The meaning of the phrase Shunyamsamya-samuchaya is that when the samuchya is the same then that samuchaya is zero.  Samuchaya means a term or a group of terms containing constant numbers and variables.   This chapter has five different parts.

• Part A:  Here Samuchaya is the term which occurs as a common factor in all the terms of the equation
• Part B:  Here Samuchaya is the product of the constant terms on both sides of the equation
• Part C:  Here Samuchya means the sum of the denominators of two fractions having the same numerator
• Part D:  Here Samuchya means the sum of the numerators and the sum on the denominators of both fractions
• Part E:  Here Samuchaya means that all the numerators are equal on both sides (LHS and RHS) of the equation

Part A:  Here the Samuchaya is the term which occurs as a common factor in all the terms of the equation. 

Example 1:     

Here we observe that x occurs as a common factor in all the terms in the equation.  Hence, x = 0 is the solution of the given linear equation.

Example 2:     

We can re-write this equation as 

Now (x+1) is the common term (Samuchaya) on both sides, hence x + 1 = 0 giving the solution x = -1.

 

Example 3:    

Step 1:  Transpose 786 to the left side and 483 to the right side

Step 2: 

Step 3: 

Step 4:  x-1 = 0  [Since (x -1) is the common factor or the samuchaya]

Step 5:  x =1

 

Exercises:                       Solution

                     

                   

              

           

           

 

Part B:  Here the Samuchaya is the product of the constant terms on both sides of the equation.

Example:  

Here we observe that the product of the constants on both sides is equal to 63 

(ie 7 x 9 = 63 and 3 x 21 = 63.  Hence, the variable term x should be equal to 0.

So the solution is x = 0.

Important note:  The coefficient of x²on both sides should be equal.                                                                            

Exercises:                         Solution

             

            

           

         

        

Part C:  Here Samuchya means the sum of the denominators of two fractions having the same numerator.

Example 1:

Step: 1  Observe that both the numerators are same(=1)

Step: 2  Add the Denominators

Step:3   Equate the sum of denominators to zero in order to obtain the final solution

Hence

  is the  required solution

 

Example 2:

Exercise                          Solution

                   

                       

                   

                   

                   

Part D : Here Samuchya means the sum of the numerators and the sum on the denominators of both fractions

Example 1:

Step 1: Add the numerators of both fractions

Step 2: Add the denominators if both fractions 

We observe that

Hence the sum 

is solution

 

Example 2:

If the sum of numerators and sum of denominators is not equal. but if one is a multiple of the other than still we can put the sum equal to zero ( after removing the multiple)

            

            

we observe that

So we can remove the multiple 2

giving us 

             

            

is the required solution

Exercise                         Solution

                    

                           

                   

 

Part E : Here If All the numerators are equal the Samuchaya here means the sum of denominators on both sides (LHS and RHS) of the equation

Example 1:

Step 1 : Observe that all the numerators are equal (=1)

Step 2 : Add Denominators on LHS 

Step 3: Add Denominator on RHS

Step 4:   (Make the sum of denominators = 0)

Step 5:  is the subtraction

Example2:

Hence

or

 

Example 3:

Here we observe that sum of denominators on both sides of equation are different

Now we need to transpose the negative terms on both sides

Now

             

                 is the solution

 

 

Exercise                                            Solution

                        

                        

                         

                     

Note here b,c,d are some constants 

                        

Corollary of 

Part E: if the numerators are not equal, then we can equalize them by taking their LCM(lowest common multiply) and than multiply the fractions accordingly

Example

Step 1: Take LCM of Numerators LCM (2,3,1,6)=6

Step 2: Make all numerators = 6

So now we can rewrite the equation as

Now take the sum of denominators on both sides of equation

Hence

     is the solution

 

Exercise                                             Solution