Perform the indicated operations and simplify.
y-4/y-9 - y+1/y+9 + y-99/y^2-81
common denominator: (y+9)*(y-9)
I think this is what you meant.
(y-4)(y+9) -(y+1)(y-9) -(y-9) all that over the denominator.
y^2+5y-36 -y^2+8y+9 -y+9 over the denominator.
12y -27 over the denominator
3(4y-9) over the denominator.
I have made a guess and put on the minimum parentheses for the problem to make sense. Do not forget to enclose numerators and denominators with parentheses when transcribing expressions from a book or other sources.
(y-4)/(y-9) - (y+1)/(y+9) + (y-99)/(y^2-81)
using (y-9)(y+9)=y²-81
=(y-4)/(y-9) - (y+1)/(y+9) + (y-99)/((y+9)(y-9))
=((y-4)(y+9)-(y+1)(y-9)+(y-99))/((y+9)(y-9))
=(y²+5y-36 - (y²-8y-9) + (y-99))/((y+9)(y-9))
=(14y-126)/((y+9)(y-9))
=14(y-9)/((y+9)(y-9))
=14/(y+9)
To perform the indicated operations and simplify the expression, we need to find a common denominator and then combine the fractions.
First, let's find the common denominator for the three fractions. The denominators are y-9, y+9, and y^2-81.
To find the common denominator, we need to factorize the last denominator, y^2-81, using the difference of squares identity.
y^2-81 = (y+9)(y-9)
Now, our common denominator is (y-9)(y+9).
Next, let's rewrite each fraction with the common denominator.
y-4/y-9 = [(y-4)(y+9)] / [(y-9)(y+9)]
y+1/y+9 = [(y+1)(y-9)] / [(y-9)(y+9)]
y-99/y^2-81 = (y-99) / [(y-9)(y+9)]
Now, we can combine the fractions by adding or subtracting the numerators and keeping the common denominator.
[(y-4)(y+9)] / [(y-9)(y+9)] - [(y+1)(y-9)] / [(y-9)(y+9)] + (y-99) / [(y-9)(y+9)]
To combine the fractions, we subtract the second fraction and add the third fraction, since the first fraction is already subtracted.
[(y-4)(y+9) - (y+1)(y-9) + (y-99)] / [(y-9)(y+9)]
Now, let's simplify the numerator.
(y-4)(y+9) - (y+1)(y-9) + (y-99)
= (y^2 + 5y - 36) - (y^2 - 8y - 9) + (y-99)
= y^2 + 5y - 36 - y^2 + 8y + 9 + y - 99
= 14y - 116
The simplified expression is 14y - 116, with a common denominator of (y-9)(y+9).