2013
Profesor: Gerson Villa Gonzรกlez
Fundamentos Matemรกticos
Fundamentos Matemรกticos Primer Departamental
INSTITUTO POLITECNICO NACIONAL GRUPO: gvilla@ipn.mx
Fundamentos Matemáticos Nombre: Grupo: Fundamentos Matemáticos
Calificación Fecha:10-04-2013
Instrucciones:
La realización de los ejercicios tiene un peso sobre la calificación del 60%
Problema 1 Traza la siguiente gráfica del sistema de ecuaciones y halle el conjunto solución si la hay
yx0 yx2 x5 Solución
Gráfica del sistema de inecuaciones
Formas Alternativas
Soluciones
Fundamentos Matemáticos
Página 2
Fundamentos Matemáticos Soluciones Reales
Numero de soluciones reales 24 Problema 2 Resolver el siguiente sistema de ecuaciones por Gauss- Jordán
2 x y 3 z w 3 3x 2 y z w 13 x 3 y z 2w 4 x y 4 z 3w 0 Solución
Fundamentos Matemáticos
Página 3
Solve the following system: -w + 2 x - y + 3 z -3 w + 3 x + 2 y - z 13 -2 w + x - 3 y + z -4 3w-x+ y+4z 0
Choose an equation and a variable to solve for.
In the second equation, look to solve for w: -w + 2 x - y + 3 z -3 w + 3 x + 2 y - z 13 -2 w + x - 3 y + z -4 3w-x+ y+4z 0
Solve for w.
Subtract 3 x + 2 y - z from both sides: -w + 2 x - y + 3 z -3 w 13 - 3 x - 2 y + z -2 w + x - 3 y + z -4 3w-x+ y+4z 0
Perform a substitution.
Substitute w -3 x - 2 y + z + 13 into the first, third and fourth equations: 5 x + y + 2 z - 13 -3 w -3 x - 2 y + z + 13
-2 H-3 x - 2 y + z + 13L + x - 3 y + z -4 3 H-3 x - 2 y + z + 13L - x + y + 4 z 0
Expand the left hand side of the equation x + z - 3 y - 2 H13 + z - 3 x - 2 yL -4.
x - 3 y + z - 2 H13 - 3 x - 2 y + zL
H6 x + 4 y - 2 z - 26L + x - 3 y + z 7 x + y - z - 26:
5 x + y + 2 z - 13 -3 w -3 x - 2 y + z + 13
7 x + y - z - 26 -4
3 H-3 x - 2 y + z + 13L - x + y + 4 z 0 Expand the left hand side of the equation y - x + 3 H13 + z - 3 x - 2 yL + 4 z 0.
y - x + 4 z + 3 H13 - 3 x - 2 y + zL
H-9 x - 6 y + 3 z + 39L - x + y + 4 z -10 x - 5 y + 7 z + 39:
5 x + y + 2 z - 13 -3 w -3 x - 2 y + z + 13 7 x + y - z - 26 -4
-10 x - 5 y + 7 z + 39 0
Choose an equation and a variable to solve for.
In the third equation, look to solve for y: 5 x + y + 2 z - 13 -3 w -3 x - 2 y + z + 13 7 x + y - z - 26 -4 -10 x - 5 y + 7 z + 39 0
Solve for y.
Subtract 7 x - z - 26 from both sides: 5 x + y + 2 z - 13 -3 w -3 x - 2 y + z + 13 y 22 - 7 x + z -10 x - 5 y + 7 z + 39 0
Perform a substitution.
Substitute y -7 x + z + 22 into the first and fourth equations: -2 x + 3 z + 9 -3 w -3 x - 2 y + z + 13 y -7 x + z + 22
-5 H-7 x + z + 22L - 10 x + 7 z + 39 0
Expand the left hand side of the equation 39 - 10 x - 5 H22 + z - 7 xL + 7 z 0.
