Universidad Nacional Abierta y a Distancia Escuela de Ciencias Administrativas, Contables, Económicas y de Negocios
Trabajo Colaborativo Algebra Lineal
Tarea 2 – Sistemas de Ecuaciones Lineales, rectas y Planos.
Presentado por:
Lina Marcela Niño Xiomara Franco Cabal Carolina González Mónica Rodríguez Melissa Andrea Papamija
Grupo: 100408_143
Tutor: Herman Belalcázar Ordoñez
Fecha: Mayo 2019
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Introducción
En este trabajo veremos los conceptos matemáticos matemático s de sistemas de ecuaciones lineales, rectas y planos, expresando soluciones a problemas básicos que impliquen el uso de los mismos, justificando sus procedimientos y resultados.
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Actividad Solución de ejercicios
Descripción del ejercicio 1:
Tema escogido para mapa conceptual:
b) Cuáles son las posibles soluciones de un sistema de ecuaciones lineales. Dé ejemplos gráficos y clasifíquelas entre soluciones consistentes e inconsistentes.
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Ejercicio 2. Aplicación de conceptos de sistemas de ecuaciones lineales en la solución de problemas básicos.
Se desea obtener un preparado semanal que cubra las necesidades mínimas de una persona en proteínas, hidratos de carbono y grasas. Suponga que el preparado debe contener una mezcla de los productos A, B y C cuyos contenidos por onzas son los que se indican en la siguiente tabla:
Proteínas
Hidratos
Grasas
Pr oducto oducto A
2
1.5
0.5
Producto Producto B
0.5
3
1.5
Pr oducto oducto C
1.5
2.2
0.5
Di s ponibi lidad
0.5 lb
0.5 lb
0.5 lb
¿Cuántas onzas de cada producto deberán mezclarse semanalmente para
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Tenemos las siguientes variables: Producto A→X Producto B→Y Producto C→Z Se tiene las siguientes ecuaciones lineales:
2 + 0.5 + 1.5 0.5 ∗ 161 1.5+3+2.20.5∗ 161 0.5+1.5+0.50.5∗ 161 Simplificamos:
2+0.5+1.58 1.5+3+2.28 0.5+1.5+0.58
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Se plantea la matriz
8 2 0 . 5 1 . 5 (10..55 13.5 20..25 |88) Simplificando por Gauss Jordán,
4 1 0 . 2 5 0 . 7 5 1 : 2 → (10..55 13.5 20..25 |88) 4 1 0 . 2 5 0 . 7 5 :: 1. 5 0.5 → (00 21..632755 10..017255 |26) 1 0 . 2 5 0 . 7 5 4 1 : 2.625 → (00 1.3175 403./112055 |16/21 ) 6 ::1.0.32755 → (10 01 4638//110055 | 80/21 16/21 ) 0 0 46/105 104/21 80/21 1 0 6 8 / 1 0 5 105 16/21 ) : 46 → (00 10 43/1105 |260/23
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: 10568 → (10 01 00 | 256/23 124/23 ) 43 : 105 0 0 1 260/23 Por lo tanto tenemos:
25613 11,13 12423 5,39 26023 11,30 Comprobamos los resultados
+.+. 211,133 +0.5 5,399 +1.5 11,300 22,26+2,69516,958,0
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Ejercicio 3. Aplicación de conceptos de sistemas de ecuaciones lineales en la solución de problemas básicos.
Considere el siguiente problema, defina el sistema de ecuaciones lineales que le describe y soluciónelo por medio de una reducción de Gauss –Jordan. Valide su resultado por medio de GeoGebra. “Un virus ha destruido parte de la información de los datos de aprobación del
curso de Álgebra Lineal (e-learning) del año 2018. Se ha logrado rescatar parte de la base de datos, sabiendo que el promedio de estudiantes del curso de Álgebra Lineal (e-learning) que entregaron y aprobaron las tareas 1, 2 y 3 del periodo 16-04 de ese año fue de 1.243 estudiantes. Se sabe que el número de estudiantes que aprobaron la Tarea 2 supera en 230 estudiantes al promedio de los que aprobaron la Tarea 1 y la Tarea 3. Así mismo, se sabe que el número de estudiantes que aprobaron la Tarea 3 es
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Solución
Tarea 1 = x Tarea 2 = y Tarea 3 = z
Se ha logrado rescatar parte de la base de datos, sabiendo que el promedio de estudiantes del curso de Álgebra Lineal (e-learning) que entregaron y aprobaron las tareas 1, 2 y 3 del periodo 16-04 de ese año fue de 1.243 estudiantes.
