Solution - Reducing fractions to their lowest terms

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "y1"   was replaced by   "y^1".  1 more similar replacement(s).

Step  1  :

            y
 Simplify   —
            d

Equation at the end of step  1  :

          y          
  (dc -  (— • c)) -  ye
          d          

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

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

           dc     dc • d
     dc =  ——  =  ——————
           1        d   

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 :

 2.2       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:

 dc • d - (cy)     d2c - cy
 —————————————  =  ————————
       d              d    

Equation at the end of step  2  :

  (d2c - cy)    
  —————————— -  ye
      d         

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

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

          ye     ye • d
    ye =  ——  =  ——————
          1        d   

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   d²c - cy  =   c • (d² - y) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  d² - y 

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 :  d²  is the square of  d¹ 

Check :  y¹   is not a square !!
Ruling : Binomial can not be factored as the difference of two perfect squares

Adding fractions that have a common denominator :

 4.3       Adding up the two equivalent fractions

 c • (d2-y) - (ye • d)     d2c - dye - cy
 —————————————————————  =  ——————————————
           d                     d       

Trying to factor a multi variable polynomial :

 4.4    Factoring    d²c - dye - cy 

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

 Factorization fails

Final result :

  d2c + dye + cy
  ——————————————
        d