First Meaning of Samucchaya

Samucchaya first means a term which occurs as a common factor in all the terms concerned.

Example 1 : 

                 Consider             11x + 4x = 13x + 9x

On taking all terms to one side and solving we get    7x = 0   which on solving yields  x = 0.

But according to the sutra , if a term (here x) occurs as a common factor in all the terms then the term is equated to zero.                       

Hence according to the sutra, the above problem can be solved  merely at sight and the solution given as  x = 0.

Example 2 :

                   Consider         5(2x + 3) = 3(2x + 3)

Here, the common term is (2x + 3).  Hence applying the above formula, we have            2x + 3 = 0.

                  Hence   x = -3/2

Second Meaning

Samucchaya also means 'the product of independent terms'.

Hence according to the sutra, if in a problem

         (x + a)(x + b) = (x + c)(x + d)    we have  a * b = c * d   

then x = 0.

This formula is practically axiomatic.

Example 1 :   (x + 4) (x - 6)  =  (x - 3) (x + 8)

                   Here   4 * (-6) = (-3) * 8 = -24  

                                Hence  x = 0.

 Example 2 :   (x + 3) (x + 12) = (x + 4) (x + 9)

                     Here  3 * 12 = 4 * 9 = 36

                              Hence  x = 0.

Third Meaning

Samucchaya thirdly means the sum of the denominators of two fractions having the same numerical numerator.

In this case the denominators are added and the sum is equated to zero and solved to obtain the value of x.

Example 1 :

                  3     +     3    =  0
              2x + 4      x - 3              

        ==>  (2x + 4) + (x - 3) = 0
                   3x + 1   = 0
                 ==>  x = -1/3   

Example 2 :

                  1      +      1      =   0
               x - 4         x + 3
               
                  ==>  x - 4 + x + 3 = 0
                        ==>  x = 1/2