Question 197366
{{{(y-2)/(y-9) - (y+1)/(y+9) + (y-135)/(y^2-81)}}} Start with the given expression.



{{{(y-2)/(y-9) - (y+1)/(y+9) + (y-135)/((y-9)(y+9))}}} Factor the last denominator.



Take note that the LCD is {{{(y-9)(y+9)}}}. In order to add/subtract these fractions, we need to get EVERY denominator equal to the LCD.




{{{((y-2)(y+9))/((y-9)(y+9)) - (y+1)/(y+9) + (y-135)/((y-9)(y+9))}}} Multiply the first fraction by {{{(y+9)/(y+9)}}}



{{{(y^2+7y-18)/((y-9)(y+9)) - (y+1)/(y+9) + (y-135)/((y-9)(y+9))}}} FOIL



{{{(y^2+7y-18)/((y-9)(y+9)) - ((y+1)(y-9))/((y+9)(y-9)) + (y-135)/((y-9)(y+9))}}} Multiply the second fraction by {{{(y-9)/(y-9)}}}
 


{{{(y^2+7y-18)/((y-9)(y+9)) - (y^2-8y-9)/((y+9)(y-9)) + (y-135)/((y-9)(y+9))}}} FOIL



Now that every denominator is equal, we can add/subtract the fractions.



{{{((y^2+7y-18)- (y^2-8y-9)+ (y-135))/((y-9)(y+9))}}} Combine the fractions.



{{{(y^2+7y-18- y^2+8y+9+ y-135)/((y-9)(y+9))}}} Distribute



{{{(16y-144)/((y-9)(y+9))}}} Combine like terms.



{{{(16(y-9))/((y-9)(y+9))}}} Factor



{{{(16*highlight((y-9)))/(highlight((y-9))(y+9))}}} Highlight the common terms.



{{{(16*cross((y-9)))/(cross((y-9))(y+9))}}} Cancel out the common terms.



{{{16/(y+9)}}} Simplify.




So {{{(y-2)/(y-9) - (y+1)/(y+9) + (y-135)/(y^2-81)}}} simplifies to {{{16/(y+9)}}}



In other words, {{{(y-2)/(y-9) - (y+1)/(y+9) + (y-135)/(y^2-81)=16/(y+9)}}}