Question

Determine a solução real de cada uma das equações: 10 pontos?

a) t/2 -2/3=5t/6

b) y\4 + 1 = 7-y\2

c) x+1\3 - x-1\6  = 5\2 + x

d)2(x-1)/3 -3(x + 1)\2 = - ( x+2)\6

Answer 1

$a)~~ \dfrac{t}{2} - \dfrac{2}{3} = \dfrac{5t}{6} \to~~mmc=6\\\\3t-4=5t\to\\\\3t-5t=4\to\\\\ -2t=4~(-1)\to\\\\ 2t=-4\to\\\\ t=- \dfrac{4}{2} \to\\\\ \large\boxed{t=-2}$




$b)~~ \dfrac{y}{4} +1=7- \dfrac{y}{2} \to~~ mmc=4\\\\ y+1=28-2y\to\\\\ y+2y=28-1\to\\\\ 3y=27\to\\\\ y= \dfrac{27}{3} \to\\\\ \large\boxed{y=9}$




$c)~~x+ \dfrac{1}{3} -x- \dfrac{1}{6} = \dfrac{5}{2} +x\to~~mmc=6\\\\6x+2-6x-1=15+6x\to\\\\ 6x-6x-6x=15-2+1\to\\\\-6x=14~(-1)\to\\\\ 6x=-14\to\\\\ x=- \dfrac{14}{6} \to\\\\ \boxed{x=- \dfrac{7}{3}}$




$d)~~ \dfrac{2(x-1)}{3} - \dfrac{3(x + 1)}{2} =- \dfrac{( x+2)}{6} \to\\\\ \dfrac{2x-2}{3} - \dfrac{3x + 3}{2} = \dfrac{-x-2}{6} \to~~ mmc=6\\\\ 4x-4-9x-9=-x-2\to\\ \\ 4x-9x+x=-2+4+9\to\\\\ -4x=11~(-1)\to\\\\ 4x=-11\to\\ \\ \large\boxed{x=- \frac{11}{4} }$

Answer 2

a) t = - 2

b) y = 10/3

c) x = -14/5

d) x = -11/2

a) t - 2 = 5t

    2     3      6

m.m.c. (2, 3, 6) = 6

3t - 4 = 5t

6    6      6

Eliminamos todos os denominadores.

3t - 4 = 5t

3t - 5t = 4

- 2t = 4

2t = - 4

t = - 4/2

t = - 2

b) y + 1 = 7 - y

     4            2

m.m.c. (4, 2) = 4

y + 4 = 2.(7 - y)

4      4         4

Eliminamos todos os denominadores.

y + 4 = 2.(7 - y)

y + 4 = 14 - 2y

y + 2y = 14 - 4

3y = 10

y = 10/3

c) x + 1 - x - 1 = 5 + x

       3       6       2

m.m.c. (2, 3, 6) = 6

2.(x + 1) - x - 1 = 3.5 + 6.x

     6         6       6       6

Eliminamos todos os denominadores.

2.(x + 1) - x - 1 = 3.5 + 6x

2x + 2 - x - 1 = 15 + 6x

x + 1 = 15 + 6x

x - 6x = 15 - 1

- 5x = 14

5x = - 14

x = - 14/5

d) 2(x - 1) - 3(x + 1) = - (x + 2)

       3             2              6

2x - 2 - 3x - 3 = - x - 2

   3           2           6

m.m.c. (3, 2, 6) = 6

2.(2x - 2) + 3.(-3x - 3) = - x - 2

      6                6               6

Eliminamos todos os denominadores.

2.(2x - 2) + 3.(-3x - 3) = - x - 2

4x - 4 - 9x - 9 = - x - 2

- 5x - 13 = - x - 2

- 5x + x = - 2 + 13

- 4x = 11

4x = - 11

x = -11/2

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