3 = 1 6 x 3 + y 4 = 7 12 { 3x2y=1 4x+3y=7 x=0 { y= 1 2 y= 7 3 y=0 { x= 1 3 x= 7 4 Sol{ x=1 y=1
    9
  10. { x+3y=6 3x+9y=10 x=0 { y=2 y= 10 9 y=0 { x=6 x= 10 3 Las rectas son paralelas el sistema es incompatible
    10
  11. { 2x+3y=13 6x+9y=39 x=0 { y= 13 3 y= 13 3 y=0 { x= 13 2 x= 13 2 Son ecuaciones equivalentes
    11
  12. { x2 2 y3 3 =4 ( 6 ) y2 2 + x3 3 = 11 3 ( 6 ) { 3(x2 ) 2(y3 ) =24 3(y2 ) +2(x3 ) =22 { 3x 6 2y+ 6 =24 3y6+2x6=22 { 3x2y=24 2x+3y=10 x=0 { y=12 y= 10 3 y=0 { x=8 x=5 Sol{ x=4 y=6
    Hallar graficamente el par de valores de x e y que satisfacen cada uno de los grupos de ecuaciones siguientes:
    12
  13. { x+y=9 xy=1 x2y=6 Sol{ x=4 y=5
    13
  14. { x+y=5 3x+4y=18 2x+3y=13 Sol{ x=2 y=3
  15. { 2x+y=1 x2y=13 3x2y=19 Sol{ x=3 y=5
    15
  16. { xy=1 2yx=4 4x5y=7 Sol{ x=2 y=3
    16

Ejercicio 184

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

Ejercicio 183

CAPITULO XXIV

Ecuaciones simultaneas con dos incognitas
Ejercicio 183
Desarrollar las determinantes:
  1. | 4 5 2 3 | =4( 3 ) 2( 5 ) =1210 =2
  2. | 2 7 3 5 | =2( 5 ) 3( 7 ) =1021 =11
  3. | 2 5 4 3 | =2( 3 ) 4( 5 ) =620 =26
  4. | 7 9 5 2 | =7(2 ) 9( 5 ) =1445 =59
  5. | 5 3 2 8 | =5(8 ) (2 ) (3 ) =406 =46
  6. | 9 11 3 7 | =9( 7 ) (3 ) (11 ) =6333 =30
  7. | 15 1 13 2 | =2(15 ) 13(1 ) =30+13 =17
  8. | 12 1 13 9 | =12(9 ) 13(1 ) =108+13 =95
  9. | 10 3 17 13 | =10( 13 ) 17( 3 ) =13051 =79
  10. | 5 8 19 21 | =5(21 ) (8 ) (19 ) =105152 =47
  11. | 8 2 3 0 | =8( 0 ) 2(3 ) =0+6 =6
  12. | 31 85 20 43 | =31( 43 ) (20 ) (85 ) =13331700 =367

