I'm looking to see if I did this right or not. ((y-5)/((x^2+5x)(x^2+5y)))+((1)/(x+y))-((2)/(x+5)) = ((y-5)/(x+y)(x+5))+((1(x+5)/(x+y)(x+5))-((2(x+y)/(x+y)(x+5)) = ((y-5)+1(x+5)-2(x+y)/(x+y)(x+5)) = y-6
To check if your simplification is correct, let's go through the steps:
1. Starting with the original equation:
((y-5)/((x^2+5x)(x^2+5y)))+((1)/(x+y))-((2)/(x+5))
2. Simplifying the denominators:
((y-5)/((x(x+5))(y(x+5))))+((1)/(x+y))-((2)/(x+5))
3. Finding a common denominator for all the terms:
For the first term: The common denominator is (x(x+5)(y(x+5)))
For the second and third terms: The common denominator is (x+5)(x+y)
4. Expanding the numerators using the common denominator:
First term: (y-5) * (x+5)
Second term: 1 * (x(x+5))
Third term: 2 * (y(x+5))
Now, the equation becomes:
((y-5)(x+5)/(x(x+5)(y(x+5)))) + ((1(x(x+5))/(x+5)(x+y))) - ((2(y(x+5))/(x+5)(x+y)))
5. Combine the numerators:
((y-5)(x+5) + (x(x+5)) - (2(y(x+5)))) / ((x(x+5)(y(x+5))) * (x+y))
Simplifying further:
((xy + 5y - 5x - 25 + x^2 + 5x - 2xy - 10y)) / ((x(x+5)(y(x+5))) * (x+y))
Simplifying the numerator:
(x^2 + 3x - 5y - 25) / ((x(x+5)(y(x+5))) * (x+y))
6. Simplify the common factors in the denominator:
(x^2 + 3x - 5y - 25) / ((x^2 + 5x)(y^2 + 5y)(x+y))
From the steps, it seems that your simplification is not correct. The simplified form should be:
(x^2 + 3x - 5y - 25) / ((x^2 + 5x)(y^2 + 5y)(x+y))
Therefore, y-6 is not the correct simplified form.