Solution - Adding, subtracting and finding the least common multiple

Step  1  :

Equation at the end of step  1  :

   x           1     1 
  (—-(y-x))————————-———)
   y(      (2•(x2)) 2y2

Step  2  :

             1 
 Simplify   ———
            2y2

Equation at the end of step  2  :

   x           1     1 
  (—-(y-x))————————-———)
   y(      (2•(x2)) 2y2
 Step  3  :

Equation at the end of step  3  :

   x                 1      1 
  (— -  (y - x)) ÷ (——— -  ———)
   y                2x2    2y2

Step  4  :

             1 
 Simplify   ———
            2x2

Equation at the end of step  4  :

   x                 1      1 
  (— -  (y - x)) ÷ (——— -  ———)
   y                2x2    2y2

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       2x² 

      The right denominator is :       2y² 

      Least Common Multiple:
      2x²y² 

Calculating Multipliers :

 5.2    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 = y²

   Right_M = L.C.M / R_Deno = x²

Making Equivalent Fractions :

 5.3      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.        y2 
   ——————————————————  =   —————
         L.C.M             2x2y2

   R. Mult. • R. Num.        x2 
   ——————————————————  =   —————
         L.C.M             2x2y2

Adding fractions that have a common denominator :

 5.4       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:

 y2 - (x2)     y2 - x2
 —————————  =  ———————
   2x2y2        2x2y2 

Equation at the end of step  5  :

   x               (y2 - x2)
  (— -  (y - x)) ÷ —————————
   y                 2x2y2  

Step  6  :

            x
 Simplify   —
            y

Equation at the end of step  6  :

  x              (y2 - x2)
  — - (y - x)) ÷ —————————
  y                2x2y2  

Step  7  :

Rewriting the whole as an Equivalent Fraction :

 7.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  y  as the denominator :

             y - x     (y - x) • y
    y - x =  —————  =  ———————————
               1            y     

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 :

 7.2       Adding up the two equivalent fractions

 x - ((y-x) • y)     xy + x - y2
 ———————————————  =  ———————————
        y                 y     

Equation at the end of step  7  :

  (xy + x - y2)   (y2 - x2)
  ————————————— ÷ —————————
        y           2x2y2  

Step  8  :

         xy+x-y2      y2-x2
 Divide  ———————  by  —————
            y         2x2y2

 8.1    Dividing fractions

To divide fractions, write the divison as multiplication by the reciprocal of the divisor :

xy + x - y2     y2 - x2       xy + x - y2       2x2y2  
———————————  ÷  ———————   =   ———————————  •  —————————
     y           2x2y2             y          (y2 - x2)

Trying to factor a multi variable polynomial :

 8.2    Factoring    xy + x - y² 

Try to factor this multi-variable trinomial using trial and error 

 Factorization fails

Trying to factor as a Difference of Squares :

 8.3      Factoring:  y² - x² 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check :  y²  is the square of  y¹ 

Check :  x²  is the square of  x¹ 

Factorization is :       (y + x)  •  (y - x) 

Dividing exponential expressions :

 8.4    y² divided by y¹ = y^{(2 - 1)} = y¹ = y

Final result :

  2x2y • (xy + x - y2)
  ————————————————————
   (x + y) • (y - x)