#color(blue)("Determining the excluded values")#
You are not mathematically 'allowed' do divide by 0. If this situation exists the equation / expression is called 'undefined'
When you get very close to a denominator of 0 the graph forms asymptotes.
So the excluded values are such that #y^2=81#
Thus #y=-9 and y=+9# are the excluded values
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Simplifying the expression")#
#color(brown)("Consider the denominator:")#
As above; #9^2=81# so #81-y^2" "->" "9^2-y^2# thus we have
#(3y-27)/(9^2-y^2)" "=" "(3y-27)/((9-y)(9+y))#
#" "#.......................................................................................
#color(brown)("Consider the numerator:")#
#3y-27# this is the same as #3y-[3xx9]#
Factor out the 3 giving: #3(y-9)#
#" "#..........................................................................................
#color(brown)("Putting it all together:")#
#(3(y-9))/((9-y)(9+y))larr" can not cancel out yet"#
Note that #(9-y)# is the same as #[-(y-9)]#
so by substitution we have:
#-(3(y-9))/((y-9)(9+y))# giving
#-(y-9)/(y-9)xx3/(9+y)#
but #(y-9)/(y-9)=1larr" This is what cancelling is all about!"#
Giving: #-1xx3/(9+y)" "=" "-3/(9+y)#
