Solution - Reducing fractions to their lowest terms

Step  1  :

            7
 Simplify   —
            y

Equation at the end of step  1  :

        7     
  (y -  —) -  13
        y     

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

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

          y     y • y
     y =  —  =  —————
          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 :

 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:

 y • y - (7)     y2 - 7
 ———————————  =  ——————
      y            y   

Equation at the end of step  2  :

  (y2 - 7)    
  ———————— -  13
     y        

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

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

          13     13 • y
    13 =  ——  =  ——————
          1        y   

Trying to factor as a Difference of Squares :

 3.2      Factoring:  y² - 7 

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 : 7 is not a square !!

Ruling : Binomial can not be factored as the difference of two perfect squares.

Adding fractions that have a common denominator :

 3.3       Adding up the two equivalent fractions

 (y2-7) - (13 • y)     y2 - 13y - 7
 —————————————————  =  ————————————
         y                  y      

Trying to factor by splitting the middle term

 3.4     Factoring  y² - 13y - 7 

The first term is,  y²  its coefficient is  1 .
The middle term is,  -13y  its coefficient is  -13 .
The last term, "the constant", is  -7 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -7 = -7 

Step-2 : Find two factors of  -7  whose sum equals the coefficient of the middle term, which is   -13 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Final result :

  y2 - 13y - 7
  ————————————
       y