Relación Ecuaciones. Ecuaciones de primer grado. Matemáticas. Resolver las siguientes ecuaciones: 5(x + 1) [1] = x + 3 5x x + 2 [2] 3 {3

Transcripción

1 Relación Ecuaciones Matemáticas Ecuaciones de primer grado Resolver las siguientes ecuaciones: 5(x + 1) [1] = x + 5x x + 8 [] [(x ) ] } = 1 [] x + 1 x + x [] 5x (x 8) = (x + ) [5] x + [] 5x [7] 5 = + 9 x = 11 x 1 7x 1 = x ( x 1 ) ( = 0 ˆ8 x 7 ) [8] x + (x 1) = [9] x (x + ) + x = 8 [10] x (x 1)(x + ) + = x [11] x x 8 = 5 x x 1 [1] 1 [ ( )] 5x x (x ) + = 1 [ (x + ) x 1 [1] + x ] (x 1)(x + ) + ( [1] x + 5 ) ( x ) (x + 5)(x ) = ( + 1 ) (x )( x) = 0 [15] x + x 1 x x 10 = (x + 1) [1] (x ) (x 1)(x + 1) x = x + [17] x + 1 x = 11 x x

2 Soluciones: [1] Identidad [] 1 [] 7 5 [] Sin solución [5] [] [7] 7 [8] [9] 5 [10] [11] 5 [1] 0 [1] [1] [15] [1] 8 [17] Ecuaciones de segundo grado 1. Resolver las siguientes ecuaciones de segundo grado: [1] 7x = [] x = 10 [] x 5 = x [] 5x = [5] x = x x [] 8x 5 = x [7] (x + 1)(x ) + = 0 [8] (x + 9)(x 9) = (x 7) [9] (x + )(x ) = 15 [10] (x ) = x(x 9) + [11] 1x 18 = 0 [1] (x ) = 1 [1] x = 0 [1] x 7x + 1 = 0 [15] x 9x + 18 = 0 [1] x 9x + 1 = 0 [17] x x + 8 = 0 [18] x x + 9 = 0 [19] x + x 7 = 0 [0] x + x + 1 = 0 [1] x(x ) = 5 [] x(x ) + 1 = 5(x ) [] x x = 1 x [5] x + 1 x = 5 8 [7] x + x = x + 1 x [] x + = x [] x = x 9 x [8] (x + 1)(x ) = 8x (x 1) [9] x(x + 1) + x + [1] x x 5 = 1 5 x = 0 [0] 8x x = x 7 [] x + ( 5 = x x + 1 ) 5 [] 5x 8 x 5 = x 8 x + 1 [] 5x 5 x + = 8 x x [5] x + 5x + = 0 [] x = 5x + [7] x 5x + = 5(8 x) [8] 5x 8x x = 0 [9] (x + 7)(x 7) = x 1 [0] x = (x + 8)(x ) [1] 5x = x + [] 8x 10x x 7 x = 0

3 . Resolver las siguientes ecuaciones de segundo grado: [] x 5 + x 5 = x x [] x + x = 9 [5] x + 9 = 10x [] x + 10x 8 = 0 [7] x x = 0 [8] x 5x + = 0 [9] x + 8x + 15 = 0 [50] x + 10x + 5 = 0 [51] x 9x = 0 [5] x 7x + 57 = 0 [5] x + x = 8 [5] x 5x + = 5(8 x) [55] x = (x + 8)(x ) [5] [ (x + ) + (x + 1) ] = 1(x + 1)(x + ) [57] 9(x ) + (x + )(x ) = (x + 7)(x ) + 9(7 x) [58] (x + 1)(x + 1) = 5(x 1)(x 1) [59] 7x(x + ) = 9(x + )(x + 7) 171(x 5) [0] (x + 1) (x ) (x + 1)(x 1)(x ) (x 1)(x ) + (x + )(x 1) = 0 [1] (x 1)(x 1)(x + ) (x 7)(x + 1)(x + ) = (x 1)(x + ) (x 1)(x 1) [] (x )(x 11) = (x + 1)(5x 1) [] x( x) = 10 x [] x + (x + 1) = 1x(x + 1) [5] (x + 1) (x 1) + 5(x + 1)(x 1) = 0 [] (x ) (x + ) = (x )(x ) [7] ( x)( + x) 18(x ) = (x + )(x )

