Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "/-5" was replaced by "/(-5)". 3 more similar replacement(s)

Step  1  :

             y
 Simplify   ——
            -5

Equation at the end of step  1  :

     x     y       x       y
  ((——+(2•——))+(3•——))-(4•——)
    -4    -4      -3      -5

Step  2  :

             x
 Simplify   ——
            -3

Equation at the end of step  2  :

     x     y       x   4y
  ((——+(2•——))+(3•——))-——
    -4    -4      -3   -5

Step  3  :

             y
 Simplify   ——
            -4

Equation at the end of step  3  :

     x          y             4y
  ((—— +  (2 • ——)) +  -x) -  ——
    -4         -4             -5

Step  4  :

             x
 Simplify   ——
            -4

Equation at the end of step  4  :

     x     y            4y
  ((—— +  ——) +  -x) -  ——
    -4    -2            -5

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       -4 

      The right denominator is :       -2 

      Least Common Multiple:
      4 

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 = -1

   Right_M = L.C.M / R_Deno = -2

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.      x • -1
   ——————————————————  =   ——————
         L.C.M               4   

   R. Mult. • R. Num.      y • -2
   ——————————————————  =   ——————
         L.C.M               4   

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:

 x • -1 + y • -2     -x - 2y
 ———————————————  =  ———————
        4               4   

Equation at the end of step  5  :

   (-x - 2y)           4y
  (————————— +  -x) -  ——
       4               -5

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  4  as the denominator :

          -x     -x • 4
    -x =  ——  =  ——————
          1        4   

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

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   -x - 2y  =   -1 • (x + 2y) 

Adding fractions that have a common denominator :

 7.2       Adding up the two equivalent fractions

 (-x-2y) + -x • 4     -5x - 2y
 ————————————————  =  ————————
        4                4    

Equation at the end of step  7  :

  (-5x - 2y)    4y
  —————————— -  ——
      4         -5

Step  8  :

Step  9  :

Pulling out like terms :

 9.1     Pull out like factors :

   -5x - 2y  =   -1 • (5x + 2y) 

Calculating the Least Common Multiple :

 9.2    Find the Least Common Multiple

      The left denominator is :       4 

      The right denominator is :       -5 

      Least Common Multiple:
      20 

Calculating Multipliers :

 9.3    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 = 5

   Right_M = L.C.M / R_Deno = -4

Making Equivalent Fractions :

 9.4      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      (-5x-2y) • 5
   ——————————————————  =   ————————————
         L.C.M                  20     

   R. Mult. • R. Num.      4y • -4
   ——————————————————  =   ———————
         L.C.M               20   

Adding fractions that have a common denominator :

 9.5       Adding up the two equivalent fractions

 (-5x-2y) • 5 - (4y • -4)     6y - 25x
 ————————————————————————  =  ————————
            20                   20   

Final result :

  6y - 25x
  ————————
     20