Ejercicio 181

CAPITULO XXIV

Ecuaciones simultaneas con dos incognitas
Ejercicio 181
Resolver los siguientes sistemas:
  1. Mathematical Equation
  2. Mathematical Equation
  3. Mathematical Equation
  4. Mathematical Equation
  5. Mathematical Equation
  6. { x b + y a =2 x a + y b = a 2 + b 2 ab { ax+by ab =2 bx+ay ab = a 2 + b 2 ab { ax+by=2ab ( 1 ) bx+ay= a 2 + b 2 ( 2 ) Despejo x de ( 1 ) ax+by =2ab ax =2abby x = 2abby a Reemplazo el valor de x en ( 2 ) b( 2abby a ) +ay = a 2 + b 2 b( 2abby a ) +ay = a 2 + b 2 2a b 2 b 2 y a +ay = a 2 + b 2 2 a b 2 a b 2 y a +ay = a 2 + b 2 y(a b 2 a ) = a 2 2 b 2 + b 2 y( a 2 b 2 a ) = a 2 b 2 y ( a 2 b 2 ) =a ( a 2 b 2 ) y =a Reemplazo el valor de y en ( 1 ) ax+by =2ab ax+ab =2ab ax =2abab a x = a b x =b Sol.{ x=b y=a
  7. Mathematical Equation
  8. Mathematical Equation
  9. Mathematical Equation
  10. Mathematical Equation
  11. Mathematical Equation
  12. Mathematical Equation
  13. Mathematical Equation
  14. Mathematical Equation
  15. Mathematical Equation
  16. Mathematical Equation
  17. Mathematical Equation
  18. { (ab ) x(a+b ) y= b 2 3ab (a+b ) x(ab ) y=ab b 2 { (ab ) x= b 2 3ab+(a+b ) y (a+b ) x=ab b 2 +(ab ) y { x= b 2 3ab+(a+b ) y ab ( 1 ) x= ab b 2 +(ab ) y a+b ( 2 ) ( 1 ) =( 2 ) b 2 3ab+(a+b ) y ab = ab b 2 +(ab ) y a+b b 2 3ab ab + (a+b ) y ab = ab b 2 a+b + (ab ) y a+b (a+b ) y ab (ab ) y a+b = ab b 2 a+b b 2 3ab ab y( a+b ab ab a+b ) = b(ab ) a+b b(b3a ) ab y[ (a+b ) 2 (ab ) 2 (ab ) (a+b ) ] =b[ ab a+b b3a ab ] y[ a 2 +2ab+ b 2 a 2 +2ab b 2 (ab ) (a+b ) ] =b[ (ab ) 2 (a+b ) (b3a ) (ab ) (a+b ) ] y(4a b ) = b [ a 2 2ab+ b 2 (ab3 a 2 + b 2 3ab ) ] 4ay = a 2 2ab + b 2 +3 a 2 b 2 + 2ab 4ay =4 a 2 y = 4 a 2 4 a y =a Reemplazo el valor de y en ( 2 ) x = ab b 2 +(ab ) y a+b x = ab b 2 +(ab ) a a+b x = ab b 2 + a 2 ab a+b x = a 2 b 2 a+b x = (a+b )(ab ) a+b x =ab Sol.{ x=ab y=a
  19. { x+b a + yb b = a+b b xa b ya a = a+b a { x+b a = a+b b yb b xa b = ya a a+b a { x=a( a+b b yb b ) b ( 1 ) x=b( ya a a+b a ) +a ( 2 ) ( 1 ) =( 2 ) a( a+b b yb b ) b =b( ya a a+b a ) +a a b (a+by+b ) b = b a (yaab ) +a a b (a+2by ) b = b a (y2ab ) +a a 2 b +2a ay b b = by a 2b b 2 a +a a 2 b +2ab+2b+ b 2 a a = ay b + by a a 2 b +a+b+ b 2 a =y( a b + b a ) a 3 + a 2 b+a b 2 + b 3 ab =y( a 2 + b 2 ab ) a 2 (a+b ) + b 2 (a+b ) =y( a 2 + b 2 ) ( a 2 + b 2 )(a+b ) =y ( a 2 + b 2 ) y =a+b Reemplazo el valor de y en ( 1 ) x =a( a+b b yb b ) b x =a( a+b b a+ b b b ) b x = a b ( a +b a ) b x = a b (b ) b x =ab Sol.{ x=ab y=a+b
  20. { x a+b + y a+b = 1 ab x b + y a = a 2 + b 2 a 2 b 2 { x a+b = 1 ab y a+b x b = a 2 + b 2 a 2 b 2 y a { x= a+b ab y ( 1 ) x= a 2 + b 2 a 2 b by a ( 2 ) ( 1 ) =( 2 ) a+b ab y = a 2 + b 2 a 2 b by a by a y = a 2 + b 2 a 2 b a+b ab y( b a 1 ) = 1 ab [ a 2 + b 2 a (a+b ) ] y( ba a ) = 1 a b [ a 2 + b 2 a(a+b ) a ] y(ba ) = 1 b [ a 2 + b 2 a 2 ab a ] y(ba ) = 1 b × b (ba ) a y = (ba ) a (ba ) y = 1 a Reemplazo el valor de y en ( 1 ) x = a+b ab y x = a+b ab 1 a x = 1 a ( a+b b 1 ) x = 1 a ( a+ b b b ) x = 1 a ( a b ) x = 1 b Sol.{ x= 1 b y= 1 a