4 Soluciones: [1] ± [] ±1 [] 0, 5 [] ± [5] 0, 5 [] ± [7] 0, [8] 0, [9] ± [10] ± [11] ± [1] ± [1] [1], [15], [1] 7, [17], [18] [19], 9 [0] 1 [1] 5, 1 [] [] 1, [], [5] 8, 1 8 [] [7], [8] 7, 1 [9] 1, [0] 7, 1 [1] 5, 1 [], 1 5 [] 5, 1 [], 5 11 [5], 1 [], 1 [7] ± [8] 0, 17 [9] ± [0] ± [1] ± [] 0, 7 [] ±5 [] [5] 9, 1 [], 8 [7] 5, [8], [9], 5 [50] 5 [51] 9, [5] 19, [5], [5] ± [55] ± [5] 1, [57] 1, 5 [58], 1 [59], [0], 5 5 [1], 1 [], 5 [] 5, [], [5] 1, [] 1, [7] ± 5 Ecuaciones bicuadradas Resolver las siguientes ecuaciones: [1] x + 8x 1 = 0 [] 5x x = 0 [] x + x = 0 [] 8x x 7 = 0 [5] 5x x 51 = 0 [] x + 5x = 0 [7] x + x + = 0 [8] x 5x 9 = 0 [9] x 7x + 1 = 0 [10] x 9x = 0 [11] x + 1x 100 = 0 [1] 9x + 1 = 0x [1] x 7x + 1 = 0 [1] x = 5 x [15] x = 1 x + 1 [1] x 10x + 9 = 0 [17] x + x 1 = 0 [18] x 1 = 0 [19] x 9x = 0 [0] x 5x + = 0 [1] x x + = 0 [] x (x 5)(x + 5) = 9(1 x)(1 + x) [] 8(x 1) (x 5)(x 1) = (x 9)(x 5) [] x ( x ) = 5 [5] x (x 1) + x ( x ) = 5 [] x 5x + 97 = 0 [7] x + x + = 0 [8] x + x 1 = 0 [9] x 10x + 9 = 0 [0] 1 [ (x + 1)(x 1) + x ] = 19x (x 1)

5 Soluciones: [1] ±1 [] ± 1, ± 5 10 [] ±1 [] ±1 [5] ± [] ± [7] Sin solución [8] ± [9] ±, ± [10] ±5, ± [11] ± [1] ±, ± [1] ±, ± [1] ±5, ± [15] ± [1] ±, ±1 [17] ± [18] ± [19] 0, ± [0] ±1, ± [1] ±1, ± [] ± [] ±, ± 1 [] ±5, ± [5] ± [] Sin solución [7] Sin solución [8] ± [9] ±, ±1 [0] ±, ± 1 7 5

6 Sistemas de dos ecuaciones con dos incógnitas Resolver los siguientes sistemas: (y + x + ) = x + y 1 [1] x + y x y = y 1 y 5x x y (x + y) + = 1 [] [5] [7] y + x 8 8 x = 9 (y x) 5x + y 7 = x y 7 = [] x + y = 0 0 1x + 0 y = 0 0 y 1 [] = 1 x = x + (x y) = + y [] y 7x 1 + = x y + x 1 x y = 1 [8] x 5 + y = 5 x y = 0 (x 1) = y + (1 + y) 8 5x y [9] (x + 1) y = 0 5x y 1 = 0 [10] (x + ) (x ) = y x + y = 8 [11] y(x 1) x( y) = 0 xy x(y 1) = [1] x + y = 10 x + y = 5 [1] x y = xy = 8 [1] x y = 5 x + y = 57 [15] 5x + 7y = 1 xy = 8 [1] x 5y = 1 xy = 7 [17] x + xy + y = 1 xy = [18] xy + y = x y = 5 [19] x + y = 90 x + y = [0] x + y = 9 x + y = [1] x y = 0 y + 5 = x [] x y = x y = 8 [] x + y = 0 xy = 1 [] x y = 1 xy = [5] x + y = 1 xy = 0