39 - 10 x + 7 z - 5 H22 - 7 x + zL H35 x - 5 z - 110L - 10 x + 7 z + 39 25 x + 2 z - 71: -2 x + 3 z + 9 -3 w -3 x - 2 y + z + 13 y -7 x + z + 22 25 x + 2 z - 71 0
Choose an equation and a variable to solve for.
In the fourth equation, look to solve for x: -2 x + 3 z + 9 -3 w -3 x - 2 y + z + 13 y -7 x + z + 22 25 x + 2 z - 71 0
Isolate terms with x to the left hand side.
Subtract 2 z - 71 from both sides: -2 x + 3 z + 9 -3 w -3 x - 2 y + z + 13 y -7 x + z + 22 25 x 71 - 2 z
Solve for x.
Divide both sides by 25: -2 x + 3 z + 9 -3 w -3 x - 2 y + z + 13 y -7 x + z + 22 71 25
x
2z 25
-
Perform a substitution.
Substitute x
71
2z -
25
3 z - 2 I 25 71
into the first equation: 25
2z M + 9 -3 25
w -3 x - 2 y + z + 13 y -7 x + z + 22 x
71 25
2z 25
-
Expand the left hand side of the equation 71 2 z 9-2 + 3 z -3. 25 25
71
2z
9-2
25
79 z 25
25 83 25
+
+3z 3z+
4z
142 -
25
25
+9
79 z
83 +
25
: 25
-3
w -3 x - 2 y + z + 13 y -7 x + z + 22 x
71 25
2z 25
-
Choose an equation and a variable to solve for.
In the first equation, look to solve for z: 79 z 25
+
83 25
-3
w -3 x - 2 y + z + 13 y -7 x + z + 22 x
71 25
-
2z 25
Isolate terms with z to the left hand side.
83 Subtract
from both sides: 25
79 z 25
-
158 25
w -3 x - 2 y + z + 13 y -7 x + z + 22 x
71 25
-
2z 25
Solve for z.
25 Multiply both sides by
: 79
z -2 w -3 x - 2 y + z + 13 y -7 x + z + 22 x
71 25
-
2z 25
Perform a back substitution.
Substitute z -2 into the second, third and fourth equations: z -2 w -3 x - 2 y + 11 y 20 - 7 x x 3
Perform a back substitution.
Substitute x 3 into the second and third equations: z -2 w 2-2 y y -1 x 3
Perform a back substitution.
Substitute y -1 into the second equation: z -2 w 4 y -1 x 3
Sort results.
Collect results in alphabetical order: Answer:
w 4 x 3 y -1 z -2
Solve the following system: -w + 2 x - y + 3 z -3 w + 3 x + 2 y - z 13 -2 w + x - 3 y + z -4 3w-x+ y+4z 0
Express the system in matrix form: w -1 2 -1 3 -3 1 3 2 -1 x 13 = y -2 1 -3 1 -4 3 -1 1 4 0 z
Write the system in augmented matrix form and use Gaussian elimination: -1 2 -1 3 -3 1 3 2 -1 13 -2 1 -3 1 -4 3 -1 1 4 0
Swap row 1 with row 4: 3 -1 1 4 0 1 3 2 -1 13 -2 1 -3 1 -4 -1 2 -1 3 -3
´ Hrow
1 Subtract 3 3
-1
0
10 3
-2 -1
1 2
1L from row 2:
1
4
0
5 3
7 -3
13
-3 -1
1 3
-4 -3
Multiply row 2 by 3: 3 -1 1 4 0 0 10 5 -7 39 -2 1 -3 1 -4 -1 2 -1 3 -3
