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 \( x = 12 \).