(Anurupye) Shunyamanyat or "If one is in ratio, the other one is zero"

This sutra is often used to solve simultaneous simple equations which may involve big numbers. But these equations in special cases can be visually solved because of a certain ratio between the coefficients. Consider the following example:

6x  +  7y = 8

19x + 14y = 16

Here the ratio of coefficients of y is same as that of the constant terms.

Therefore, the "other" is zero, i.e., x = 0. Hence the solution of the

equations is x = 0 and y = 8/7.

This sutra is easily applicable to more general cases with any number of variables. For instance

ax + by + cz = a

bx + cy + az = b

cx + ay + bz = c

 which yields x = 1, y = 0, z = 0.

A corollary (upsutra) of this sutra says Sankalana-Vyavakalanaabhyam or By addition and by subtraction. It is applicable in case of simultaneous linear equations where the x- and y-coefficients are interchanged. For instance:

45x - 23y = 113

23x - 45y = 91

 By addition: 68x - 68 y = 204 => 68(x-y) = 204 => x - y = 3