matha namo namaha


ĀNURŨPYE ŚŨNYAMANYAT
The Sutra Anurupye Sunyamanyat says : 'If one is in ratio, the other one is zero'.
We use this Sutra in solving a special type of simultaneous simple equations in which the coefficients of 'one' variable are in the same ratio to each other as the independent terms are to each other. In such a context the Sutra says the 'other' variable is zero from which we get two simple equations in the first variable (already considered) and of course give the same value for the variable.
Example 1:
                            3x +  7y = 2
                            4x + 21y = 6
Observe that the y-coefficients are in the ratio 7 : 21 i.e., 1 : 3, which is same as the ratio of independent terms i.e., 2 : 6 i.e., 1 : 3. Hence the other variable x = 0 and 7y = 2 or 21y = 6 gives y = 2 / 7
Example 2:
                            323x + 147y = 1615
                            969x + 321y = 4845
The very appearance of the problem is frightening. But just an observation and anurupye sunyamanyat give the solution x = 5, because coefficient of x ratio is
        323 : 969 = 1 : 3 and constant terms ratio is 1615 : 4845 = 1 : 3.
             y = 0 and 323 x = 1615 or 969 x = 4845 gives x = 5.
Solve the following by anurupye sunyamanyat.

        1.   12x + 78y = 12                     2.    3x + 7y = 24
              16x + 96y = 16                           12x + 5y = 96

        3.    4x – 6y = 24                         4.    ax + by = bm
               7x – 9y = 36                                 cx + dy = dm
In solving simultaneous quadratic equations, also we can take the help of the ‘sutra’ in the following way:
Example 3 :
                   Solve for x and y
                                                                 x + 4y = 10
                                     x2 + 5xy + 4y2 + 4x - 2y = 20

              x2 + 5xy + 4y2 + 4x - 2y = 20 can be written as
                ( x + y ) ( x + 4y ) + 4x – 2y = 20
             10 ( x + y ) + 4x – 2y = 20 ( Since x + 4y = 10 )
               10x + 10y + 4x – 2y = 20
                              14x + 8y = 20

                  Now x + 4y = 10
             14x + 8y = 20 and 4 : 8 :: 10 : 20

      from the Sutra, x = 0 and 4y = 10, i.e.,, 8y = 20 y = 10/4 = 2½
         Thus x = 0 and y = 2½ is the solution.

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