Determine the solutions (if any) of the given equation.

Question

4/x + 4/x-15=1                                   and                             x-5/x-4 - x-7/x^2-5x+4= 1/5

Answer 1

Hi Tyler,

 

Problem #1 (solution and check):

 

4/x + 4/x - 15 = 1

4/x + 4/x = 1 + 15

(4 + 4)/x = 16

8/x = 16

8 = 16x

x = 1/2

 

4/(1/2) + 4/(1/2) - 15 = 1

4(2) + 4(2) - 15 = 1

8 + 8 - 15 = 1

16 - 15 = 1

1 = 1

 

Problem #2 (solution and check):

 

(x-5)/(x-4) - (x-7)/(x²-5x+4) = 1/5

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

[(x-4)(x-1)][(x-5)/(x-4)] - 

[(x-4)(x-1)][(x-7)/(x-4)(x-1)] = [(x-4)(x-1)][1/5]

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

x² - 6x + 5 - x + 7 = (x²-5x+4)/5

5(x²-7x+12) = 5(x²-5x+4)/5

5x² - 35x + 60 = x² -  5x + 4

4x² - 30x + 56 = 0

2(2x² - 15x + 28) = 0

2x² -15x + 28 = 0

2x² + (-8-7)x + 28 = 0

2x² -8x - 7x + 28 = 0

2x(x-4) - 7(x-4) = 0

(x-4)(2x-7) = 0

x - 4 = 0

x = 4 (reject)

2x - 7 = 0

2x = 7

x = 7/2 (only answer)

 

(4-5)/(4-4) 

-1/0 (undefined)

(7/2 - 5)/(7/2 - 4) - (7/2 - 7)/[(7/2)²-5(7/2)+4)] = 1/5

(7/2 - 10/2)/(7/2 - 8/2) -

(7/2 - 14/2)/(49/4 - 35/2 + 4) = 1/5

(-3/2)/(-1/2) - (-7/2)/(49/4 - 70/4 + 16/4) = 1/5

3 - (-7/2)/(-5/4) = 1/5

3 - [(-7/2)(-4/5)] = 1/5

3 - 28/10 = 1/5

30/10 - 28/10 = 1/5

2/10 = 1/5

1/5 = 1/5 

   

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.