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Fundamentos Matemáticos Primer Departamental
INSTITUTO POLITECNICO NACIONAL GRUPO:
Fundamentos Matemáticos
Fundamentos Matemáticos Página 2
Nombre: Calificación
Grupo: Fecha:10-04-2013
Fundamentos Matemáticos
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
0
2
5
y x
y x
x
Solución
Gráfica del sistema de inecuaciones
Formas Alternativas
Soluciones
Fundamentos Matemáticos
Fundamentos Matemáticos Página 3
Soluciones Reales
Numero de soluciones reales 24
Problema 2
Resolver el siguiente sistema de ecuaciones por Gauss- Jordán
2 3 3
3 2 13
3 2 4
4 3 0
x y z w
x y z w
x y z w
x y z w
Solución
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
3 w - x + y + 4 z � 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
3 w - x + y + 4 z � 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
3 w - x + y + 4 z � 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
x �
71
25-
2 z
25
Perform a substitution.
Substitute x �
71
25
-
2 z
25
into the first equation:
3 z - 2 I 71
25-
2 z
25M + 9 � -3
w � -3 x - 2 y + z + 13
y � -7 x + z + 22
x �
71
25-
2 z
25
Expand the left hand side of the equation
9 - 2
71
25
-
2 z
25
+ 3 z � -3.
9 - 2
71
25
-
2 z
25
+ 3 z � 3 z +
4 z
25
-
142
25
+ 9 �
79 z
25
+
83
25
:
79 z
25+
83
25� -3
w � -3 x - 2 y + z + 13
y � -7 x + z + 22
x �
71
25-
2 z
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-
2 z
25
Isolate terms with z to the left hand side.
Subtract
83
25
from both sides:
79 z
25� -
158
25
w � -3 x - 2 y + z + 13
y � -7 x + z + 22
x �
71
25-
2 z
25
Solve for z.
Multiply both sides by
25
79
:
z � -2
w � -3 x - 2 y + z + 13
y � -7 x + z + 22
x �
71
25-
2 z
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
3 w - x + y + 4 z � 0
Express the system in matrix form:
-1 2 -1 3
1 3 2 -1
-2 1 -3 1
3 -1 1 4
w
x
y
z
=
-3
13
-4
0
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
Subtract
1
3
´ Hrow 1L from row 2:
3 -1 1 4 0
010
3
5
3-
7
313
-2 1 -3 1 -4
-1 2 -1 3 -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
Add
2
3
´ Hrow 1L to row 3:
3 -1 1 4 0
0 10 5 -7 39
01
3-
7
3
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
Add
1
3
´ Hrow 1L to row 4:
3 -1 1 4 0
0 10 5 -7 39
0 1 -7 11 -12
05
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
Subtract
1
10
´ Hrow 2L from row 3:
3 -1 1 4 0
0 10 5 -7 39
0 0 -15
2
117
10-
159
10
0 5 -2 13 -9
Multiply row 3 by
10
3
:
3 -1 1 4 0
0 10 5 -7 39
0 0 -25 39 -53
0 5 -2 13 -9
Subtract
1
2
´ Hrow 2L from row 4:
3 -1 1 4 0
0 10 5 -7 39
0 0 -25 39 -53
0 0 -9
2
33
2-
57
2
Multiply row 4 by
2
3
:
3 -1 1 4 0
0 10 5 -7 39
0 0 -25 39 -53
0 0 -3 11 -19
Subtract
3
25
´ Hrow 3L from row 4:
3 -1 1 4 0
0 10 5 -7 39
0 0 -25 39 -53
0 0 0158
25-
316
25
Multiply row 4 by
25
158
:
3 -1 1 4 0
0 10 5 -7 39
0 0 -25 39 -53
0 0 0 1 -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
w + 3 x + 2 y - z � 13 Hequation 2L
