por

Taller de
Ejercicios de
Ecuaciones
Lineales

Erika del Cristo Ramos Aguilar

Gestion de la Produccion
Industrial Ficha 1243380
Nicolas Rafael Licero Arzuza
RESOLVER LOS SIGUIENTES SISTEMAS DE ECUACIONES

El objetivo de esta primera parte de la ACTIVIDAD es ejercitar las técnicas de

solución de sistemas de ecuaciones lineales.

1. -10y = -7x + 0,1 2. 2.3x + 4.7y - 70 = 0

11x = 0,1 + 16y 3.4x + 5.6y – 10 = 80

y = -7x + 0,1 ecuacion 1 2.3x = -4.7y + 70

-10 3.4x = -5.6y + 10 + 80

y = 11x – 0,1 ecuacion 2 x = -4.7y + 70 ecuacion 1

16 2.3
-7x + 0,1 = 11x – 0,1
-10 16 x = -5.6y + 90 ecuacion 2
16(-7x + 0,1) = -10 (11x - 0,1) 3.4
-112x + 1.6 = -110x + 1 -4.7y + 70 = -5.6y + 90
-112x + 110x = 1 - 1.6 2.3 3.4
-2x = -0.6 3.4(-4.7y + 70 ) = 2.3 (-5.6y + 90)
x = 0.3 -15.98y + 238 = -12.88y + 207
-15.98y + 12.88y = 207 - 238
-3.1y = -31
y = -7x + 0,1 ecuacion 1 y = 10
-10

y = -7(0.3) + 0.1 = -2/-10 x = -4.7y + 70 ecuacion 1

-10 2.3
x = -4.7 (10) + 70
y = 0.2 2.3

x= 23
2.3

x = 10

1
 6
3x + 9 + 2y + 10 = 6*(7)
3. 2x/15 + 7y/12 = 3 3x + 9 + 2y + 10 = 42
7x/25 – 5y/16 = 3/20 3x + 2y + 19 = 42

3x + 2y = 23 ecuacion 1

2x + 7y = 3 ; 8x + 35y = 180 x+4/3 – 2y-3/5 = 2

60
5(x+4) - 3 (2y-3) = 2
15
7x – 5y = 3 ; 112x - 125y = 60
5x + 20 - 6y + 9 = 15*(2)
400 400 5x - 6y + 29 = 30
5x - 6y = 30-29
8x + 35y = 180 (-14)
112x - 125y = 60 (1) 5x - 6y = 1 ec. 2

-112x – 490y = -2520 3x + 2y = 23 (3)

112x - 125y = 60 5x - 6y = 1 (1)
-615y = -2460
y = -2460/-615 9x + 6y = 69
5x - 6y = 1
y=4 14x = 70
x = 70/14

8x + 35 (4) = 180 x=5

8x + 140 = 180
8x = 180 – 140
8x = 40 3(5) + 2y = 23
x = 40/8 15 + 2y = 23
x=5 2y = 23-15
2y = 8
y = 8/2

y=4

5. 3x+2/4 – 6y-1/5 = 1
5x-3/7 + 9y+1/2 = 6
4. x+3/2 + y+5/3 = 7
x+4/3 – 2y-3/5 = 2
3x+2/4 – 6y-1/5 = 1
x+3/2 + y+5/3 = 7
5 (3x+2) - 4 (6y-1) = 1
3(x+3) + 2 (y+5) = 7 20

2
15x + 10 - 24y + 4 = 20*(1) 4x - y = 24- 9
15x - 24y + 14 = 20
15x - 24y = 20- 14 4x - y = 15
ecuacion 1
15x - 24y = 6 ecuacion 1
2x-1/3 – 3y+4/4 = 7/12
5x-3/7 + 9y+1/2 = 6
4 (2x-1) - 3 (3y+4) = 7
2 (5x-3) + 7 (9y+1) = 6 12 12
14 4 (2x-1) - 3 (3y+4) = 7
10x - 6 + 63y + 7 = 14(6) 8x – 4 – 9y – 12 = 7
10x + 63y + 1 = 84 8x – 9y – 16 = 7
10x + 63y = 84 – 1 8x – 9y = 7+ 16

10x + 63y = 83 ecuacion 2 8x – 9y = 23 ecuacion 2

4x - y = 15 (-2)
15x - 24y = 6 x= 6 + 24y / 15 8x – 9y = 23 (1)
10x + 63y = 83 x= 83 – 63y / 10
-8x + 2y = -30
6 + 24y = 83 – 63y 8x – 9y = 23
15 10 -7y = -7
10 (6 + 24y) = 15 (83 – 63y) y=1
60 + 240y = 1245 – 945y
240y + 945y = 1245 – 60
1185y = 1185 4x – (1) = 15
4x = 15 + 1
y=1 4x= 16
x = 16
4
15x – 24 (1) = 6 x=4
15x – 24 = 6
15x = 6 + 24
15x = 30
x = 30/15

x=2

6. x+4/2 – y+7/8 = 3 7. 3x + 1/ 4 – 2y = 4/3

2x-1/3 – 3y+4/4 = 7/12 x + y =1

x+4/2 – y+7/8 = 3 3x + 1/ 4 – 2y = 4/3

3 (3x + 1) = 4 (4 – 2y)
4 (x+4) - (y + 7) = 3 9x + 3 = 16 – 8y
8
4x + 16 - y - 7 = 8* (3) 9x + 8y = 13 ecuacion 1
4x - y + 9 = 24

