Question 1036165
<pre><b>
Here's the best way to do it.  The other tutor's method 
will work, but it is too messy.  That's probably why he 
didn't bother to finish it. LOL

We are asked to simplify this fraction:

{{{(a/(a-x)+b/(b-x)+c/(c-x))/(3/x-1/(x-a)-1/(x-b)-1/(x-c))}}}

Let each of the bottom denominators x-a, x-b, x-c 
equal to single letters:

Let u = x-a then a = x-u
Let v = x-b then b = x-v
Let w = x-c then c = x-w

Substitute all those:

{{{((x-u)/(x-u-x)+(x-v)/(x-v-x)+(x-w)/(x-w-x))/(3/x-1/u-1/v-1/w)}}}{{{""=""}}}{{{((x-u)/(-u)+(x-v)/(-v)+(x-w)/(-w))/(3/x-1/u-1/v-1/w)}}}
 
Get rid of so many - signs by multiplying 
top and bottom through by -1

{{{((x-u)/u+(x-v)/v+(x-w)/w)/(-3/x+1/u+1/v+1/w)}}}

Make two fractions out of each fraction in the top:

{{{(x/u-u/u+x/v-v/v+x/w-w/w)/(-3/x+1/u+1/v+1/w)}}}{{{""=""}}}{{{(x/u-1+x/v-1+x/w-1)/(-3/x+1/u+1/v+1/w)}}}{{{""=""}}}{{{(-3+x/u+x/v+x/w)/(-3/x+1/u+1/v+1/w)}}}

Notice that the terms in the top are x times
the terms on the bottom.  [So we can tell here that
the answer is just x.]  However, Multiply 
top and bottom by x:

{{{(x(-3+x/u+x/v+x/w))/(x(-3/x+1/u+1/v+1/w))}}}

Leave the top as is, distribute the x in the bottom:

{{{(x(-3+x/u+x/v+x/w))/(-3+x/u+x/v+x/w))}}}

Cancel

{{{x(cross(-3+x/u+x/v+x/w))/cross(-3+x/u+x/v+x/w)}}}

{{{x}}}   <--- the answer!  Just x.

Edwin</pre></b>