solve:
y-3/y-7 - y+1/y+7 + y-63/y2-49
since y^2-49 = (y-7)(y+7), use that as a common denominator. The numerators then satisfy
(y-3)(y+7) - (y+1)(y-7) + (y-63)
y^2 + 4y - 21 - y^2 + 6y + 7 + y - 63
11y - 77
11(y-7)
So, the final fraction is
11(y-7)/(y^2-49)
hey! That's just 11/(y+7)
To simplify the expression, let's start by finding a common denominator for all the fractions.
In this case, the common denominator will be (y - 7)(y + 7)(y + 1).
Now, we can rewrite each fraction with the common denominator:
(y - 3)/(y - 7) = (y - 3)(y + 7)(y + 1)/[(y - 7)(y + 7)(y + 1)]
(y + 1)/(y + 7) = (y + 1)(y - 7)(y + 1)/[(y - 7)(y + 7)(y + 1)]
(y - 63)/(y^2 - 49) = (y - 63)/[(y - 7)(y + 7)(y + 1)]
Now we can combine the fractions:
[(y - 3)(y + 7)(y + 1) - (y + 1)(y - 7)(y + 1) + (y - 63)]/[(y - 7)(y + 7)(y + 1)]
Next, let's simplify the numerator:
[(y - 3)(y + 7)(y + 1) - (y + 1)(y - 7)(y + 1) + (y - 63)] =
(y^2 - 2y - 21)(y + 1) - (y^2 - 6y - 8)(y + 1) + (y - 63)
= (y^3 - y^2 - 23y - 21) - (y^3 - 7y^2 - 15y - 8) + y - 63
= y^3 - y^2 - 23y - 21 - y^3 + 7y^2 + 15y + 8 + y - 63
= (6y^2 - 7y - 76)/[(y - 7)(y + 7)(y + 1)]
Finally, the expression simplifies to:
(6y^2 - 7y - 76)/[(y - 7)(y + 7)(y + 1)]
Therefore, the simplified expression is (6y^2 - 7y - 76)/[(y - 7)(y + 7)(y + 1)].