Ejercicio 177

CAPITULO XXIV

Ecuaciones simultaneas con dos incognitas
Ejercicio 177
Resolver por el método de igualación:
  1. { x+3y=6 ( 1 ) 5x2y=13 ( 2 ) Despejando x de ( 1 ) x =63y Reemplazo x en ( 2 ) 5x2y =13 5(63y ) 2y =13 3015y2y =13 17y =30+13 17 y = 17 y =1 Reemplazo y en ( 1 ) x =63y x =63( 1 ) x =3 Sol.{ x=3 y=1
  2. { 5x+7y=1 ( 1 ) 3x+4y=24 ( 2 ) Despejando x de ( 1 ) 5x =7y1 x = 7y1 5 x = 7y+1 5 Reemplazo el valor de x en ( 2 ) 3x+4y =24 3( 7y+1 5 ) +4y =24 3(7y+1 ) 5 =4y24 21y+3 =5(4y24 ) 21y+3 =20y120 21y+20y =3120 41y =123 y = 41 y =3 Reemplazo y en ( 2 ) 3x+4y =24 3x+4(3 ) =24 3x12 =24 3x =1224 3x =12 x = 3 x =4 Sol.{ x=4 y=3
  3. { 4y+3x=8 ( 1 ) 8x9y=77 ( 2 ) Despejando x de ( 1 ) 3x =84y 3x =4(2y ) x = 4(2y ) 3 Reemplazando el valor de x en ( 2 ) 8x9y =77 8 4(2y ) 3 9y =77 32(2y ) 3 =9y77 32(2y ) =3(9y77 ) 6432y =27y231 27y32y =64231 59y =295 y = 59 y =5 Reemplazo y en ( 1 ) 4y+3x =8 4( 5 ) +3x =8 20+3x =8 3x =820 3x =12 x = 3 x =4 Sol.{ x=4 y=5
  4. { x5y=8 ( 1 ) 7x+8y=25 ( 2 ) Despejando x de ( 1 ) x5y =8 x =8+5y Reemplazo el valor de x en ( 2 ) 7x+8y =25 7(8+5y ) +8y =25 5635y+8y =25 27y =56+25 27y =81 y = 27 y =3 Reemplazo y en ( 1 ) x5y =8 x5(3 ) =8 x+15 =8 x =815 x =7 Sol.{ x=7 y=3
  5. { 15x+11y=32 ( 1 ) 7y9x=8 ( 2 ) Despejando y de ( 1 ) 15x+11y =32 11y =3215x y = 3215x 11 Reemplazando el valor de y en ( 2 ) 7y9x =8 7( 3215x 11 ) 9x =8 7( 3215x 11 ) =9x+8 7(3215x ) =11(9x+8 ) 224105x =99x+88 105x99x =224+88 204x =136 x = x = 2 3 Reemplazo x en ( 2 ) 7y9x =8 7y( 2 3 ) =8 7y6 =8 7y =6+8 7y =14 y = 7 y =2 Sol.{ x= 2 3 y=2
