1/(x-5)-1/(x-7)=1/(x-1)-1/(x-3)

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To solve this equation, we can first find a common denominator for all the fractions on both sides. The least common denominator here is (x-5)(x-7)(x-1)(x-3).

Multiplying each term by this common denominator, we get:

(x-7)(x-1)(x-3) - (x-5)(x-1)(x-3) = (x-5)(x-7)(x-1) - (x-5)(x-7)(x-3)

Expanding each term, we get:

(x^3 - 8x^2 + 15x - 21) - (x^3 - 8x^2 + 16x - 15) = (x^3 - 13x^2 + 42x - 35) - (x^3 - 8x^2 - 12x + 105)

Simplifying both sides, we get:

-6x + 6 = 32x - 140

Combining like terms:

32x + 6x = 140 + 6
38x = 146
x = 146/38
x = 73/19

Therefore, the solution to the equation 1/(x-5) - 1/(x-7) = 1/(x-1) - 1/(x-3) is x = 73/19.