5/x-7 = 7/x-5

To solve the equation \( \frac{5}{x-7} = \frac{7}{x-5} \), we first need to find a common denominator to simplify the equation:

\( \frac{5}{x-7} = \frac{7}{x-5} \)

To find the common denominator, multiply the denominators together:

\( (x-7)(x-5) = x^2 -7x -5x +35 = x^2 -12x +35 \)

Now rewrite the equation with the common denominator:

\( \frac{5}{x-7} = \frac{7}{x-5} \)

\( \frac{5(x-5)}{(x-7)(x-5)} = \frac{7(x-7)}{(x-7)(x-5)} \)

\( \frac{5x-25}{x^2 -12x +35} = \frac{7x-49}{x^2 -12x +35} \)

Now we can cancel out the denominators:

\( 5x - 25 = 7x - 49 \)

Now, solve for x:

\( 5x - 25 = 7x - 49 \)

Subtract 5x from both sides:

\( -25 = 2x - 49 \)

Add 49 to both sides:

\( 24 = 2x \)

Divide by 2:

\( x = 12 \)

Therefore, the solution to the equation \( \frac{5}{x-7} = \frac{7}{x-5} \) is \( x = 12 \).