  6. { 10x+18y=11 ( 1 ) 16x9y=5 ( 2 ) Despejando x de ( 1 ) 10x+18y =11 10x =18y11 x = 18y+11 10 Reemplazando el valor de x en ( 2 ) 16x9y =5 ( 18y+11 ) 9y =5 8( 18y+11 5 ) =9y5 8(18y+11 ) =5(9y5 ) 144y88 =45y25 144y45y =8825 189y =63 y = 63 y = 1 3 Reemplazo y en ( 2 ) 16x9y =5 16x( 1 3 ) =5 16x+3 =5 16x =53 x = 8 x = 1 2 Sol.{ x= 1 2 y= 1 3
  7. { 4x+5y=5 ( 1 ) 10y4x=7 ( 2 ) Despejando x de ( 1 ) 4x+5y =5 4x =55y 4x =5(1y ) x = 5(1y ) 4 Reemplazando el valor de x en ( 2 ) 10y4x =7 10y 4 [ 5(1y ) 4 ] =7 10y5(1y ) =7 10y5+5y =7 5y =57 5y =2 y = 2 5 Reemplazo y en ( 1 ) 4x+5y =5 4x+ 5 ( 2 5 ) =5 4x+2 =5 4x =52 4x =3 x = 3 4 Sol.{ x= 3 4 y= 2 5
  8. { 32x25y=13 ( 1 ) 16x+15y=1 ( 2 ) Despejando x de ( 2 ) 16x+15y =1 16x =115y x = 115y 16 Reemplazando el valor de x en ( 1 ) 32x25y =13 ( 115y 16 ) 25y =13 2(115y ) =25y+13 230y =25y+13 25y30y =132 55y =11 y = 11 y = 1 5 Reemplazo y en ( 2 ) 16x+15y =1 16x+( 1 5 ) =1 16x3 =1 16x =4 x = 4 x = 1 4 Sol.{ x= 1 4 y= 1 5
  9. { 13y+11x=163 ( 1 ) 8x+7y=94 ( 2 ) Despejando x de ( 2 ) 8x+7y =94 8x =947y x = 7y94 8 Reemplazando el valor x en ( 1 ) 13y+11x =163 13y+11( 7y94 8 ) =163 11( 7y94 8 ) =13y163 11(7y94 ) =8(13y163 ) 77y1034 =104y1304 77y104y =10341304 27y =270 y = 27 y =10 Reemplazo y en ( 2 ) 8x+7y =94 8x+7( 10 ) =94 8x+70 =94 8x =9470 8x =24 x = 8 x =3 Sol.{ x=3 y=10

