Ecuaciones Lineales

EJERCICIOS UNIDAD III

Resuelve las siguientes ecuaciones.

a. -3x + 7 = 5x + 13

-3x- 5x=13-7

-8x=6

x=- 6/8 b. 101x +102 = 103x + 104

102-104 = 103x-101x

-2 = 2x

x= -1

c. 0.3x – 0.24 = 0.2x + 0.09

0.3x – 0.2x =0.09-0.24

0.1x=-0.15

x=- 0.15/0.1

x=-1.5 d. 0.02 x + 3.75 = 0.8x – 0.15

0.8x-0.02 x= 3.75+0.15

0.78x=3.9

x= 3.9/0.78

x=5

e. 3(x – 4) = - 4

3x-12=-4

3x=12-4

3x=8

x= 8/3 f. -2(x+5)=30 – x

-2x-10=30-x

-2x+x=30+10

x=-40

g. ax – b = cx + d

x= (cx+d+b)/a

h. ax + bx + c = dx + ex – f

x= ((-c-f))/((a+b-d-e) )

i. 8(3x – 5 )– 4(2x + 3 )= 12

24x-40-8x-12=12

16x=40+24

x= 64/16

x=4 j. 5 (x + 4)= - 2(x - 3)

5x+20=-2x+6

5x+2x=6-20

7x=-14

x=- 14/7

x=-2

k. A = ½ h(B + x)

A=½ hB+½ hx

½ hx=A-½ hB

x= 2(A-½ hB)/h

x= (2A-hB)/h

x= 2A/h-B l. s = 4nx + 8x

s=x(4n+8)

x= s/(4n+8)

m. 3x – 9 = x + 3

2x=9+3

x= 12/2

x=6 n. 2x – 5 = 5x + 4

5x-2x=-5-4

3x=-9

x=- 9/3

x=-3

o. -2 (x – 1) – 5 = 3(x – 1 ) – 10

-2x+1-5=3x-3-10

3x+2x=4-3-10

5x=-9

x=- 9/5

Resuelve los siguientes sistemas de ecuaciones por los 5 métodos. (Suma y resta, igualación, sustitución, determinantes y grafico).

a)

x + y=10

x – y = 4

Suma y Resta

x + y=10

x – y = 4

2x=14

x= 14/2

x=7

7+y=10

y=3

Igualación

x + y+4=10+ x-y

2y=10-4

y= 6/2

y=3

x-3=4

x=7 Sustitución

x=10-y

10-y-y=4

-2y=4-10

y= (-6)/(-2)

y=3

x=10-3

x=7

Determinantes

x + y=10

x – y = 4

x= [■(10& 1@4&-1)]/[■(1& 1@1&-1)]

x= ((10)(-1)-(4)(1))/((1)(-1)-(1)(1))

x= (-10-4)/(-1-1)

x= (-14)/(-2)

x=7

y= [■(1& 10@1& 4)]/[■(1& 1@1&-1)]

y= ((1)(4)-(1)(10))/((1)(-1)-(1)(1))

y= (4-10)/(-1-1)

y= (-6)/(-2)

y=3

x y=10-x

9 1

8 2

7 3

6 4

5 5

4 6

3 7

2 8

1 9

0 10

-1 11

-2 12

-3 13

-4 14

-5 15

-6 16

-7 17

-8 18

-9 19

x y=x-4

9 5

8 4

7

3

6 2

5 1

4 0

3 -1

2 -2

1 -3

0 -4

x -5

-2 -6

-3 -7

-4 -8

-5 -9

-6 -10

-7 -11

-8 -12

-9 -13

b) 2x – y = 7

3x + 4y = 16

Suma y Resta

-3(2x – y = 7 )

2(3x + 4y = 16)

-6x+3y = -21

6x + 8y = 32

11y = 11

y=1

2x-1=7

x= 8/2

x=4

Igualación

2x – 7=4- 3/4 x

2x+ 3/4 x=11

11/4 x=11

x= (11)4/11

x=4

y=2(4)-7

y=1 Sustitución

2x – y = 7

y=2x-7

3x + 4(2x-7) = 16

3x+8x-28=16

11x=28+16

x= 44/11

x=4

y=2(4)-7

y=1

Determinantes

2x – y = 7

3x + 4y = 16

x= [■(7& -1@16& 4)]/[■(2& -1@3& 4)]

x= ((7)(4)-(16)(-1))/((2)(4)-(3)(-1))

x= (28+16)/(8+3)

x= 44/11

x=4

y= [■(2& 7@3& 16)]/[■(2& -1@3& 4)]

y= ((2)(16)-(3)(7))/((2)(4)-(3)(-1))

y= (32-21)/(8+3)

y= 11/11

y=1

El sistema es consistente, independiente y con una única solución en (4,1)

Gráfico

2x – y_1 = 7

y_1=2x-7

3x + 4y_2 = 16

y_2=4-3/4 x

c)

3x + y = 7

4x – 5y =3

Suma y Resta

-4(3x + y = 7)

3(4x – 5y =3)

-12x-4y=-28

12x-15y=9

0-19y=-19

y= (-19)/(-19)

y=1

3x + y = 7

x= (7-y)/3

x= 6/3

x=2

Igualación

7-3x= 4/5 x- 3/5

3x+ 4/5 x=7+ 3/5

19/5 x= 38/5

x= 38/19

x=2

y=7-3(2)

y=1 Sustitución

3x + y = 7

y=7-3x

4x – 5(7-3x) =3

4x-35+15x=3

19x=38

x= 38/19

x=2

y=7-3(2)

y=1

Determinantes

3x + y = 7

4x – 5y =3

x= [■(7& 1@3& -5)]/[■(3& 1@4& -5)]

x= ((7)(-5)-(3)(1))/((3)(-5)-(4)(1))

x= (-35-3)/(-15-4)

x= (-38)/(-19)

x=2

y= [■(3& 7@4& 3)]/[■(3& 1@4& -5)]

y= ((3)(3)-(4)(7))/((3)(-5)-(4)(1))

y= (9-28)/(-15-4)

y= (-19)/(-19)

y=1

El sistema es consistente, independiente y con una única solución en (2,1)

Gráfico

3x + y = 7 4x – 5y =3

y=7-3x y= 4/5 x- 3/5

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