Solution - Reducing fractions to their lowest terms

Step  1  :

            y
 Simplify   —
            x

Equation at the end of step  1  :

              y      
  (9x -  (9 • —)) -  y
              x      

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

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

           9x     9x • x
     9x =  ——  =  ——————
           1        x   

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:

 9x • x - (9y)     9x2 - 9y
 —————————————  =  ————————
       x              x    

Equation at the end of step  2  :

  (9x2 - 9y)    
  —————————— -  y
      x         

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

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

         y     y • x
    y =  —  =  —————
         1       x  

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   9x² - 9y  =   9 • (x² - y) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  x² - 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 :  x²  is the square of  x¹ 

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

 9 • (x2-y) - (y • x)     9x2 - xy - 9y
 ————————————————————  =  —————————————
          x                     x      

Trying to factor a multi variable polynomial :

 4.4    Factoring    9x² - xy - 9y 

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

 Factorization fails

Final result :

  9x2 + xy + 9y
  —————————————
        x