7 Soluciones: En los pares que siguen, la primera coordenada es la x y la segunda la y. [1] (9, 0) [] (5, ) [] (, ) ( [] 0, 1) 19 [5] ( 7, ) [] (5, ) [7] (0, 0 1) [8] (, 8) [9] (5, ) [10] (, ) [11] (, ) [1] (0, 5), (, )} [1] (8, ), (, 8)} [1] ( 7, ), ( 7, ), ( 7, ), ( 7, )} [15] (1, 8), ( 5, ) ( } [1] (9, 8), 0, ) 5 5 } [17] (, ), (, )} [18] (, 1), ( 7, 1) } [19] (1, 11), (11, 1)} [0] (0, ), ( 1 5, 9 5) } [1] (, 1), ( 5 18, 5 ) } [] (, ), (, ), (, ), (, )} [] (, ), (, ), (, ), (, )} [] (, ), (, ), (, ), (, )} [5] (, 5), (5, ), ( 5, ), (, 5)} Sistemas de tres ecuaciones con tres incógnitas Resolver los siguientes sistemas: x + y + z = 1 x + y z = 11 [1] x + y z = 5 [] x y z = x y z = 9 x + 5y + z = 5 x + y z = 1 x y z = 1 [] x y z = [5] x 5y + 10z = 5 x y + z = 17 x + y 11z = 10 x y + z = 8 x y + z = 1 [7] 5x + y z = 1 [8] y x z = 18 x y z = 11 x + y z = 0 [] [] [9] x + y + z = x y + z = 1 x + y + z = 11 x + y z = 5 5x y + z = x y + z = x + y z = x y + 5z = 18 x + 7y z = Soluciones: En las ternas que siguen, la primera coordenada es la x, la segunda la y, y la tercera es la z Problemas [1] (0, 8, ) [] (,, ) [] (, 1, ) [] (,, ) [5] (5,, 1) [] (1,, ) [7] (,, 1) [8] (, 1, 5) [9] (, 7, ) 1. La suma de los catetos de un triángulo rectángulo es de 9 cm. Determinar los lados del triángulo y el área sabiendo que la hipotenusa es cm. mayor que el cateto 7

8 menor. Solución: lados = 9, 0, 1 cm., Área = 180 cm.. Cuál es el número cuyos más 9 multiplicados por los Solución: ± menos 9 dan por resultado. Hallar un número cuyo cuadrado lo sobrepase en unidades. Solución: 115, 11. La suma de dos números es y la diferencia de sus cuadrados es 50. Cuáles son estos números?. Solución: 7, El perímetro de un triángulo rectángulo mide 0 cm. y la hipotenusa cm. Calcular las longitudes de los dos catetos. Solución:, 10. Un librero vende 8 libros a dos precios distintos: unos a 5 pts. y otros a pts., obteniendo por la venta 105 pts. Cuántos libros vendió de cada clase?. Solución: 9 libros de 5 pts., 75 libros de pts. 7. Una habitación rectangular tiene una superficie de 8 m y su perímetro es de m. Hallar las dimensiones de la habitación. Solución:, 7 m. 8. La diagonal de un rectángulo mide 55 cm. Cuánto miden sus lados, si la base es los de la altura?. Solución:, m. 9. El perímetro de un rectángulo es de 0 m. y su área 5 m. Cuánto miden sus lados?. Solución: 9, m. 10. El área de un rectángulo es de 1 m y su perímetro es 1 m. Hallar las dimensiones. Solución:, m. 11. El área de un rectángulo es 0 m. Si cada dimensión aumenta en 1 m., el área aumenta en 5 m. Hallar las dimensiones. Solución: 0, 1 m. 1. Hallar la altura de un cono sabiendo que su generatriz mide 1 cm. y el área de su base 5π cm. Solución: 1 cm. 1. Hallar la altura y el radio de la base de un cono, sabiendo que la generatriz mide 50 m., y que la altura es 10 m. más larga que el radio de la base. Solución: altura = 0 m., radio = 0 m. 8

