Taller No 5
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TALLER No. 5 1)
y 2 −8 x −4 =0
2 x + y −10 =0 2 x + y =10 2 x =10 − y
(−4 )2 x
8 x = y 2 −4
(−4 )
=10 −y
8 x =−4
y 2 −4 =8 x −8 x = −40 +4 y 8 x = −4
+y 2 0
+y2
−44 +4 y + y 2
y 2 +4 y −44 a =1 b =4 c =−44 y = y =
2 x =10 − y
−b ± b 2 −4 ac 2a −4 ± 4
10 − y x= 2 10 −4,9 x= 2 5,1 x= 2 x =2,55
−4(1)(−44 ) 2(1)
2
−4 ± 16 −4(−44 ) 2 −4 ± 16 +176 = 2 −4 +13,8 9,8 = = 2 2 =4,9 −4 −13,8 −17,8 = = 2 2 =−8,9
y = y y y y y
10 −y x = 2 X Y 4 2 3 4 2 6 1 8
x =
X 1.5 4 7.5 12
y 2 −4 8 Y 4 6 8 10
Punto de equilibrio
X 2.55
Y 4.9
10 9 8 7 6
Column C Column G
10 9 8 7 Column C
6
Column G
5 4 3 2 1
1.5
2
2.5
3
3.5
4
Column C Column G
4
Column C Column G
2)
x 2 +5 x −y +1 =0
2 x 2 + y −9 =0
y = x 2 +5 x +1
− y =2 x 2 −9
y = x 2 +5 x +1 − y =2 x 2
−9
3 x 2 +5 x −8
3 x 2 +5 y −8 (3 x +8)( x −1) 3 x 2 +5 x −8
a =3 b =5 c =−8 y = x 2 +5 x +1
y =12 +5(1) +1 y =1 +5 +1 y =7
3 x +8 =0 x +1 =0 3 x =−8 −8 x = 3 x =1
y = x 2 +5 x +1 X 0.5 1 5 7 10
160 150 140
Y 3.75 7 51 85 151
− y =2 x 2 −9 y =
X 0.5 1 5 7 10
2 x −9 −1
Punto de equilibrio
2
Y 8.5 7 -41 -81 -191
X 1
Y 7
160 150 140 130 120 110 100 90 80
C
70 60 50 40 30 20 10 0 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
9
Column C
9.5
10
3)
y 2 + y + x −20
2 y 2 −x −3 y −4
−x = y 2 + y −20
x = 2 y 2 −3 y −4
x = 2 y 2 −3 y −4 −x = y 2 +
y −20
3 y 2 −2 y −24 a =3 b =2 c = −24 y = y =
−b ±
b 2 −4ac 2a
−( −2 ) ±
x x x x
2 −4(3)( −24 ) 2(3) 2
4 −12 ( −24 ) 6 2 ± 4 +288 y = 6 2 +17 ,08 19 ,08 y = = 6 6 y =3,18 y =
−( −2 ) ±
x = 2 y 2 −3 y −4 = 2(3,18 ) −3(3,18 ) −4 = 2(10 ,1) −9,54 −4 = 20 , 2 −9,54 −4 =6,66 2
2 −17 ,08 −15 ,08 = 6 6 y =−2,55 y =
x =
y 2 + y −20 −1
X 14.0 0.0 -22.0 -52.0
Punto de equlibrio X Y 6.7 3.18
x = 2 y 2 − 3y − 4
Y 2 4 6 8
X -2 16 50 100
Y 2 4 6 8
8 7 6 Column C
5
Column G 4 3 2 -60.0
-50.0
-40.0
-30.0
-20.0
-10.0
0.0
10.0
20.0
−4
3(3,18 ) −4 54 −4 −4
unto de equlibrio
Column C Column G
10.0
20.0
4)
( x +12 )( y +6 ) =169
x − y +6 =0 y = x +6
( x +12 )( x +6 +6 ) =169 ( x +12 )( x +12 ) =169
x +25
=0
x =− 25
x 2 +12 x +12 x +144 =169 x 2 +24 x +144 =169 x 2 +24 x =169 −144
x −1 =0 x =1 y = x +6
x 2 +24 x = 25
y =1 +6
x 2 +24 x −25 =0 ( x +25)( x −1)