-2 w + x - 3 y + z � -4 Hequation 3L
3 w - x + y + 4 z � 0 Hequation 4L
Swap equation 1 with equation 4:
3 w - x + y + 4 z � 0 Hequation 1L
w + 3 x + 2 y - z � 13 Hequation 2L
-H2 wL + x - 3 y + z � -4 Hequation 3L
-w + 2 x - y + 3 z � -3 Hequation 4L
Subtract
1
3
´ Hequation 1L from equation 2:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x
3+
5 y
3-
7 z
3� 13 Hequation 2L
-H2 wL + x - 3 y + z � -4 Hequation 3L
-w + 2 x - y + 3 z � -3 Hequation 4L
Multiply equation 2 by 3:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x + 5 y - 7 z � 39 Hequation 2L
-H2 wL + x - 3 y + z � -4 Hequation 3L
-w + 2 x - y + 3 z � -3 Hequation 4L
Add
2
3
´ Hequation 1L to equation 3:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x + 5 y - 7 z � 39 Hequation 2L
0 w +x
3-
7 y
3+
11 z
3� -4 Hequation 3L
-w + 2 x - y + 3 z � -3 Hequation 4L
Multiply equation 3 by 3:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x + 5 y - 7 z � 39 Hequation 2L
0 w +x - 7 y + 11 z � -12 Hequation 3L
-w + 2 x - y + 3 z � -3 Hequation 4L
Add
1
3
´ Hequation 1L to equation 4:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x + 5 y - 7 z � 39 Hequation 2L
0 w +x - 7 y + 11 z � -12 Hequation 3L
0 w +5 x
3-
2 y
3+
13 z
3� -3 Hequation 4L
Multiply equation 4 by 3:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x + 5 y - 7 z � 39 Hequation 2L
0 w +x - 7 y + 11 z � -12 Hequation 3L
0 w +5 x - 2 y + 13 z � -9 Hequation 4L
Subtract
1
10
´ Hequation 2L from equation 3:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x + 5 y - 7 z � 39 Hequation 2L
0 w +0 x -
15 y
2+
117 z
10� -
159
10Hequation 3L
0 w +5 x - 2 y + 13 z � -9 Hequation 4L
Multiply equation 3 by
10
3
:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x + 5 y - 7 z � 39 Hequation 2L
0 w +0 x - 25 y + 39 z � -53 Hequation 3L
0 w +5 x - 2 y + 13 z � -9 Hequation 4L
Subtract
1
2
´ Hequation 2L from equation 4:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x + 5 y - 7 z � 39 Hequation 2L
0 w +0 x - 25 y + 39 z � -53 Hequation 3L
0 w +0 x -
9 y
2+
33 z
2� -
57
2Hequation 4L
Multiply equation 4 by
2
3
:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x + 5 y - 7 z � 39 Hequation 2L
0 w +0 x - 25 y + 39 z � -53 Hequation 3L
0 w +0 x - 3 y + 11 z � -19 Hequation 4L
Subtract
3
25
´ Hequation 3L from equation 4:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x + 5 y - 7 z � 39 Hequation 2L
0 w +0 x - 25 y + 39 z � -53 Hequation 3L
0 w +0 x +0 y +158 z
25� -
316
25Hequation 4L
Multiply equation 4 by
25
158
:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x + 5 y - 7 z � 39 Hequation 2L
0 w +0 x - 25 y + 39 z � -53 Hequation 3L
0 w +0 x +0 y +z � -2 Hequation 4L
Subtract 39 ´ Hequation 4L from equation 3:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x + 5 y - 7 z � 39 Hequation 2L
0 w +0 x - 25 y +0 z � 25 Hequation 3L
0 w +0 x +0 y +z � -2 Hequation 4L