9 1. La suma de dos números es 8 y su producto es 15. Calcular dichos números. Solución:, La hipotenusa de un triángulo mide 5 cm. Cuánto miden los catetos sabiendo que uno es 1 cm. mayor que el otro?. Solución:, cm. 1. La hipotenusa de un triángulo rectángulo vale cm. Hallar las longitudes de los catetos sabiendo que uno de ellos es 1 cm. mayor que el otro. Solución: 1, 0 cm. 17. Hallar las dimensiones de un rectángulo cuyo perímetro es 50 cm. y su área 150 cm. Solución: 10, 15 cm. 18. Hallar un número sabiendo que la suma del triple del mismo con el doble de su recíproco es igual a 5. Solución: 1 ó. 19. Hallar dos números positivos sabiendo que uno de ellos es igual al triple del otro más 5 y que el producto de ambos es 8. Solución:, 17 Ecuaciones de grado superior e irracionales 1. Resolver las siguientes ecuaciones: [1] x + x + x + 1 = 0 [] x + x x = 0 [] x 5 = 0 [] x x 1x 0x = 0 [5] x x 11x + 9x 80 = 0 [] x + x x 1 = 0 [7] x + x + x 10 = 0 [8] x 5x + 5x = 0 [9] x + 7x 9x + = 0 [10] x + x x 1 = 0 [11] x + x x = 0 [1] x 5x + = 0 [1] 8x 1x 9x + 11x = 0 [1] x 5 + x + x + x x 1 = 0 [15] x + x 1x x + 8 = 0 Soluciones: [1] 1 [] 1, 1, } [] [] 0, 5, } [5] 1, 5,, } [] 1, 7, } [7] 1,, 5} [8] 1, 1,, 1 } [9], 1, 1 } [10],, } [11] 1, 1, } [1] 1, ± } [1] 1, 1,, 1} [1] 1, 1, 1 } [15],,, } 9

10 . Resolver las siguientes ecuaciones: [1] x = 0 [] x + 1 = x 1 [] 7 x x = 7 [] x = x [5] x + = x 1 [] x 1 + x + = [7] x + x + 7 = 5 [8] x x + = [9] x 1 = x 5 + x 9 [10] x = x 1 [11] x + 1 = x [1] x + 7 = x + [1] x 7 x = [1] x = x [15] + x + = x 7 [1] 5 x = x [17] x 5 + = x + [18] x + 1 = x + [19] x + 10 = 1 + x + [0] x + 5 x + = 1 [1] (x ) x = (x 5) [] x + x + = x [] x + + x + 1 = 7x + [] 15 x = x [5] 5 + x = x + 1 [] x + 1 x = 1 [7] x 1 + x = 1 [8] x 1 = x 5 + x 9 x + 0 [9] + x = x [0] x + 1 x = x [1] x x = 0 [] x + 1 = (x 1) [] x = x + 1 [] x + 1 = 5 x 7 [5] x + x = [] x + 5 = x + 10x + 5 5(x + ) [7] ( x 1 1 ) + = x 1 [8] x 85 = x 7 Soluciones: [1] [] [] [] 8, 1 } [5] 1 9 [] 5 [7], } [8], 1} [9] 5 [10], 1} [11] 9 [1] 9, 1} [1] 8, } [1] 1 [15] 1 [1], 1 } [17] 7 [18] [19] [0] 1 [1] 5 [] [] [] [5] [] 0, } [7] Sin solución [8] 5 [9] [0] 0 [1] 11, 5} [] [] 1 [] [5] [] 1, 1} [7] 10, } [8]