Ejercicio 176

CAPITULO XXIV

Ecuaciones simultaneas con dos incognitas
Ejercicio 176
Resolver por el método de igualación:
  1. { x+6y=27 ( 1 ) 7x3y=9 ( 2 ) Despejando x de ( 1 ) x =276y Despejando x de ( 2 ) 7x =3y+9 x = 3y+9 7 Igualando ambas ecuaciones 276y = 3y+9 7 7(276y ) =3y+9 18942y =3y+9 42y3y =9189 45y =180 y = 45 y =4 Reemplazo y en ( 1 ) x =276( 4 ) x =2724 x =3 Sol.{ x=3 y=4
  2. { 3x2y=2 ( 1 ) 5x+8y=60 ( 2 ) Despejando x de ( 1 ) 3x =2y2 x = 2y2 3 x = 2(y1 ) 3 Despejando x de ( 2 ) 5x =608y x = 608y 5 x = 4(15+2y ) 5 Igualando ambas ecuaciones 2 (y1 ) 3 = (15+2y ) 5 5(y1 ) =6(15+2y ) 5y5 =9012y 5y+12y =590 17y =85 y = 17 y =5 Reemplazo y en ( 1 ) 3x =2(5 ) 2 3x =102 3x =12 x = 3 x =4 Sol.{ x=4 y=5
  3. { 3x+5y=7 ( 1 ) 2xy=4 ( 2 ) Despejando x de ( 1 ) 3x =75y x = 75y 3 Despejando x de ( 2 ) 2x =y4 x = y4 2 Igualando ambas ecuaciones 75y 3 = y4 2 2(75y ) =3(y4 ) 1410y =3y12 3y10y =1412 13y =26 y = 13 y =2 Reemplazo y en ( 1 ) 3x =75( 2 ) 3x =710 3 x = 3 x =1 Sol.{ x=1 y=2
  4. { 7x4y=5 ( 1 ) 9x+8y=13 ( 2 ) Despejando x de ( 1 ) 7x =4y+5 x = 4y+5 7 Despejando x de ( 2 ) 9x =138y x = 138y 9 Igualando ambas ecuaciones 4y+5 7 = 138y 9 9(4y+5 ) =7(138y ) 36y+45 =9156y 36y+56y =9145 92y =46 y = 46 y = 1 2 Reemplazo y en ( 1 ) 7x =( 1 2 ) +5 7 x = 7 x =1 Sol.{ x=1 y= 1 2
  5. { 9x+16y=7 ( 1 ) 4y3x=0 ( 2 ) Despejando x de ( 1 ) 9x =716y x = 716y 9 Despejando x de ( 2 ) 3x =4y x = 4y 3 Igualando ambas ecuaciones 716y 9 = 4y 3 3(716y ) =9(4y ) 2148y =36y 48y36y =21 84y =21 y = 21 y = 1 4 Reemplazo y en ( 2 ) 3x =4y 3x = 4 ( 1 4 ) x = 1 3 Sol.{ x= 1 3 y= 1 4
  6. { 14x11y=29 ( 1 ) 13y8x=30 ( 2 ) Despejando x de ( 1 ) 14x =11y29 x = 11y29 14 Despejando x de ( 2 ) 8x =3013y x = 13y30 8 Igualando ambas ecuaciones 11y29 = 13y30 4(11y29 ) =7(13y30 ) 44y116 =91y210 44y91y =116210 47y =94 y = 47 y =2 Reemplazo y en ( 2 ) 14x =11y29 14x =11( 2 ) 29 14x =2229 14x =7 x = 7 x = 1 2 Sol.{ x= 1 3 y= 1 4
  7. { 15x11y=87 ( 1 ) 12x5y=27 ( 2 ) Despejando x de ( 1 ) 15x =11y87 x = 11y87 15 Despejando x de ( 2 ) 12x =5y27 x = 5y27 12 Igualando ambas ecuaciones 11y87 = 5y27 4(11y87 ) =5(5y27 ) 44y348 =25y+135 44y+25y =348+135 69y =483 y = 69 y =3 Reemplazo y en ( 2 ) 12x =5y27 12x =5( 3 ) 27 12x =1527 12 x = 12 x =1 Sol.{ x=1 y=3
  8. { 7x+9y=42 ( 1 ) 12x+10y=4 ( 2 ) Despejando x de ( 1 ) 7x =429y x = 429y 7 Despejando x de ( 2 ) 12x =410y x = 2 (2+5y ) x = 2+5y 6 Igualando ambas ecuaciones 429y 7 = 2+5y 6 6(429y ) =7(2+5y ) 25254y =1435y 35y54y =25214 19y =266 y = 19 y =14 Reemplazo y en ( 1 ) 7x =429y 7x =429( 14 ) 7x =42126 7x =84 x = 7 x =12 Sol.{ x=12 y=14
  9. { 6x18y=85 ( 1 ) 24x5y=5 ( 2 ) Despejando x de ( 1 ) 6x =18y85 x = 18y85 6 Despejando x de ( 2 ) 24x =5y5 x = 5y5 24 x = 5(y1 ) 24 Igualando ambas ecuaciones 18y85 6 = 5(y1 ) 4(18y85 ) =5y5 72y340 =5y5 72y5y =3405 67y =335 y = 67 y =5 Reemplazo y en ( 2 ) 24x =5y5 24x =5( 5 ) 5 24x =255 24x =20 x = x = 5 6 Sol.{ x= 5 6 y=5