3
 -4x +9y = 1 (1)
9x + 8y = 13 (1) -x + y = 4 (-4)
x+y =1 (-9)
-4x +9y = 1
9x + 8y = 13
-9x - 9y = -9 4x – 4y = -16
-1y = 4 5y = -15
y = 4/-1
y = -3
y = -4

x + (-4) = 1
-4x + 9y = 1
x = 1+4
-4x + 9 *(-3) = 1
x=5
-4x – 27 = 1
-4x = 1 + 27
x = 28/-4
x = -7

8. 7 – 2x / 5 – 3y = 3/2 9. x – 3/y + 2 = 2/3

y–x=4 x + 1/y – 2 = 3/2

7 – 2x / 5 – 3y = 3/2 x – 3/y + 2 = 2/3

2 (7 – 2x) = 3 (5 – 3y)
14 – 4x = 15 – 9y 3 (x-3) = 2 (y+2)
-4x +9y = 15-14 3x – 9 = 2y + 4
-4x +9y = 1 ecuacion 1 3x – 2y = 9 + 4
3x – 2y = 13 ecuacion 1

4
 -3x + 4y = 1 ecuacion 1
x + 1/y – 2 = 3/2
3x – y + 1 / x – y + 3 = 5
2 (x+1) = 3 (y-2) 3x – y + 1 = 5 (x – y + 3)
2x + 2 = 3y - 6
2x - 3y = - 6 – 2 3x – y + 1 = 5x – 5y + 15
2x - 3y = - 8 ecuacion 2
3x – 5x + 5y - y + = 15 –
1
3x – 2y = 13 2x - 3y = - 8
-2x + 4y = 14 ecuacion 2
x = 2y + 13 x = 3y - 8
3 2 -3x + 4y = 1 (-1)
-2x + 4y = 14 (1)
2y + 13 = 3y - 8
3 2 3x - 4y = - 1
-2x + 4y = 14
2 (2y + 13) = 3 (3y - 8)
1x = 13
4y + 26 = 9y – 24
4y - 9y = - 24 – 26 x = 13
- 5y = - 50
y = 10
-3 (13) + 4y = 1
x = 3 * (10) – 8
2 -39 + 4y = 1
x = 30 – 8 /2
4y = 39+11
x = 22/2
x = 11 y = 40/4

y = 10

11. x + 3y + 13/ 4x + 5y – 25 = 3
8x + y + 6/ 5x + 3y – 23 = 5
10. x + 2y + 1/2x – y +1 = 2
3x – y + 1 / x – y + 3 = 5 x + 3y + 13/ 4x + 5y – 25 = 3

x + 2y + 1/2x – y +1 = 2 x + 3y + 13 = 3* (4x + 5y – 25)

x + 2y + 1 = 2 (2x – y +1) x + 3y + 13 = 12x + 15y – 75

x + 2y + 1 = 4x – 2y + 2 x – 12x + 3y – 15y = -75 – 13

x - 4x + 2y + 2y = 2 – 1 -11x – 12y = -88 ecuacion 1

5
8x + y + 6/ 5x + 3y – 23 = 5 20(x+1) – 15(y+2) = 12(2x – 2y)
60 60
8x + y + 6 = 5* (5x + 3y – 23) 20x + 20 – 15y – 30 = 24x – 24y
8x + y + 6 = 25x + 15y - 115 20x - 24x – 15y + 24y = -20 + 30

8x - 25x + y - 15y = - 115 – 6 -4x + 9y = 10 ecuacion 1

-17x – 14y = - 121 ecuacion 2

x-3/4 – y-3/3 = 2y – x

3(x-3) – 4(y-3) = 2y – x
-11x – 12y = -88 (-17) 12
-17x – 14y = - 121 (11) 3x - 9 – 4y + 12 = 12(2y – x)
3x - 9 – 4y + 12 = 24y – 12x
187x + 204y = 1496 3x + 12x – 4y – 24y = 9 – 12
-187x – 154y = -1331 15x – 28y = -3 ec. 2
50y = 165
y = 165/50
y = 3.3 -4x + 9y = 10 (15)

15x – 28y = -3 (4)

-60x + 135y = 150

-11x – 12 (3.3) = -88
60x – 112y = -12
-11x – 39.6 = -88
23y = 138
-11x = -88 + 39.6
y=6
x = -48.4 /-11
-4x + 9y = 10
x = 4.4 -4x + 9 * (6) = 10
12. x+1/3 –y+2/4 = 2(x-y)/5 -4x + 54 = 10
x-3/4 – y-3/3 = 2y – x -4x = 10 – 54
x = - 44/-4
x+1/3 –y+2/4 = 2(x-y)/5 x = 11

BIBLIOGRAFÍA

1. Lay David, Algebra lineal, editorial Pearson, primera edición, 2012, México.

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2. Salas Guerrero Humberto, Programación Lineal aplicada, Ecoediciones,
1999. Colombia.

3. Muñoz, Rodolfo, Investigación de Operaciones. 1ra Edic., Mc Graw Hill,

2011, México.

4. Apuntes de Clase.

5. Algebra de A. Baldor.