´ Hrow 1L to row 3: 3 3 -1 1 4 0 0 10 5 -7 39
2
Add
0
1 3
-3
7
11 3
-4
-1
2
-1
3
-3
Multiply row 3 by 3: 3 -1 1 4 0 0 10 5 -7 39 0 1 -7 11 -12 -1 2 -1 3 -3
´ Hrow 1L to row 4: 3 3 -1 1 4 0 0 10 5 -7 39 0 1 -7 11 -12
1
Add
0
5 3
2
-3
13 3
-3
Multiply row 4 by 3: 3 -1 1 4 0 0 10 5 -7 39 0 1 -7 11 -12 0 5 -2 13 -9
1 Subtract 10 3 -1 1 0 10 5
´ Hrow
2L from row 3:
4 -7
0
0
-
15 2
117 10
0
5
-2
13
0 39 -
159 10
-9
10 Multiply row 3 by
: 3 3 -1 1 4 0 0 10 5 -7 39 0 0 -25 39 -53 0 5 -2 13 -9
´ Hrow 2L from row 4: 2 3 -1 1 4 0 0 10 5 -7 39 0 0 -25 39 -53
1
Subtract
0
0
9
-2
33 2
-
57 2
2 Multiply row 4 by
: 3
3 -1 1 4 0 0 10 5 -7 39 0 0 -25 39 -53 0 0 -3 11 -19
´ Hrow 3L from row 4: 25 3 -1 1 4 0 0 10 5 -7 39 0 0 -25 39 -53
3
Subtract
0
0
0
158 25
-
316 25
25 Multiply row 4 by 3 -1 1 4 0 10 5 -7 0 0 -25 39 0 0 0 1
: 158 0 39 -53 -2
Subtract 39 ´ Hrow 4L from row 3: 3 -1 1 4 0 0 10 5 -7 39 0 0 -25 0 25 0 0 0 1 -2
Divide row 3 by -25: 3 -1 1 4 0 0 10 5 -7 39 0 0 1 0 -1 0 0 0 1 -2 Subtract 5 ´ Hrow 3L from row 2: 3 -1 1 4 0 0 10 0 -7 44 0 0 1 0 -1 0 0 0 1 -2 Add 7 ´ Hrow 4L to row 2: 3 -1 1 4 0 0 10 0 0 30 0 0 1 0 -1 0 0 0 1 -2
Divide row 2 by 10: 3 -1 1 4 0 0 1 0 0 3 0 0 1 0 -1 0 0 0 1 -2
Add row 2 to row 1: 3 0 1 4 3 0 1 0 0 3 0 0 1 0 -1 0 0 0 1 -2
Subtract row 3 from row 1: 3 0 0 4 4 0 1 0 0 3 0 0 1 0 -1 0 0 0 1 -2 Subtract 4 ´ Hrow 4L from row 1: 3 0 0 0 12 0 1 0 0 3 0 0 1 0 -1 0 0 0 1 -2
Divide row 1 by 3: 1 0 0 0 4 0 1 0 0 3 0 0 1 0 -1 0 0 0 1 -2
Collect results: Answer:
w 4 x 3 y -1 z -2
Solve the following system: -w + 2 x - y + 3 z -3 Hequation 1L Hequation 2L w + 3 x + 2 y - z 13 -2 w + x - 3 y + z -4 Hequation 3L Hequation 4L 3w-x+ y+4z 0 Swap equation 1 with equation 4: Hequation 1L
3w-x+ y+4z 0
Hequation 2L
w + 3 x + 2 y - z 13 -H2 wL + x - 3 y + z -4
Hequation 3L Hequation 4L
-w + 2 x - y + 3 z -3
´ Hequation 1L from equation 2: 3 Hequation 1L 3w-x+ y+4z 0
1
Subtract
0w+
10 x 3
5y
+
7z 3
-
3
Hequation 2L
13
Hequation 3L
-H2 wL + x - 3 y + z -4 -w + 2 x - y + 3 z -3
Hequation 4L
Multiply equation 2 by 3: 3w-x+ y+4z 0
Hequation 1L Hequation 2L
0 w +10 x + 5 y - 7 z 39
Hequation 3L
-H2 wL + x - 3 y + z -4 -w + 2 x - y + 3 z -3
Hequation 4L
´ Hequation 1L to equation 3: 3 Hequation 1L 3w-x+ y+4z 0 Hequation 2L 0 w +10 x + 5 y - 7 z 39
2
Add
x
0 w +3 -
7y
+
3
11 z 3
Hequation 3L
-4
Hequation 4L
