SOLUTION: Solve the equation: Numerator: a/(a-x)+b/(b-x)+c/(c-x) Denominator: 3/x-1/(x-a)-1/(x-b)-1/(x-c). Find the value of Numerator/Denominator

Algebra ->  Equations -> SOLUTION: Solve the equation: Numerator: a/(a-x)+b/(b-x)+c/(c-x) Denominator: 3/x-1/(x-a)-1/(x-b)-1/(x-c). Find the value of Numerator/Denominator      Log On


   

Question 1036165: Solve the equation: Numerator: a/(a-x)+b/(b-x)+c/(c-x)
Denominator: 3/x-1/(x-a)-1/(x-b)-1/(x-c).
Find the value of Numerator/Denominator
Found 2 solutions by Aldorozos, Edwin McCravy:
Answer by Aldorozos(172) About Me  (Show Source):
You can put this solution on YOUR website!
Please note that it is not an equation. What you are doing here is trying to find the value of Numerator/Denominator as you have indicated
[(a/a-x)+(b/b-x)+(c/c-x)] / [(3/x)-(1/x-a)-(1/x-b)-(1/x-c)] =
[(a/a-x)+(b/b-x)+(c/c-x)] / [(3/x)+ (1/a-x)+(1/b-x)+(1/c-x))] We did this by factoring out
[a(b-x)(c-x) + b(a-x)(c-z)+c(a-x)(b-x)]/[(a-x)(b-x)(c-x)] = Numerator
3[(a-x)(b-x)(c-x)+x(b-x)+x(x-a)(x-c)+x(x-a)(x-c)]/x[(a-x)(b-x)(c-x) = Denominator
The last step is to multiply the numbers.

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

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:



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:

%22%22=%22%22
 
Get rid of so many - signs by multiplying 
top and bottom through by -1

%28%28x-u%29%2Fu%2B%28x-v%29%2Fv%2B%28x-w%29%2Fw%29%2F%28-3%2Fx%2B1%2Fu%2B1%2Fv%2B1%2Fw%29

Make two fractions out of each fraction in the top:

%28x%2Fu-u%2Fu%2Bx%2Fv-v%2Fv%2Bx%2Fw-w%2Fw%29%2F%28-3%2Fx%2B1%2Fu%2B1%2Fv%2B1%2Fw%29%22%22=%22%22%28x%2Fu-1%2Bx%2Fv-1%2Bx%2Fw-1%29%2F%28-3%2Fx%2B1%2Fu%2B1%2Fv%2B1%2Fw%29%22%22=%22%22%28-3%2Bx%2Fu%2Bx%2Fv%2Bx%2Fw%29%2F%28-3%2Fx%2B1%2Fu%2B1%2Fv%2B1%2Fw%29

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:

%28x%28-3%2Bx%2Fu%2Bx%2Fv%2Bx%2Fw%29%29%2F%28x%28-3%2Fx%2B1%2Fu%2B1%2Fv%2B1%2Fw%29%29

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

%28x%28-3%2Bx%2Fu%2Bx%2Fv%2Bx%2Fw%29%29%2F%28-3%2Bx%2Fu%2Bx%2Fv%2Bx%2Fw%29%29

Cancel

x%28cross%28-3%2Bx%2Fu%2Bx%2Fv%2Bx%2Fw%29%29%2Fcross%28-3%2Bx%2Fu%2Bx%2Fv%2Bx%2Fw%29

x   <--- the answer!  Just x.

Edwin