Divide equation 3 by -25:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x + 5 y - 7 z � 39 Hequation 2L
0 w +0 x +y +0 z � -1 Hequation 3L
0 w +0 x +0 y +z � -2 Hequation 4L
Subtract 5 ´ Hequation 3L from equation 2:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x + 0 y - 7 z � 44 Hequation 2L
0 w +0 x +y +0 z � -1 Hequation 3L
0 w +0 x +0 y +z � -2 Hequation 4L
Add 7 ´ Hequation 4L to equation 2:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +10 x +0 y +0 z � 30 Hequation 2L
0 w +0 x +y +0 z � -1 Hequation 3L
0 w +0 x +0 y +z � -2 Hequation 4L
Divide equation 2 by 10:
3 w - x + y + 4 z � 0 Hequation 1L
0 w +x +0 y +0 z � 3 Hequation 2L
0 w +0 x +y +0 z � -1 Hequation 3L
0 w +0 x +0 y +z � -2 Hequation 4L
Add equation 2 to equation 1:
3 w + 0 x +y + 4 z � 3 Hequation 1L
0 w +x +0 y +0 z � 3 Hequation 2L
0 w +0 x +y +0 z � -1 Hequation 3L
0 w +0 x +0 y +z � -2 Hequation 4L
Subtract equation 3 from equation 1:
3 w + 0 x +0 y +4 z � 4 Hequation 1L
0 w +x +0 y +0 z � 3 Hequation 2L
0 w +0 x +y +0 z � -1 Hequation 3L
0 w +0 x +0 y +z � -2 Hequation 4L
Subtract 4 ´ Hequation 4L from equation 1:
3 w +0 x +0 y +0 z � 12 Hequation 1L
0 w +x +0 y +0 z � 3 Hequation 2L
0 w +0 x +y +0 z � -1 Hequation 3L
0 w +0 x +0 y +z � -2 Hequation 4L
Divide equation 1 by 3:
w +0 x +0 y +0 z � 4 Hequation 1L
0 w +x +0 y +0 z � 3 Hequation 2L
0 w +0 x +y +0 z � -1 Hequation 3L
0 w +0 x +0 y +z � -2 Hequation 4L
Collect results:
Answer:
w � 4
x � 3
y � -1
z � -2
Fundamentos Matemáticos
Fundamentos Matemáticos Página 4
Problema 3
Resuelva por fracciones parciales lo siguiente
3 2
2
4 4 4 2
2 1
x x x
x x
Solución
Realizando primero la división polinomica tenemos
2
2 2 2 2
2 2 2 2 2
2 2 2 2
3
2 2 2
2
2
2
2 2 2
2
1 ( 1) ( 1)1 1 1
2 ( 1)( 1) ( 1) ( 1) ( 1) ( 1)
( 1) ( 1)1 1
2 ( 1)( 2 1) ( 2 1)
( 2 1)( 1) ( 2 1
2
)
22 2
x x A B C D
x x xx x x
x x A x x B x C x x D x
x xx x
x x A x x x B x x
C x x x D x x
Ax AxAAx x Ax BA Bxx x
2
2
2
2
3
2
3
2
2 ( ) ( ) ( 2 2 )
2 2
B
C D
x x x A C x A B C D x A
Cx
Cx CCx
B C D
A B
x DCx
x
x x
C
D
D
Tendríamos 2 ecuaciones las cuales son las siguientes
0
2
2 2 1
0
A C
A B C D
A B C D
A B C D
Por lo tanto
Fundamentos Matemáticos
Fundamentos Matemáticos Página 5
1 3 1 1, , ,
2 4 2 4A B C D
La descomposición en fracciones parciales seria la siguiente
2
2 2 2 2
2 1 3 1 1
2 1 2( 1) 4( 1)1 1 4 1
x x
x x xx x x
Problema 4
Resuelva por Cramer el siguiente sistema ecuaciones
3 0
5 3
2 1
x y
y z
x z
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
1
0 3 0
3 1 5
1 0 1 6 6
1 3 0 29 29
0 1 5
2 0 1
x
Calculando el determinante 2
Fundamentos Matemáticos
Fundamentos Matemáticos Página 6
1 0 0
0 3 5 2
2 1 1
Ahora calculamos el valor de la variable y
2
1 0 0
0 3 5
2 1 1 2 2
1 3 0 29 29
0 1 5
2 0 1
y
Calculando el determinante 3
1 3 0
0 1 3 19
2 0 1
Ahora calculamos el valor de la variable z
3
1 3 0
0 1 3
2 0 1 17 17
1 3 0 29 29
0 1 5
2 0 1
z