-w + 2 x - y + 3 z -3
Multiply equation 3 by 3: 3w-x+ y+4z 0 0 w +10 x + 5 y - 7 z 39
Hequation 1L Hequation 2L Hequation 3L
0 w +x - 7 y + 11 z -12
Hequation 4L
-w + 2 x - y + 3 z -3
´ Hequation 1L to equation 4: 3 Hequation 1L 3w-x+ y+4z 0 Hequation 2L 0 w +10 x + 5 y - 7 z 39 Hequation 3L 0 w +x - 7 y + 11 z -12
1
Add
0w+
5x 3
-
2y 3
+
13 z 3
Hequation 4L
-3
Multiply equation 4 by 3: 3w-x+ y+4z 0 0 w +10 x + 5 y - 7 z 39 0 w +x - 7 y + 11 z -12
Hequation 1L Hequation 2L Hequation 3L Hequation 4L
0 w +5 x - 2 y + 13 z -9
´ Hequation 2L from equation 3: 10 Hequation 1L 3w-x+ y+4z 0 Hequation 2L 0 w +10 x + 5 y - 7 z 39
1
Subtract
0 w +0 x -
15 y 2
+
117 z 10
Hequation 3L
159 10
-
Hequation 4L
0 w +5 x - 2 y + 13 z -9
10 Multiply equation 3 by
: 3
Hequation 1L
3w-x+ y+4z 0 0 w +10 x + 5 y - 7 z 39
Hequation 2L Hequation 3L
0 w +0 x - 25 y + 39 z -53
Hequation 4L
0 w +5 x - 2 y + 13 z -9
´ Hequation 2L from equation 4: 2 Hequation 1L 3w-x+ y+4z 0 Hequation 2L 0 w +10 x + 5 y - 7 z 39 Hequation 3L 0 w +0 x - 25 y + 39 z -53
1
Subtract
0 w +0 x -
9y 2
+
33 z 2
-
Hequation 4L
57 2
2 Multiply equation 4 by
: 3
3w-x+ y+4z 0 0 w +10 x + 5 y - 7 z 39 0 w +0 x - 25 y + 39 z -53
Hequation 1L Hequation 2L Hequation 3L Hequation 4L
0 w +0 x - 3 y + 11 z -19
´ Hequation 3L from equation 4: 25 Hequation 1L 3w-x+ y+4z 0 Hequation 2L 0 w +10 x + 5 y - 7 z 39 Hequation 3L 0 w +0 x - 25 y + 39 z -53
3
Subtract
0 w +0 x +0 y +
158 z 25
-
Hequation 4L
316 25
25 Multiply equation 4 by
: 158
3w-x+ y+4z 0 0 w +10 x + 5 y - 7 z 39 0 w +0 x - 25 y + 39 z -53
Hequation 1L Hequation 2L Hequation 3L Hequation 4L
0 w +0 x +0 y +z -2
Subtract 39 ´ Hequation 4L from equation 3: Hequation 1L 3w-x+ y+4z 0 Hequation 2L 0 w +10 x + 5 y - 7 z 39 Hequation 3L
0 w +0 x - 25 y +0 z 25
Hequation 4L
0 w +0 x +0 y +z -2
Divide equation 3 by -25: 3w-x+ y+4z 0 0 w +10 x + 5 y - 7 z 39
Hequation 1L Hequation 2L Hequation 3L
0 w +0 x + y +0 z -1
Hequation 4L
0 w +0 x +0 y +z -2
Subtract 5 ´ Hequation 3L from equation 2: Hequation 1L 3w-x+ y+4z 0 Hequation 2L
0 w +10 x + 0 y - 7 z 44
Hequation 3L
0 w +0 x + y +0 z -1 0 w +0 x +0 y +z -2
Hequation 4L
Add 7 ´ Hequation 4L to equation 2: Hequation 1L 3w-x+ y+4z 0 Hequation 2L
0 w +10 x +0 y +0 z 30
Hequation 3L
0 w +0 x + y +0 z -1 0 w +0 x +0 y +z -2
Divide equation 2 by 10: 3w-x+ y+4z 0