y =7
x 2 −x +25 x −25 x 2 +24 x −25 =0
( x +12 )( y +6 ) =169
y = x +6 X 1 2 3 4
Y 7 6.1 5.3 4.6
X 1 2 4 6
Y 7 8 10 12
Punto de equlibrio X 1
Y 7
12 11 10 9 8
Colum Colum
7 6 5 4 1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
3.25
3.5
3.75
4
5
Column C Column G
3.5
3.75
4
5)
P1 =80
P2 =60
Q1 =10
Q2 = 20
m =
P2 −P1 Q2 −Q1
60 −80 20 −10 −20 m = 10 m =− 2 m =
P −P1 =m(Q −Q1 )
P −80 =−2(Q −10 ) P −80 =−2Q +20 P =−20 +20 +80 P =−2Q +100 Comprobación P =−2(10 ) +100 P =−20 +100 P =80 6) P =100 1
P2 = 0
Q1 = 0 m =
Q2 =50
P2 −P1 Q 2 −Q1
0 −100 50 −0 −100 m = 50 m =− 2 m =
P −P1 = m(Q −Q1 )
P −100 = −2(Q −0 ) P −100 = −2Q +0 P = −2Q +100 Comprobaci
ón
P = −2(0 ) +100 P =0 +100 P =100
7)
P1 =50
P2 = 75
Q1 =50
Q2 =100
m =
P2 −P1 Q 2 −Q1
75 −50 m = 100 −50 25 m = 50 m =0,5 P −P1 = m(Q −Q1 )
P −50 =0,5(Q −50 ) P −50 =0,5Q −25 P =0,5Q −25 +50 P =0,5Q +25 Comprobaci ón
P =0,5(50 ) +25 P = 25 +25 P =50
8) P = 25 1
Q1 = 0 m =
P2 −P1 Q 2 −Q1
35 −25 20 −0 10 m = 20 m =0,5 m =
P −P1 = m(Q −Q1 )
P −25 =0,5(Q −0 ) P −25 =0,5Q −0 P =0,5Q +25 Comprobaci ón P =0,5(0 ) +25 P =0 +25 P = 25
P2 =35 Q2 = 20
y 2 −8 x −4 =0
2 x + y −10 =0 2 x + y =10 2 x =10 − y
(−4 )2 x
8 x = y 2 −4
(−4 )
=10 −y
8 x =−4
y 2 −4 =8 x −8 x = −40 +4 y 8 x = −4
+y 2 0
+y2
−44 +4 y + y 2
y 2 +4 y −44 a =1 b =4 c =−44 y = y =
2 x =10 − y
−b ± b 2 −4 ac 2a −4 ± 4
10 − y x= 2 10 −4,9 x= 2 5,1 x= 2 x =2,55
−4(1)(−44 ) 2(1)
2
−4 ± 16 −4(−44 ) 2 −4 ± 16 +176 = 2 −4 +13,8 9,8 = = 2 2 =4,9 −4 −13,8 −17,8 = = 2 2 =−8,9
y = y y y y y
10 −y x = 2 X Y 4 2 3 4 2 6 1 8
x =
X 1.5 4 7.5 12
y 2 −4 8 Y 4 6 8 10
Punto de equilibrio
X 2.55
Y 4.9
10 9 8 7 6
Column C Column G
10 9 8 7 Column C
6
Column G
5 4 3 2 1
1.5
2
2.5
3
3.5
4
Column C Column G
4
Column C Column G
2)
x 2 +5 x −y +1 =0
2 x 2 + y −9 =0
y = x 2 +5 x +1
− y =2 x 2 −9
y = x 2 +5 x +1 − y =2 x 2
−9
3 x 2 +5 x −8
3 x 2 +5 y −8 (3 x +8)( x −1) 3 x 2 +5 x −8
a =3 b =5 c =−8 y = x 2 +5 x +1
y =12 +5(1) +1 y =1 +5 +1 y =7
3 x +8 =0 x +1 =0 3 x =−8 −8 x = 3 x =1
y = x 2 +5 x +1 X 0.5 1 5 7 10
160 150 140
Y 3.75 7 51 85 151
− y =2 x 2 −9 y =
X 0.5 1 5 7 10
2 x −9 −1
Punto de equilibrio
2
Y 8.5 7 -41 -81 -191
X 1
Y 7
160 150 140 130 120 110 100 90 80
C
