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Step by Step Solution

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

a/(a*x-1)+b/(b*x-1)-(a+b)=0

Step 1 :

               b  
 Simplify   ——————
            xb - 1

Equation at the end of step 1 :

       a          b       
  (———————— +  ——————) -  (a + b)  = 0 
   (ax - 1)    xb - 1

Step 2 :

               a  
 Simplify   ——————
            ax - 1

Equation at the end of step 2 :

      a         b       
  (—————— +  ——————) -  (a + b)  = 0 
   ax - 1    xb - 1

Step 3 :

Calculating the Least Common Multiple :

3.1    Find the Least Common Multiple

      The left denominator is :       ax-1

      The right denominator is :       xb-1

Number of times each Algebraic Factor
appears in the factorization of:
Algebraic Factor	Left Denominator	Right Denominator	L.C.M = Max {Left,Right}
ax-1	1	0	1
xb-1	0	1	1

Least Common Multiple:
(ax-1) • (xb-1)

Calculating Multipliers :

3.2    Calculate multipliers for the two fractions

    Denote the Least Common Multiple by L.C.M
    Denote the Left Multiplier by Left_M
    Denote the Right Multiplier by Right_M
    Denote the Left Deniminator by L_Deno
    Denote the Right Multiplier by R_Deno

   Left_M = L.C.M / L_Deno = xb-1

   Right_M = L.C.M / R_Deno = ax-1

Making Equivalent Fractions :

3.3 Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example : 1/2 and 2/4 are equivalent, y/(y+1)² and (y²+y)/(y+1)³ are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

   L. Mult. • L. Num.         a • (xb-1)  
   ——————————————————  =   ———————————————
         L.C.M             (ax-1) • (xb-1)

   R. Mult. • R. Num.         b • (ax-1)  
   ——————————————————  =   ———————————————
         L.C.M             (ax-1) • (xb-1)

Adding fractions that have a common denominator :

3.4 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 a • (xb-1) + b • (ax-1)         2axb - a - b   
 ———————————————————————  =  ———————————————————
     (ax-1) • (xb-1)         (ax - 1) • (xb - 1)

Equation at the end of step 3 :

     (2axb - a - b)      
  ——————————————————— -  (a + b)  = 0 
  (ax - 1) • (xb - 1)

Step 4 :

Rewriting the whole as an Equivalent Fraction :

4.1 Subtracting a whole from a fraction

Rewrite the whole as a fraction using (ax-1) • (xb-1) as the denominator :

             a + b     (a + b) • (ax - 1) • (xb - 1)
    a + b =  —————  =  —————————————————————————————
               1            (ax - 1) • (xb - 1)

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

4.2 Adding up the two equivalent fractions

 (2axb-a-b) - ((a+b) • (ax-1) • (xb-1))     -a²x²b + a²x - ax²b² + 4axb - 2a + xb² - 2b 
 ——————————————————————————————————————  =  ———————————————————————————————————————————
          1 • (ax-1) • (xb-1)                         1 • (ax - 1) • (xb - 1)

Step 5 :

Pulling out like terms :

5.1     Pull out like factors :

   -a²x²b + a²x - ax²b² + 4axb - 2a + xb² - 2b  =

  -1 • (a²x²b - a²x + ax²b² - 4axb + 2a - xb² + 2b)

Equation at the end of step 5 :

  -a²x²b + a²x - ax²b² + 4axb - 2a + xb² - 2b 
  ———————————————————————————————————————————  = 0 
              (ax - 1) • (xb - 1)

Step 6 :

When a fraction equals zero :

 6.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

  -a²x²b+a²x-ax²b²+4axb-2a+xb²-2b 
  ——————————————————————————————— • (ax-1)•(xb-1) = 0 • (ax-1)•(xb-1)
           (ax-1)•(xb-1)

Now, on the left hand side, the (ax-1) • (xb-1) cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
-a²x²b+a²x-ax²b²+4axb-2a+xb²-2b = 0

Solving a Single Variable Equation :

6.2 Solve -a²x²b+a²x-ax²b²+4axb-2a+xb²-2b = 0

In this type of equations, having more than one variable (unknown), you have to specify for which variable you want the equation solved.

We shall not handle this type of equations at this time.