Hequation 4L
0 w +x +0 y +0 z 3
Hequation 1L
0 w +0 x + y +0 z -1 0 w +0 x +0 y +z -2
Hequation 3L
Hequation 2L Hequation 4L
Add equation 2 to equation 1: 3 w + 0 x +y + 4 z 3 0 w +x +0 y +0 z 3 0 w +0 x + y +0 z -1 0 w +0 x +0 y +z -2
Hequation 1L Hequation 2L Hequation 3L Hequation 4L
Subtract equation 3 from equation 1: 3 w + 0 x +0 y +4 z 4 0 w +x +0 y +0 z 3 0 w +0 x + y +0 z -1 0 w +0 x +0 y +z -2
Hequation 1L Hequation 2L Hequation 3L Hequation 4L
Subtract 4 ´ Hequation 4L from equation 1: 3 w +0 x +0 y +0 z 12
0 w +x +0 y +0 z 3 0 w +0 x + y +0 z -1 0 w +0 x +0 y +z -2
Hequation 1L Hequation 2L Hequation 3L Hequation 4L
Divide equation 1 by 3: w +0 x +0 y +0 z 4 0 w +x +0 y +0 z 3 0 w +0 x + y +0 z -1 0 w +0 x +0 y +z -2
Collect results: Answer:
w 4 x 3 y -1 z -2
Hequation 1L Hequation 2L Hequation 3L Hequation 4L
Fundamentos Matemáticos
Problema 3 Resuelva por fracciones parciales lo siguiente
4 x3 4 x 2 4 x 2 2 x2 x 1 Solución Realizando primero la división polinomica tenemos
2 x2 x
x 1 x 1 2
2
2 x2 x
x 1 x 1 2
2
A B C D 2 x 1 x 1 ( x 1) ( x 1)2
A( x 1)( x 1) 2 B( x 1) 2 C ( x 1) 2 ( x 1) D( x 1) 2 ( x 1) 2 ( x 1) 2
2 x 2 x A( x 1)( x 2 2 x 1) B( x 2 2 x 1) C ( x 2 2 x 1)( x 1) D( x 2 2 x 1) 2 x 2 x Ax3 2 Ax 2 Ax Ax 2 2 Ax A Bx 2 2 Bx B Cx3 Cx 2 2Cx 2 2Cx Cx C Dx 2 2 Dx D 2 x 2 x x 3 ( A C ) x 2 ( A B C D ) x ( A 2 B C 2 D) A B C D Tendríamos 2 ecuaciones las cuales son las siguientes
AC 0 A B C D 2 A 2B C 2D 1 A B C D 0 Por lo tanto Fundamentos Matemáticos
Página 4
Fundamentos Matemáticos 1 3 1 1 A , B ,C , D 2 4 2 4 La descomposición en fracciones parciales seria la siguiente
2 x2 x
x 1 x 1 2
2
1 3 1 1 2 2 x 1 4 x 1 2( x 1) 4( x 1) 2
Problema 4 Resuelva por Cramer el siguiente sistema ecuaciones
x 3y 0 y 5z 3 2x z 1 Solución Calculando el
1 3
0
0 1 5 29 2 0
1
Calculando el determinante 1
0 3 0 3 1 5 6 1 0 1 Ahora calculamos el valor de la variable x
0 3 1 x 1 1 0 2
3 0 1 5 0 1 6 6 3 0 29 29 1 5 0 1
Calculando el determinante 2 Fundamentos Matemáticos
Página 5
Fundamentos Matemáticos 1
0
0
0
3
5 2
2 1 1 Ahora calculamos el valor de la variable y
1 0 2 y 2 1 0 2
0 3 1 3 1 0
0 5 1 2 2 0 29 29 5 1
Calculando el determinante 3
1 3 0 1
0 3 19
2 0 1 Ahora calculamos el valor de la variable z
1 0 2 z 3 1 0 2
3 0 1 3 0 1 17 17 3 0 29 29 1 5 0 1
Fundamentos Matemáticos
Página 6