70 60 50 40 30 20 10 0 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
9
Column C
9.5
10
3)
y 2 + y + x −20
2 y 2 −x −3 y −4
−x = y 2 + y −20
x = 2 y 2 −3 y −4
x = 2 y 2 −3 y −4 −x = y 2 +
y −20
3 y 2 −2 y −24 a =3 b =2 c = −24 y = y =
−b ±
b 2 −4ac 2a
−( −2 ) ±
x x x x
2 −4(3)( −24 ) 2(3) 2
4 −12 ( −24 ) 6 2 ± 4 +288 y = 6 2 +17 ,08 19 ,08 y = = 6 6 y =3,18 y =
−( −2 ) ±
x = 2 y 2 −3 y −4 = 2(3,18 ) −3(3,18 ) −4 = 2(10 ,1) −9,54 −4 = 20 , 2 −9,54 −4 =6,66 2
2 −17 ,08 −15 ,08 = 6 6 y =−2,55 y =
x =
y 2 + y −20 −1
X 14.0 0.0 -22.0 -52.0
Punto de equlibrio X Y 6.7 3.18
x = 2 y 2 − 3y − 4
Y 2 4 6 8
X -2 16 50 100
Y 2 4 6 8
8 7 6 Column C
5
Column G 4 3 2 -60.0
-50.0
-40.0
-30.0
-20.0
-10.0
0.0
10.0
20.0
−4
3(3,18 ) −4 54 −4 −4
unto de equlibrio
Column C Column G
10.0
20.0
4)
( x +12 )( y +6 ) =169
x − y +6 =0 y = x +6
( x +12 )( x +6 +6 ) =169 ( x +12 )( x +12 ) =169
x +25
=0
x =− 25
x 2 +12 x +12 x +144 =169 x 2 +24 x +144 =169 x 2 +24 x =169 −144
x −1 =0 x =1 y = x +6
x 2 +24 x = 25
y =1 +6
x 2 +24 x −25 =0 ( x +25)( x −1)
y =7
x 2 −x +25 x −25 x 2 +24 x −25 =0
( x +12 )( y +6 ) =169
y = x +6 X 1 2 3 4
Y 7 6.1 5.3 4.6
X 1 2 4 6
Y 7 8 10 12
Punto de equlibrio X 1
Y 7
12 11 10 9 8
Colum Colum
7 6 5 4 1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
3.25
3.5
3.75
4
5
Column C Column G
3.5
3.75
4
5)
P1 =80
P2 =60
Q1 =10
Q2 = 20
m =
P2 −P1 Q2 −Q1
60 −80 20 −10 −20 m = 10 m =− 2 m =
P −P1 =m(Q −Q1 )
P −80 =−2(Q −10 ) P −80 =−2Q +20 P =−20 +20 +80 P =−2Q +100 Comprobación P =−2(10 ) +100 P =−20 +100 P =80 6) P =100 1
P2 = 0
Q1 = 0 m =
Q2 =50
P2 −P1 Q 2 −Q1
0 −100 50 −0 −100 m = 50 m =− 2 m =
P −P1 = m(Q −Q1 )
P −100 = −2(Q −0 ) P −100 = −2Q +0 P = −2Q +100 Comprobaci
ón
P = −2(0 ) +100 P =0 +100 P =100
7)
P1 =50
P2 = 75
Q1 =50
Q2 =100
m =
P2 −P1 Q 2 −Q1
75 −50 m = 100 −50 25 m = 50 m =0,5 P −P1 = m(Q −Q1 )
P −50 =0,5(Q −50 ) P −50 =0,5Q −25 P =0,5Q −25 +50 P =0,5Q +25 Comprobaci ón
P =0,5(50 ) +25 P = 25 +25 P =50
8) P = 25 1
Q1 = 0 m =
P2 −P1 Q 2 −Q1
35 −25 20 −0 10 m = 20 m =0,5 m =
P −P1 = m(Q −Q1 )
P −25 =0,5(Q −0 ) P −25 =0,5Q −0 P =0,5Q +25 Comprobaci ón P =0,5(0 ) +25 P =0 +25 P = 25
P2 =35 Q2 = 20
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