11 Inecuaciones 1. Resolver las siguientes desigualdades: [1] x > 0 [] x > 0 [] x + > [] x [5] (x ) < [] (x + ) > (x + ) [7] x 1 x + > x 1 x [8] (x ) (x + ) < 5 [ ( )] x(x 1) [9] (x )( + x) > ( x) [10] x 5 < x( x) [11] [ x ( )] x + x x (x )(x ) [1] x + > x [1] (x 7) (x + ) + 1 [1] 7x x + 5(x + ) 1 [15] x (x ) < (x + ) [1] x x x Soluciones: [1] x > [] x < [] x > 1 [] x [5] x < [] x < 0 [7] x > 9 5 [8] x > 0 [9] x > [10] x > 0 [11] x 15 [1] x < 5 [1] x 1 [1] x 5 [15] x > 1 [1] x 51 11

12 . Resolver las siguientes desigualdades: x x > x [1] [] + x 5 < 8 x 7 < x 1 x x 9 < 5 (x 1) + (x + ) (x ) > x 1 < x + 7 [] (x + 1) (x ) < (x + ) [] x + x 5 x > x 5(x + 1) < x + 19 x + 5( x) > x 8(x + ) [5] x(x + 1) < + (x 1) [] x(7 x) > x(5 x) + 10x (x + )(x 1) + 11 (x 1) + x(x + ) [7] [8] [10] Soluciones: x 5 x + 7(x ) + 7 > (x + 9) (5x + 5) x x x < 1 > [9] x + 1 x + > 1 x x + 1 x 1x 1 < x + x (x + 1) 8 (x 1) (x + 1)(x 1) > (x ) > 0 [1] 1 < x < [] x < 15 [] 1 1 [] 8 < x 7 [5] x < 5 [] 7 < x < 0 [7] Sin solución [8] 1 < x < [9] 1 < x 1 [10] < x < 1. Resolver las siguientes desigualdades: [1] x 5x + > 0 [] (x + 1) + x + (x + )(x ) + x [] x + 5x + > 0 [] x x + < 0 [5] x x 8 0 [] x x 0 0 [7] x + x + 1 > 0 [8] x x + 9 < 0 [9] x x + 0 [10] x + 10x [11] x + x + > 0 [1] x x [1] x x + 9 < 0 [1] x 8x [15] (x + 1) x + 5 > x(x + ) + 10 [1] (x + )(x 1) + x 5x [17] (x ) (5x + )(x + 1) x + 1 [18] (x + ) + x 5 < 8(x + )

13 Soluciones: [1] x < óó x > [] x 5 [] x < óó x > [] 1 < x < [5] x óó x [] x 5 [7] R 1} [8] Sin solución [9] R [10] x = 5 [11] R [1] R [1] Sin solución [1] Sin solución [15] x < 1óó x > [1] x = 1 [17] x 0óó x 9 [18] R. Resolver las siguientes desigualdades: [1] x + x [] x + 5 x + 0 [] x + 1 x 0 > 0 [] 9x 7 5 x 0 [5] x x [] x (x + )(x ) < 0 [7] x 7x + 7x < 0 [8] x + x 0 x + 1 [9] x 5 > x + 1 [10] 1 ( x 1 ) + x < x + (x 1) [11] + x 5 > x 5 5(x ) [1] 7 ( 1 x ) < x [1] ( x 5 ) 1 x x x 0 [1] 1 (x 1) 8 (x + 1) (1 x) Soluciones: [1] x [, ] [] x ] ] ], 1, + [ [] x ] ], ] 5, + [ [] x ] ] ], 7 9 5, + [ [5] x ] 5, ] [] x ], [ ]0, [ [7] x ], 1[ ]1, [ [8] x [, 1] [1, + [ [9] x ], + [ [10] x ] [, [11] x ] [, 1 [1] x ] [, 17 5 [1] x ], 1[ ]0, 1[ [, + [ [1] x [ 1, + [ 1