x+y+z 3 1243 y x +2 z +230
Se sabe que el número de estudiantes que aprobaron la Tarea 2 supera en 230 estudiantes al promedio de los que aprobaron la Tarea 1 y la Tarea 3.
Así mismo, se sabe que el número de estudiantes que aprobaron la Tarea 3 es menor en 90 estudiantes al promedio de los estudiantes que aprobaron las Tareas 1 y 2.
z x +2 y 90
Ordenamos las ecuaciones de tal forma que se puedan representar en una matriz
++3729
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1 1 1 3729 + (00 30 03 |3549 4189 ) 3 3 0 7638 3 + (00 30 03 |3549 4189 ) 3 0 0 3449 (00 30 03 |3549 4189 ) 6 6 3 , 3 , 3 (10 01 00 |1149, ) 1396, 3 3 0 0 1 1183 1149,66 1396,33 1183 Estudiantes que aprobaron la Tarea 1: 1149,66 Estudiantes que aprobaron la Tarea 2: 1396,33 Estudiantes que aprobaron la Tarea 3: 1183
Ejercicio 4.
Defina las ecuaciones vectoriales, paramétricas y simétricas, de las siguientes rectas, y grafíquelas con ayuda de GeoGebra (u otras herramientas como Scilab, Octave o Matlab): 1. De la recta que pasa por los puntos P= (-3,4,8) y Q=(2,5,7).
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La gráfica en el GeoGebra es la siguiente:
Por lo tanto, el vector director de la recta es,
⃗ , , v 2+3,54,78 2+3,54,78 ⃗ 5 , 1 , 1
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Las ecuaciones simétricas son,
La ecuación vectorial de la recta,
+5 3 1 4 1 8
, , , , +∝ , , ,, ,,+∝ +∝ ,,
2. De la recta que pasa por el punto R=(-5, 4,-3) y que es paralela a la recta que pasa por los puntos A=(-2,4,6) y B=(1,-3,5).
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La ecuación del vector director de la recta paralela es,
v x x, y y, z z v 1+2,34,56 1+2,34,56 3,7,11
Las ecuaciones paramétricas de la recta son,
++ ∝∝ + ∝
Reemplazamos
5+3∝ 47∝ 3∝
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3. De la recta que pasa por el punto S= (-9, 6,11) y cuyo vector director es V= (-2, 7,-6). GeoGebra:
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Las ecuaciones simétricas son,
2+ 9 7 6 11 6 La ecuación vectorial de la recta,
, , , , +∝ , , ,, ,,+∝ +∝ ,,
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( (8i )(15 )(15i )) )) – ((-40i ) (-3i )) )) = 0i ( (3 j )(15 )(15 j )) )) – ((-15 j ) (-3 j )) )) = 0 j ( (3k )(-40 )(-40k )) )) – ((-15k ) (8k )) )) = 0k Conclusión: Los vectores son paralelos ya que el producto cruz de sus vectores son normales = 0
0i + 0j + 0k = 0i + 0j + 0k
b) ¿Cuál es la ecuación del plano que contiene los puntos A(-3,7,9), B(2,4,6) y C(2,2,-1)? Desarrolle claramente el paso a paso necesario para llegar a dicha ecuación y grafique el plano correspondiente.
1. Realizar vector
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35 130 55 310+ 55 35 30 15 50 15 + 25 15 15+3510 15+3510
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GeoGebra:
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Conclusión
Podemos concluir que con este trabajo adquirirnos más conocimientos sobre los sistemas de ecuaciones lineales y sus diferentes procesos en la soluciòn de problemas
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Bibliografías
https://www.geogebra.org/classic