Respuesta :

5/(x-5)=x/(x-5)-5/4.
Multiply through by 4(x-5):
20=4x-5(x-5).
20=4x-5x+25=25-x, x=5.
This is an extraneous solution because when x=5, the denominators become zero so division by zero occurs, and the result is meaningless.
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ANSWER

The solution
[tex]x = 5[/tex]
is extraneous.

EXPLANATION

The given equation is

[tex] \frac{5}{x - 5} = \frac{x}{x - 5} - \frac{5}{4} [/tex]

We multiply the above equation with the least common multiple of denominators which is
[tex]4(x - 5)[/tex]


This implies that,

[tex] 4(x - 5) \times \frac{5}{x - 5} = 4(x - 5) \times \frac{x}{x - 5} - \frac{5}{4} \times 4(x - 5)[/tex]



This simplifies to

[tex] 20 = 4x- 5x + 25[/tex]

[tex]20 - 25 = 4x - 5x[/tex]


This implies that,


[tex] - 5 = - x[/tex]



Therefore
[tex]x = 5[/tex]


But this solution does not satisfy the original equation.



[tex] \frac{5}{5 - 5} \ne \frac{5}{5 - 5} - \frac{5}{4} [/tex]


[tex] \frac{5}{0} \ne \frac{5}{0} - \frac{5}{4} [/tex]


Hence
[tex]x = 5[/tex]
is an extraneous solution.